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Understanding size effects on the strength of single crystals through high-temperature micropillar compression
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ScienceDirect Acta Materialia 81 (2014) 50–57 www.elsevier.com/locate/actamat
Understanding size effects on the strength of single crystals through high-temperature micropillar compression Rafael Soler a, Jeffrey M. Wheeler b, Hyung-Jun Chang a, Javier Segurado a,c, Johann Michler b, Javier Llorca a,c,⇑, Jon M. Molina-Aldareguia a,⇑ b
a IMDEA Materials Institute, c/Eric Kandel 2, 28906 Getafe, Madrid, Spain EMPA – Swiss Federal Laboratories for Materials Science and Technology, Feuerwerkerstrasse 39, Thun CH-3602, Switzerland c Department of Materials Science, Polytechnic University of Madrid, E.T.S. de Ingenieros de Caminos, 28040 Madrid, Spain
Received 29 April 2014; received in revised form 27 June 2014; accepted 2 August 2014
Abstract Compression tests of h1 1 1i-oriented LiF single-crystal micropillars 1–5 lm in diameter were carried out from 25 °C to 250 °C. While the flow stress at ambient temperature was independent of the micropillar diameter, a strong size effect developed with elevated temperature. This behavior was explained by rigorously accounting for the different contributions to the flow stress of the micropillars as a function of temperature and pillar diameter: the lattice resistance, the forest hardening; and the size-dependent contribution as a result of the operation of single-arm dislocation sources. This was possible because the micropillars were obtained by chemically etching away the surrounding matrix in directionally solidified LiF–NaCl and LiF–KCl eutectics, avoiding any use of focused ion beam methods, yielding micropillars with a controlled dislocation density, independent of the sample preparation technique. In particular, the role of the lattice resistance on the size effect of micrometer-size single crystals was demonstrated unambiguously for the first time. This result rationalizes the different values of power-law exponent for the size effect found in the literature for face-centered cubic and body-centered cubic metals as well as for covalent and ionic solids. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Mechanical size effects; Micropillar compression; Ionic compounds
1. Introduction
1% and 10% followed a power-law dependence on micropillar diameter D of the form
The analysis of size effects in the mechanical strength of micrometer-size single crystals has received a lot of attention from the scientific community in recent years. Most of the work has focused on face-centered cubic (fcc) pure metals, such as Ni, Cu and Au [1–3], and was obtained by compressing micropillars with diameters in the range between 1 and 10 lm at room temperature. It was found [4–6] that the strength at a given plastic strain between
r ¼ ADn
⇑ Corresponding authors at: IMDEA Materials Institute, c/Eric Kandel 2, 28906 Getafe, Madrid, Spain. Tel.: +34 91 549 3422. E-mail addresses: [email protected] (J. Llorca), jon.molina@ imdea.org (J.M. Molina-Aldareguia).
ð1Þ
where A is a constant, and n is the power-law exponent. These micropillars were not dislocation free (i.e. their behavior was not controlled by dislocation nucleation and/or dislocation starvation processes), and size effects arose from the operation of single-arm dislocation sources [7–9], leading to power-law exponents in the range 0.61– 0.97 for fcc metals [4]. However, the strong size effects reported in fcc metals were not found in stronger materials where a significant contribution of lattice resistance to plastic strength is expected. For instance, the power-law exponent of Eq. (1) corresponding to body-centered cubic (bcc)
http://dx.doi.org/10.1016/j.actamat.2014.08.007 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
R. Soler et al. / Acta Materialia 81 (2014) 50–57
single crystals was in the range 0.21–0.48 [10–12], while size effects were small, and sometimes even negligible in strong solids such as GaAs [13], InSb [14] and Si [15–17]. Moreover, different power-law exponents were reported in single crystals that presented a strong plastic anisotropy, such as MgO [18] and LiF [19], depending whether plastic deformation was controlled by the “soft” slip system (high n) or the hard slip system (negligible size effect with n 0). Several studies have tried to understand the effect of lattice resistance on size effects. The stress necessary to activate single-arm dislocation sources can be determined by adding the lattice resistance to the contribution of the single-arm sources, forest dislocations and the line tension stress [7,20–21]. Following Parthasarathy et al. [7], the critical resolved shear stress sCRSS for the activation of a single-arm dislocation source is given by alb pffiffiffiffiffiffiffi sCRSS ¼ s0 þ 0:5lb qtot þ kmax ðD; qtot ; bÞ
ð2Þ
where s0 is the lattice resistance to dislocation glide, l is the anisotropic shear modulus, b is the Burgers vector, qtot is the total dislocation density, a is a constant of the order kmax is the statistical average length of the of unity, and weakest single-arm dislocation source, which is a function of the micropillar diameter D, the dislocation density qtot and the slip plane orientation b. The first two terms in Eq. (2) contribute to the bulk strength of the material sbulk and are size independent. The third term, ssize, gives rise to the size effect, since the average length of the weakest single-arm dislocation source scales with the micropillar diameter, as described statistically by Parthasarathy et al. [7]. Korte and Clegg [18] demonstrated that a consequence of Eq. (2) is that ssize can be much higher than sbulk in materials with negligible bulk strength, such as fcc metals, while the opposite behavior can be the origin of the limited size effect encountered on micropillar compression of bcc metals and ceramics. Since then, several authors have found good correlation between the room-temperature size exponent n and the critical temperature for bcc metals [12], high-entropy bcc alloys [22] as well as a range of ionic crystals [23], indicating that the magnitude of the power law exponent does indeed decrease with an increase in the lattice resistance of the material. More recently, Lee and Nix [24] carried out a thorough study of the effect of lattice resistance s0, the shear modulus l and the Burgers vector b on the size effect. They proposed the following universal scaling law sCRSS sbulk ¼ ADn lb
ð3Þ
which rationalizes the experimental data for fcc and bcc metals for typical dislocation densities of the order of 1012 m2. Moreover, they proposed that size effects in strong solids should depend on temperature, because the lattice resistance s0 decreases with temperature. However, the contribution of forest hardening and of single-arm dislocation source operation should be essentially “athermal”.
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In addition, the magnitude of the Burgers vector b will increase with temperature, owing to thermal expansion, while the shear modulus, l will decrease. The influence of these variations on the size effect is expected to be much smaller than the contribution of the temperature-dependent lattice resistance. Thus, available models for predicting the size effect on the strength of micropillars are able to predict the experimental trends, but lack sound experimental validation. This is due to the impossibility of estimating the lattice resistance and the forest hardening contribution in Eq. (2) in those materials for which micropillar compression data were available. These limitations arise from the uncertainties in the actual dislocation density and the strain rate of the tests, associated with micropillar sample preparation by focused ion beam (FIB) methods and compression using load-controlled mechanical testing systems, respectively [18]. The main objective of this investigation is to overcome these limitations, and this is achieved by means of compression tests on h1 1 1i oriented LiF single crystal micropillars with diameters in the range 1–5 lm from ambient temperature to 250 °C, using a displacement-controlled microcompression system to achieve constant strain rate compression tests. High-temperature micropillar compression is a novel experimental technique [17,19,25,26,29,30], which allows systematic variation in the lattice resistance using temperature control, while the athermal contributions to strength remain unchanged. Moreover, LiF [1 1 1] constitutes a model system because micropillars with varying diameters can be obtained by chemically etching away the surrounding matrix in directionally solidified LiF–NaCl and LiF– KCl eutectics. This avoids any use of FIB methods, and yields micropillars with a controlled dislocation density, independent of the sample preparation technique. This allowed the precise determination of the contribution of lattice resistance, forest hardening, and single-arm dislocation source operation to the strength of the pillars as a function of both pillar size and temperature, contributing to a sound experimental validation of existing models on size effects in mechanical strength. 2. Experimental procedure LiF micropillars were obtained without the use of FIB, using the novel approach previously employed by Bei et al. for Mo [31] and by the present authors for LiF [19]. This was a critical point for this investigation, because the precise estimation of the different contributions to the critical resolved shear stress in Eq. (2) requires reliable values of the initial dislocation density in the micropillar. FIB machining, the usual technique to manufacture micropillars from single crystals, introduces defects in the micropillars such as the formation of Ga+ implanted amorphous layers [32], lattice distortions and dislocation loops [33]. Therefore, even if the initial dislocation density of the crystal is measured, the defect structure after FIB milling can
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be dramatically different, introducing many uncertainties in Eq. (2). Circular LiF rods grow in the [1 1 1] direction during directional-solidification of LiF–NaCl and LiF–KCl eutectics [27,28]. The rods are distributed in a hexagonal pattern, and the rod diameter can be varied from 1 lm to 5 lm by changing the growth rate from 50 mm h1 to 3 mm h1. LiF micropillars with an aspect ratio (length to diameter ratio) of 2:1, ideal for micropillar compression to avoid buckling and constraint effects, were obtained by controlled chemical etching of the matrix with CH4O, on a mechanically polished cross section. The single crystal nature of the micropillars and their orientation, parallel to the [1 1 1] axis, were checked by transmission electron microscopy [19]. The as-grown LiF rods contained a large initial dislocation density, of the order of 2.5 1013 m2, as revealed by etch pit density analysis of extracted LiF fibers etched with H2O (Fig. 1). The micropillar compression tests at ambient and elevated temperature were carried out in a Zeiss DSM 962 scanning electron microscope equipped with an Alemnis in situ indenter [29,30]. The system is inherently displacement-controlled and allows temperature control of both the indenter tip and the sample on independent thermocouple-controlled feedback loops. A flat punch diamond tip was used to apply the load on the micropillars. All test temperatures reported in this work correspond to the sample surface temperatures as measured by the thermally calibrated indenter as a temperature probe, using a technique described previously [34]. This technique greatly reduces any uncertainty in surface temperature owing to thermal gradients in samples with low thermal conductivity. The precision in the surface temperature measurement is on the order of 1 °C. Compression tests were performed at temperatures of 25, 75, 130 and 250 °C at a constant strain rate of 103 s1, unless indicated otherwise. The measured load– displacement curves were corrected for the extra compliance associated with the elastic deflection of the matrix at
Fig. 1. Dislocation pits on the surface of a LiF rod after etching with H2O: etch pit density 2.5 1013 m2.
the base of the pillar, following a standard shear lag model for load transfer between the matrix and an isolated fiber embedded in an infinite matrix [19,35]. The initial micropillar diameter and the gauge length were measured in situ to compute the stress and strain from corrected load–displacement curves. Slight variations in the pillar diameter (<10%) were found from pillar to pillar. Regarding the stress–strain response, the elastic modulus corresponding to the initial elastic loading was always lower than that associated with elastic unloading at the end of the test, regardless of micropillar diameter or test temperature. These discrepancies can be attributed to the impossibility of attaining a perfect parallelism between the top surface of the pillar and the indenter flat punch tip [36]. The slight misalignment between them induces a stress concentration at the initial contact point of the micropillar, which leads to early plastification and the apparent reduction of the initial loading stiffness. This behavior made it difficult to establish the onset of plastic yielding from the stress–strain curves. To avoid uncertainties in the yield stress determination, the flow stress at 10% strain, r10%, was used throughout this investigation as the reference value for the compressive strength as a function of temperature and micropillar diameter, as is routinely done in other micropillar compression works. In any case, the results obtained were not affected by this choice, and similar conclusions could be drawn if the flow stresses were determined at lower plastic strains of 3%. Additionally, load resolution of the experimental setup became an issue for the smallest micropillars tested at 250 °C, as the maximum load carried by a LiF micropillar 1 lm in diameter at high temperature was of the order of 15 lN, very close to the resolution of the load cell. This limitation hindered the study of the effect of temperature on the size effect above 250 °C. 3. Experimental results The stress–strain curves corresponding to micropillars 5, 2.5 and 1 lm in diameter tested in the temperature range 25–250 °C are plotted in Fig. 2a–c, respectively. The stress–strain behavior becomes more stochastic as the diameter of the micropillar decreases, especially for the 1 lm micropillars. This is in agreement with other micropillar compression experiments [20,37,38], where the larger scatter and the more jerky behavior of the smallest pillars were attributed to the reduction in the number of the available dislocation sources as the diameter decreased. Plastic flow in LiF is highly anisotropic, owing to its ionic character. The preferred slip system belongs to the f1 1 0g “soft” {1 1 0}h1 1 0i family, whose lattice resistance s0 is negligible at room temperature and only increases at critf1 1 0g ical temperatures T 0 below 233 °C [37]. The Schmid factor of the {1 1 0}h1 1 0i slip system is 0.5 during compression of h1 0 0i oriented single crystals and, as a result, the compressive strength of bulk h1 0 0i single crystals is in the range 5–40 MPa, depending on the dislocation density
R. Soler et al. / Acta Materialia 81 (2014) 50–57
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and impurity level. LiF single crystals can also deform plastically in the “hard” {1 0 0}h1 1 0i slip system when loaded along the h1 1 1i direction. In this orientation, the Schmid factor for the “soft” {1 1 0}h1 1 0i slip system is 0, while the Schmid factor for the “hard” {1 0 0} slip is 0.47. The f1 1 0g lattice resistance s0 on the “hard” {1 0 0} slip system at room temperature is on the order of a few hundreds of f1 1 0g megapascals [40], substantially larger than s0 at room temperature, and only decreases for critical temperatures f100g T0 250 C [39,41]. Previous investigations of the strength of h1 0 0i oriented LiF micropillars showed that plastic deformation was governed by the soft slip system [23,42]. A strong size effect was found, with power-law exponents of the order of 0.7–0.8, independently of initial dislocation density. However, in the case of h1 1 1i-oriented micropillars, plastic deformation occurs on the hard slip system [19,36], and no size effect on the flow stress was reported at ambient temperature [19]. The magnitude of the critical resolved shear stress at 10% strain (obtained from the flow stress and the Schmid factor) is plotted in Fig. 3 as a function of micropillar diameter for temperatures in the range 25–250 °C. The plot shows the emergence of a size effect with temperature and the power-law exponent (obtained by fitting the experimental results to Eq. (1)) increases from n 0 at 25 °C, to n 0.53 at 250 °C, and the size effect at 250 °C becomes similar to that displayed by most fcc metals, and also LiF oriented in the h1 0 0i direction, at room temperature. 4. Discussion on temperature dependent size effects In order to check the validity of Eq. (2), it is necessary to estimate the bulk lattice resistance of h1 1 1i LiF as a function of temperature. Gilman [40] performed torsion tests
Fig. 2. Compressive stress–strain curves of h1 1 1i LiF micropillars (a) 5 lm, (b) 2.5 lm, and (c) 1 lm in diameter, respectively, for temperatures of 25 °C (blue lines), 75 °C (green lines), 130 °C (red lines), and 250 °C (gray lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. sCRSS as a function of the micropillar diameter in the temperature range 25–250 °C. The power law exponent n at each temperature obtained by fitting the experimental results to Eq. (1) is indicated in the plot.
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on macrosocopic LiF (1 0 0) single crystals from 350 °C to 650 °C. The brittle nature of LiF when deformed along the hard {1 0 0} slip system led to cracking below 350 °C, and no data could be obtained. A Peierls’ model [43,44] for the dependence of the shear flow stress on temperature was applied in this work to extend the data of the bulk yield stress to lower temperatures, with the aid of the room temperature micropillar compression results. The Orowan equation relates the shear strain rate c_ to the mobile dislocation density qm, the Burgers vector b and dislocation velocity v, according to c_ ¼ qm bm
ð4Þ
The dislocation velocity v can be estimated using a standard approach for a stress-activated process: ðsp sa ÞV ðsp þ sa ÞV v ¼ t b exp exp ð5Þ kT kT where t is the attempt frequency, sp is the Peierls stress, V is the activation volume, T is the absolute temperature, sa is the applied shear stress, and k Boltzmann’s constant. From Eq. (5) and assuming that sa = s0, the thermal dependence of the lattice resistance s0 (first term in Eq. (2)) of the hard {1 0 0}h1 1 0i slip system can be expressed as kT sp V c_ 1 s0 ¼ sinh exp ð6Þ V kT 2qm vb2
Fig. 4. Influence of strain rate on the critical resolved shear stress of [1 1 1] LiF micropillar single crystals at ambient temperature. The activation volume V was determined from the linear slope between sCRSS and the natural logarithm of the shear strain rate.
This relationship forms the basis for extending the available bulk yield stress data for LiF [1 1 1] single crystals to lower temperatures, with the aid of the room temperature micropillar compression results. Substitution of Eq. (6) into Eq. (2) leads to kT sp V c_ pffiffiffiffiffiffiffi 1 sCRSS ¼ sinh exp þ 0:5lb qtot 2 V kT 2qm vb alb þ kmax ðD; qtot ; bÞ
ð7Þ
where the first two terms contribute to the bulk strength of the material, sbulk, and the third term gives rise to the size effect ssize. It is worth noting that, in contrast to the bulk crystals, LiF [1 1 1] micropillars can indeed be plastically deformed at room temperature, because fracture is suppressed owing to the small dimensions of the specimens [45]. Taking advantage of the fact that the flow stress of LiF [1 1 1] at room temperature is not affected by size effects, as shown in Fig. 2, micropillars were tested at different strain rates in the range 0.00033–0.033 s1 at room temperature. Assuming that the activation volume is independent of the strain rate, that sV remains large with respect to kT, and neglecting ssize, Eq. (7) can be simplified to kT dc kT pffiffiffiffiffiffiffi ð8Þ sbulk ¼ sp þ ln ln qm vb2 þ 0:5lb qtot V dt V i.e. the bulk shear stress sbulk varies linearly with the natural logarithm of shear strain rate. Fig. 4 shows that this is
Fig. 5. Theoretical predictions of the bulk strength sbulk of the hard {1 0 0}h1 1 0i slip system in LiF as a function of temperature for two different dislocation densities and imposed shear strain rates, as summarized in Table 1, following Eq. (8). The dots represent the experimental sCRSS data for micropillars of different diameter (1 lm in red, 2.5 lm in green and 5 lm in blue) as well as for bulk LiF (gray dots) from Gilman [40]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
indeed the case and validates the Peierls’ model to analyze the strain rate sensitivity of LiF [1 1 1] micropillars at room temperature. From the linear slope kT , it is possible to V determine the activation volume V, yielding a value of 2.6 1028 m3 or 11.41 b3. It should be noted that this value is in good agreement with the activation volume expected for lattice resistance governed dislocation glide [46].
R. Soler et al. / Acta Materialia 81 (2014) 50–57 Table 1 Material parameters and testing conditions in Eq. (6) to determine the evolution of sCRSS of the hard {1 0 0}h1 1 0i slip system in bulk LiF as a function of temperature.
Testing conditions Bulk [38] This work Material constants v (s1) 8.0 1011
b (nm) 0.285
c_ ðs1 Þ
qtot (m2)
105 2.13 103
108 2.5 1013
V (b3) 11.41
sp (MPa) 530
Moreover, the room temperature microcompression data provide the basis for extending the available bulk yield stress data for LiF [1 1 1] single crystals to lower temperatures, with the aid of Eq. (8). Fig. 5 plots the experimental critical resolved shear stress on the {1 0 0} hard slip system
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of LiF coming from microcompression in this work together with the available bulk shear strengths in the literature [40,47] from testing single crystals at high temperature. Both sets of results could be fit by Eq. (8) using a Peierls stress sp = 530 MPa and the appropriate experimental parameters of dislocation density and shear strain rate. The corresponding curves are shown as solid lines in Fig. 5. The actual materials and testing parameters used in Eq. (8) are summarized in Table 1, where the attempt frequency v was calculated from Debye’s frequency equation. The {1 0 0} shear modulus l, including temperature dependence, was obtained from Hart [48]. The shear strain rate and the total dislocation density for each data set are reported, and the mobile dislocation density qm was considered 3/12 of the total dislocation density, assuming slip along the three “hard” slip systems preferentially oriented for slip out of the 12 total slip systems.
Fig. 6. Evolution of sCRSS according to Eq. (7) (solid line) and comparison with experimental data: (a) 25 °C; (b) 75 °C; (c) 130 °C; (d) 250 °C. The dashed lines represent the size-independent contribution sbulk at each temperature (lattice resistance and forest hardening), while the dotted line represents the sizedependent contribution ssize, due to the operation of single-arm dislocation sources.
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It should be noted that Eq. (8) successfully captured the influence of temperature on the critical resolved shear stress of micropillars at ambient temperature (blue curve) and bulk crystals (gray curve) at high temperature. However, it failed to capture the behavior of micropillars at high temperature, and this was due to the increasing importance of the athermal size-dependent contribution ssize, in the case of micropillars as the lattice resistance decreased. Once the first two terms of Eq. (7), accounting for the variation in bulk shear strength sbulk with temperature have been obtained, it is possible to estimate the third term, i.e. the expected size-dependent contribution ssize, arising from the operation of single-arm dislocation sources. For this, the statistical average length of the weakest single-arm diskmax as a function of micropillar diameter location source was estimated using the statistical model developed by Parthasarathy et al. [7] for each micropillar diameter: Z R kmax ¼ kmax pðkmax Þdkmax 0
¼
p1 pðR kmax Þðb kmax Þ pRb 0 p½ðR kmax Þ þ ðb kmax Þ pkmax dkmax pRb Z
R
1
ð9Þ
where R is the radius of the pillar, and p is the number of pinning points, given by p ¼ Integer½qm pRh
ð10Þ
where qm is the mobile dislocation density, h is the height of the pillar, and the rest of the symbols have the usual meaning. Finally, the variation in the shear modulus l with temperature was obtained from Hart [48], while the temperature dependence of the Burgers vector was neglected in the analysis. The predictions of Eq. (7) for the sCRSS as a function of temperature are plotted in Fig. 6, together with the experimental high-temperature micropillar compression data. The agreement between the experimental results and the predictions in Eq. (7) is remarkable. The bulk strength dominates over the size-dependent contribution at 25 °C and 75 °C in the range of micropillar diameters tested, and no size effects are found. However, both contributions are of the same order at 250 °C, and a strong size effect develops at high temperatures. These results show that lattice resistance influences the power-law exponents for the size effect of different materials, as proposed by Korte and Clegg [18] and Lee and Nix [24]. Recent results in Mo micropillars [26], a bcc metal with a high lattice resistance, have also shown similar trends, with the size exponent increasing at temperatures close to 250 °C, the critical temperature for Mo at which the lattice resistance is expected to become negligible. However, the micropillars in that study were machined by FIB milling, and the uncertainty in the initial dislocation density precluded any estimation of the size-dependent and size-independent terms at each temperature.
5. Conclusions The influence of temperature on the compressive strength of [1 1 1] LiF single crystals was measured by means of micropillar compression tests. Micropillars of diameters in the range 1–5 lm were obtained by means of directional solidification and surface etching of eutectic compounds, leading to single crystals with a controlled initial dislocation density of 2.5 1013 m2. The micropillar flow strength was independent of the micropillar diameter at ambient temperature, but a strong size effect developed with temperature: micropillars 1 lm in diameter were twice as strong as those 5 lm in diameter at 250 °C. The different contributions to the flow stress of the micropillars, namely the lattice resistance, forest hardening and the sizedependent contribution as a result of operation of singlearm dislocation sources were rigorously accounted for as a function of both temperature and micropillar diameter. It was demonstrated that the size effect observed during micropillar compression comes about as a result of the relative weights of the size-independent (lattice resistance plus forest hardening) and size-dependent contributions to strength. At room temperature, the former contribution dominated, and no size effect was found, while at 250 °C, both contributions were of the same order for the micropillar diameters studied, leading to a strong size effect. Thus, the role of the lattice resistance on the size effect of micrometer-size single crystals was demonstrated unambiguously for the first time. This result rationalizes the different values of power-law exponent for the size effect found in the literature for fcc and bcc metals, as well as covalent and ionic solids. Acknowledgements This investigation was supported by the Spanish Ministry of Economy and Competitiveness through project MAT2012-31889. HYC acknowledges the support of a Juan del Cierva post-doctoral fellowship from the Spanish Ministry of Economy and Competitiveness. The authors are indebted to Prof. V.M. Orera and Dr. R.I. Merino from the Institute of Materials Science of Arago´n (CSIC), who grew and provided the eutectic crystals. References [1] Uchic MD, Dimiduk DM, Florando JN, Nix WD. Science 2004;305:986–9. [2] Greer JR, Oliver WC, Nix WD. Acta Mater 2005;53:1821–30. [3] Volkert CA, Lilleodden ET. Philos Mag 2006;86:5567–79. [4] Uchic MD, Shade PA, Dimiduk DM. Annu Rev Mater Res 2009;39:361–86. [5] Dehm G. Prog Mater Sci 2009;54:664–88. [6] Greer JR, De Hosson JTM. Prog Mater Sci 2011;56:654–724. [7] Parthasarathy TA, Rao SI, Dimiduk DM, Uchic MD, Trinkle DR. Scr Mater 2007;56:313–6. [8] Rao SI, Dimiduk DM, Parthasarathy TA, Uchic MD, Tang M, Woodward C. Acta Mater 2008;56:3245–59.
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ScienceDirect Acta Materialia 81 (2014) 50–57 www.elsevier.com/locate/actamat
Understanding size effects on the strength of single crystals through high-temperature micropillar compression Rafael Soler a, Jeffrey M. Wheeler b, Hyung-Jun Chang a, Javier Segurado a,c, Johann Michler b, Javier Llorca a,c,⇑, Jon M. Molina-Aldareguia a,⇑ b
a IMDEA Materials Institute, c/Eric Kandel 2, 28906 Getafe, Madrid, Spain EMPA – Swiss Federal Laboratories for Materials Science and Technology, Feuerwerkerstrasse 39, Thun CH-3602, Switzerland c Department of Materials Science, Polytechnic University of Madrid, E.T.S. de Ingenieros de Caminos, 28040 Madrid, Spain
Received 29 April 2014; received in revised form 27 June 2014; accepted 2 August 2014
Abstract Compression tests of h1 1 1i-oriented LiF single-crystal micropillars 1–5 lm in diameter were carried out from 25 °C to 250 °C. While the flow stress at ambient temperature was independent of the micropillar diameter, a strong size effect developed with elevated temperature. This behavior was explained by rigorously accounting for the different contributions to the flow stress of the micropillars as a function of temperature and pillar diameter: the lattice resistance, the forest hardening; and the size-dependent contribution as a result of the operation of single-arm dislocation sources. This was possible because the micropillars were obtained by chemically etching away the surrounding matrix in directionally solidified LiF–NaCl and LiF–KCl eutectics, avoiding any use of focused ion beam methods, yielding micropillars with a controlled dislocation density, independent of the sample preparation technique. In particular, the role of the lattice resistance on the size effect of micrometer-size single crystals was demonstrated unambiguously for the first time. This result rationalizes the different values of power-law exponent for the size effect found in the literature for face-centered cubic and body-centered cubic metals as well as for covalent and ionic solids. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Mechanical size effects; Micropillar compression; Ionic compounds
1. Introduction
1% and 10% followed a power-law dependence on micropillar diameter D of the form
The analysis of size effects in the mechanical strength of micrometer-size single crystals has received a lot of attention from the scientific community in recent years. Most of the work has focused on face-centered cubic (fcc) pure metals, such as Ni, Cu and Au [1–3], and was obtained by compressing micropillars with diameters in the range between 1 and 10 lm at room temperature. It was found [4–6] that the strength at a given plastic strain between
r ¼ ADn
⇑ Corresponding authors at: IMDEA Materials Institute, c/Eric Kandel 2, 28906 Getafe, Madrid, Spain. Tel.: +34 91 549 3422. E-mail addresses: [email protected] (J. Llorca), jon.molina@ imdea.org (J.M. Molina-Aldareguia).
ð1Þ
where A is a constant, and n is the power-law exponent. These micropillars were not dislocation free (i.e. their behavior was not controlled by dislocation nucleation and/or dislocation starvation processes), and size effects arose from the operation of single-arm dislocation sources [7–9], leading to power-law exponents in the range 0.61– 0.97 for fcc metals [4]. However, the strong size effects reported in fcc metals were not found in stronger materials where a significant contribution of lattice resistance to plastic strength is expected. For instance, the power-law exponent of Eq. (1) corresponding to body-centered cubic (bcc)
http://dx.doi.org/10.1016/j.actamat.2014.08.007 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
R. Soler et al. / Acta Materialia 81 (2014) 50–57
single crystals was in the range 0.21–0.48 [10–12], while size effects were small, and sometimes even negligible in strong solids such as GaAs [13], InSb [14] and Si [15–17]. Moreover, different power-law exponents were reported in single crystals that presented a strong plastic anisotropy, such as MgO [18] and LiF [19], depending whether plastic deformation was controlled by the “soft” slip system (high n) or the hard slip system (negligible size effect with n 0). Several studies have tried to understand the effect of lattice resistance on size effects. The stress necessary to activate single-arm dislocation sources can be determined by adding the lattice resistance to the contribution of the single-arm sources, forest dislocations and the line tension stress [7,20–21]. Following Parthasarathy et al. [7], the critical resolved shear stress sCRSS for the activation of a single-arm dislocation source is given by alb pffiffiffiffiffiffiffi sCRSS ¼ s0 þ 0:5lb qtot þ kmax ðD; qtot ; bÞ
ð2Þ
where s0 is the lattice resistance to dislocation glide, l is the anisotropic shear modulus, b is the Burgers vector, qtot is the total dislocation density, a is a constant of the order kmax is the statistical average length of the of unity, and weakest single-arm dislocation source, which is a function of the micropillar diameter D, the dislocation density qtot and the slip plane orientation b. The first two terms in Eq. (2) contribute to the bulk strength of the material sbulk and are size independent. The third term, ssize, gives rise to the size effect, since the average length of the weakest single-arm dislocation source scales with the micropillar diameter, as described statistically by Parthasarathy et al. [7]. Korte and Clegg [18] demonstrated that a consequence of Eq. (2) is that ssize can be much higher than sbulk in materials with negligible bulk strength, such as fcc metals, while the opposite behavior can be the origin of the limited size effect encountered on micropillar compression of bcc metals and ceramics. Since then, several authors have found good correlation between the room-temperature size exponent n and the critical temperature for bcc metals [12], high-entropy bcc alloys [22] as well as a range of ionic crystals [23], indicating that the magnitude of the power law exponent does indeed decrease with an increase in the lattice resistance of the material. More recently, Lee and Nix [24] carried out a thorough study of the effect of lattice resistance s0, the shear modulus l and the Burgers vector b on the size effect. They proposed the following universal scaling law sCRSS sbulk ¼ ADn lb
ð3Þ
which rationalizes the experimental data for fcc and bcc metals for typical dislocation densities of the order of 1012 m2. Moreover, they proposed that size effects in strong solids should depend on temperature, because the lattice resistance s0 decreases with temperature. However, the contribution of forest hardening and of single-arm dislocation source operation should be essentially “athermal”.
51
In addition, the magnitude of the Burgers vector b will increase with temperature, owing to thermal expansion, while the shear modulus, l will decrease. The influence of these variations on the size effect is expected to be much smaller than the contribution of the temperature-dependent lattice resistance. Thus, available models for predicting the size effect on the strength of micropillars are able to predict the experimental trends, but lack sound experimental validation. This is due to the impossibility of estimating the lattice resistance and the forest hardening contribution in Eq. (2) in those materials for which micropillar compression data were available. These limitations arise from the uncertainties in the actual dislocation density and the strain rate of the tests, associated with micropillar sample preparation by focused ion beam (FIB) methods and compression using load-controlled mechanical testing systems, respectively [18]. The main objective of this investigation is to overcome these limitations, and this is achieved by means of compression tests on h1 1 1i oriented LiF single crystal micropillars with diameters in the range 1–5 lm from ambient temperature to 250 °C, using a displacement-controlled microcompression system to achieve constant strain rate compression tests. High-temperature micropillar compression is a novel experimental technique [17,19,25,26,29,30], which allows systematic variation in the lattice resistance using temperature control, while the athermal contributions to strength remain unchanged. Moreover, LiF [1 1 1] constitutes a model system because micropillars with varying diameters can be obtained by chemically etching away the surrounding matrix in directionally solidified LiF–NaCl and LiF– KCl eutectics. This avoids any use of FIB methods, and yields micropillars with a controlled dislocation density, independent of the sample preparation technique. This allowed the precise determination of the contribution of lattice resistance, forest hardening, and single-arm dislocation source operation to the strength of the pillars as a function of both pillar size and temperature, contributing to a sound experimental validation of existing models on size effects in mechanical strength. 2. Experimental procedure LiF micropillars were obtained without the use of FIB, using the novel approach previously employed by Bei et al. for Mo [31] and by the present authors for LiF [19]. This was a critical point for this investigation, because the precise estimation of the different contributions to the critical resolved shear stress in Eq. (2) requires reliable values of the initial dislocation density in the micropillar. FIB machining, the usual technique to manufacture micropillars from single crystals, introduces defects in the micropillars such as the formation of Ga+ implanted amorphous layers [32], lattice distortions and dislocation loops [33]. Therefore, even if the initial dislocation density of the crystal is measured, the defect structure after FIB milling can
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be dramatically different, introducing many uncertainties in Eq. (2). Circular LiF rods grow in the [1 1 1] direction during directional-solidification of LiF–NaCl and LiF–KCl eutectics [27,28]. The rods are distributed in a hexagonal pattern, and the rod diameter can be varied from 1 lm to 5 lm by changing the growth rate from 50 mm h1 to 3 mm h1. LiF micropillars with an aspect ratio (length to diameter ratio) of 2:1, ideal for micropillar compression to avoid buckling and constraint effects, were obtained by controlled chemical etching of the matrix with CH4O, on a mechanically polished cross section. The single crystal nature of the micropillars and their orientation, parallel to the [1 1 1] axis, were checked by transmission electron microscopy [19]. The as-grown LiF rods contained a large initial dislocation density, of the order of 2.5 1013 m2, as revealed by etch pit density analysis of extracted LiF fibers etched with H2O (Fig. 1). The micropillar compression tests at ambient and elevated temperature were carried out in a Zeiss DSM 962 scanning electron microscope equipped with an Alemnis in situ indenter [29,30]. The system is inherently displacement-controlled and allows temperature control of both the indenter tip and the sample on independent thermocouple-controlled feedback loops. A flat punch diamond tip was used to apply the load on the micropillars. All test temperatures reported in this work correspond to the sample surface temperatures as measured by the thermally calibrated indenter as a temperature probe, using a technique described previously [34]. This technique greatly reduces any uncertainty in surface temperature owing to thermal gradients in samples with low thermal conductivity. The precision in the surface temperature measurement is on the order of 1 °C. Compression tests were performed at temperatures of 25, 75, 130 and 250 °C at a constant strain rate of 103 s1, unless indicated otherwise. The measured load– displacement curves were corrected for the extra compliance associated with the elastic deflection of the matrix at
Fig. 1. Dislocation pits on the surface of a LiF rod after etching with H2O: etch pit density 2.5 1013 m2.
the base of the pillar, following a standard shear lag model for load transfer between the matrix and an isolated fiber embedded in an infinite matrix [19,35]. The initial micropillar diameter and the gauge length were measured in situ to compute the stress and strain from corrected load–displacement curves. Slight variations in the pillar diameter (<10%) were found from pillar to pillar. Regarding the stress–strain response, the elastic modulus corresponding to the initial elastic loading was always lower than that associated with elastic unloading at the end of the test, regardless of micropillar diameter or test temperature. These discrepancies can be attributed to the impossibility of attaining a perfect parallelism between the top surface of the pillar and the indenter flat punch tip [36]. The slight misalignment between them induces a stress concentration at the initial contact point of the micropillar, which leads to early plastification and the apparent reduction of the initial loading stiffness. This behavior made it difficult to establish the onset of plastic yielding from the stress–strain curves. To avoid uncertainties in the yield stress determination, the flow stress at 10% strain, r10%, was used throughout this investigation as the reference value for the compressive strength as a function of temperature and micropillar diameter, as is routinely done in other micropillar compression works. In any case, the results obtained were not affected by this choice, and similar conclusions could be drawn if the flow stresses were determined at lower plastic strains of 3%. Additionally, load resolution of the experimental setup became an issue for the smallest micropillars tested at 250 °C, as the maximum load carried by a LiF micropillar 1 lm in diameter at high temperature was of the order of 15 lN, very close to the resolution of the load cell. This limitation hindered the study of the effect of temperature on the size effect above 250 °C. 3. Experimental results The stress–strain curves corresponding to micropillars 5, 2.5 and 1 lm in diameter tested in the temperature range 25–250 °C are plotted in Fig. 2a–c, respectively. The stress–strain behavior becomes more stochastic as the diameter of the micropillar decreases, especially for the 1 lm micropillars. This is in agreement with other micropillar compression experiments [20,37,38], where the larger scatter and the more jerky behavior of the smallest pillars were attributed to the reduction in the number of the available dislocation sources as the diameter decreased. Plastic flow in LiF is highly anisotropic, owing to its ionic character. The preferred slip system belongs to the f1 1 0g “soft” {1 1 0}h1 1 0i family, whose lattice resistance s0 is negligible at room temperature and only increases at critf1 1 0g ical temperatures T 0 below 233 °C [37]. The Schmid factor of the {1 1 0}h1 1 0i slip system is 0.5 during compression of h1 0 0i oriented single crystals and, as a result, the compressive strength of bulk h1 0 0i single crystals is in the range 5–40 MPa, depending on the dislocation density
R. Soler et al. / Acta Materialia 81 (2014) 50–57
53
and impurity level. LiF single crystals can also deform plastically in the “hard” {1 0 0}h1 1 0i slip system when loaded along the h1 1 1i direction. In this orientation, the Schmid factor for the “soft” {1 1 0}h1 1 0i slip system is 0, while the Schmid factor for the “hard” {1 0 0} slip is 0.47. The f1 1 0g lattice resistance s0 on the “hard” {1 0 0} slip system at room temperature is on the order of a few hundreds of f1 1 0g megapascals [40], substantially larger than s0 at room temperature, and only decreases for critical temperatures f100g T0 250 C [39,41]. Previous investigations of the strength of h1 0 0i oriented LiF micropillars showed that plastic deformation was governed by the soft slip system [23,42]. A strong size effect was found, with power-law exponents of the order of 0.7–0.8, independently of initial dislocation density. However, in the case of h1 1 1i-oriented micropillars, plastic deformation occurs on the hard slip system [19,36], and no size effect on the flow stress was reported at ambient temperature [19]. The magnitude of the critical resolved shear stress at 10% strain (obtained from the flow stress and the Schmid factor) is plotted in Fig. 3 as a function of micropillar diameter for temperatures in the range 25–250 °C. The plot shows the emergence of a size effect with temperature and the power-law exponent (obtained by fitting the experimental results to Eq. (1)) increases from n 0 at 25 °C, to n 0.53 at 250 °C, and the size effect at 250 °C becomes similar to that displayed by most fcc metals, and also LiF oriented in the h1 0 0i direction, at room temperature. 4. Discussion on temperature dependent size effects In order to check the validity of Eq. (2), it is necessary to estimate the bulk lattice resistance of h1 1 1i LiF as a function of temperature. Gilman [40] performed torsion tests
Fig. 2. Compressive stress–strain curves of h1 1 1i LiF micropillars (a) 5 lm, (b) 2.5 lm, and (c) 1 lm in diameter, respectively, for temperatures of 25 °C (blue lines), 75 °C (green lines), 130 °C (red lines), and 250 °C (gray lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. sCRSS as a function of the micropillar diameter in the temperature range 25–250 °C. The power law exponent n at each temperature obtained by fitting the experimental results to Eq. (1) is indicated in the plot.
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R. Soler et al. / Acta Materialia 81 (2014) 50–57
on macrosocopic LiF (1 0 0) single crystals from 350 °C to 650 °C. The brittle nature of LiF when deformed along the hard {1 0 0} slip system led to cracking below 350 °C, and no data could be obtained. A Peierls’ model [43,44] for the dependence of the shear flow stress on temperature was applied in this work to extend the data of the bulk yield stress to lower temperatures, with the aid of the room temperature micropillar compression results. The Orowan equation relates the shear strain rate c_ to the mobile dislocation density qm, the Burgers vector b and dislocation velocity v, according to c_ ¼ qm bm
ð4Þ
The dislocation velocity v can be estimated using a standard approach for a stress-activated process: ðsp sa ÞV ðsp þ sa ÞV v ¼ t b exp exp ð5Þ kT kT where t is the attempt frequency, sp is the Peierls stress, V is the activation volume, T is the absolute temperature, sa is the applied shear stress, and k Boltzmann’s constant. From Eq. (5) and assuming that sa = s0, the thermal dependence of the lattice resistance s0 (first term in Eq. (2)) of the hard {1 0 0}h1 1 0i slip system can be expressed as kT sp V c_ 1 s0 ¼ sinh exp ð6Þ V kT 2qm vb2
Fig. 4. Influence of strain rate on the critical resolved shear stress of [1 1 1] LiF micropillar single crystals at ambient temperature. The activation volume V was determined from the linear slope between sCRSS and the natural logarithm of the shear strain rate.
This relationship forms the basis for extending the available bulk yield stress data for LiF [1 1 1] single crystals to lower temperatures, with the aid of the room temperature micropillar compression results. Substitution of Eq. (6) into Eq. (2) leads to kT sp V c_ pffiffiffiffiffiffiffi 1 sCRSS ¼ sinh exp þ 0:5lb qtot 2 V kT 2qm vb alb þ kmax ðD; qtot ; bÞ
ð7Þ
where the first two terms contribute to the bulk strength of the material, sbulk, and the third term gives rise to the size effect ssize. It is worth noting that, in contrast to the bulk crystals, LiF [1 1 1] micropillars can indeed be plastically deformed at room temperature, because fracture is suppressed owing to the small dimensions of the specimens [45]. Taking advantage of the fact that the flow stress of LiF [1 1 1] at room temperature is not affected by size effects, as shown in Fig. 2, micropillars were tested at different strain rates in the range 0.00033–0.033 s1 at room temperature. Assuming that the activation volume is independent of the strain rate, that sV remains large with respect to kT, and neglecting ssize, Eq. (7) can be simplified to kT dc kT pffiffiffiffiffiffiffi ð8Þ sbulk ¼ sp þ ln ln qm vb2 þ 0:5lb qtot V dt V i.e. the bulk shear stress sbulk varies linearly with the natural logarithm of shear strain rate. Fig. 4 shows that this is
Fig. 5. Theoretical predictions of the bulk strength sbulk of the hard {1 0 0}h1 1 0i slip system in LiF as a function of temperature for two different dislocation densities and imposed shear strain rates, as summarized in Table 1, following Eq. (8). The dots represent the experimental sCRSS data for micropillars of different diameter (1 lm in red, 2.5 lm in green and 5 lm in blue) as well as for bulk LiF (gray dots) from Gilman [40]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
indeed the case and validates the Peierls’ model to analyze the strain rate sensitivity of LiF [1 1 1] micropillars at room temperature. From the linear slope kT , it is possible to V determine the activation volume V, yielding a value of 2.6 1028 m3 or 11.41 b3. It should be noted that this value is in good agreement with the activation volume expected for lattice resistance governed dislocation glide [46].
R. Soler et al. / Acta Materialia 81 (2014) 50–57 Table 1 Material parameters and testing conditions in Eq. (6) to determine the evolution of sCRSS of the hard {1 0 0}h1 1 0i slip system in bulk LiF as a function of temperature.
Testing conditions Bulk [38] This work Material constants v (s1) 8.0 1011
b (nm) 0.285
c_ ðs1 Þ
qtot (m2)
105 2.13 103
108 2.5 1013
V (b3) 11.41
sp (MPa) 530
Moreover, the room temperature microcompression data provide the basis for extending the available bulk yield stress data for LiF [1 1 1] single crystals to lower temperatures, with the aid of Eq. (8). Fig. 5 plots the experimental critical resolved shear stress on the {1 0 0} hard slip system
55
of LiF coming from microcompression in this work together with the available bulk shear strengths in the literature [40,47] from testing single crystals at high temperature. Both sets of results could be fit by Eq. (8) using a Peierls stress sp = 530 MPa and the appropriate experimental parameters of dislocation density and shear strain rate. The corresponding curves are shown as solid lines in Fig. 5. The actual materials and testing parameters used in Eq. (8) are summarized in Table 1, where the attempt frequency v was calculated from Debye’s frequency equation. The {1 0 0} shear modulus l, including temperature dependence, was obtained from Hart [48]. The shear strain rate and the total dislocation density for each data set are reported, and the mobile dislocation density qm was considered 3/12 of the total dislocation density, assuming slip along the three “hard” slip systems preferentially oriented for slip out of the 12 total slip systems.
Fig. 6. Evolution of sCRSS according to Eq. (7) (solid line) and comparison with experimental data: (a) 25 °C; (b) 75 °C; (c) 130 °C; (d) 250 °C. The dashed lines represent the size-independent contribution sbulk at each temperature (lattice resistance and forest hardening), while the dotted line represents the sizedependent contribution ssize, due to the operation of single-arm dislocation sources.
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R. Soler et al. / Acta Materialia 81 (2014) 50–57
It should be noted that Eq. (8) successfully captured the influence of temperature on the critical resolved shear stress of micropillars at ambient temperature (blue curve) and bulk crystals (gray curve) at high temperature. However, it failed to capture the behavior of micropillars at high temperature, and this was due to the increasing importance of the athermal size-dependent contribution ssize, in the case of micropillars as the lattice resistance decreased. Once the first two terms of Eq. (7), accounting for the variation in bulk shear strength sbulk with temperature have been obtained, it is possible to estimate the third term, i.e. the expected size-dependent contribution ssize, arising from the operation of single-arm dislocation sources. For this, the statistical average length of the weakest single-arm diskmax as a function of micropillar diameter location source was estimated using the statistical model developed by Parthasarathy et al. [7] for each micropillar diameter: Z R kmax ¼ kmax pðkmax Þdkmax 0
¼
p1 pðR kmax Þðb kmax Þ pRb 0 p½ðR kmax Þ þ ðb kmax Þ pkmax dkmax pRb Z
R
1
ð9Þ
where R is the radius of the pillar, and p is the number of pinning points, given by p ¼ Integer½qm pRh
ð10Þ
where qm is the mobile dislocation density, h is the height of the pillar, and the rest of the symbols have the usual meaning. Finally, the variation in the shear modulus l with temperature was obtained from Hart [48], while the temperature dependence of the Burgers vector was neglected in the analysis. The predictions of Eq. (7) for the sCRSS as a function of temperature are plotted in Fig. 6, together with the experimental high-temperature micropillar compression data. The agreement between the experimental results and the predictions in Eq. (7) is remarkable. The bulk strength dominates over the size-dependent contribution at 25 °C and 75 °C in the range of micropillar diameters tested, and no size effects are found. However, both contributions are of the same order at 250 °C, and a strong size effect develops at high temperatures. These results show that lattice resistance influences the power-law exponents for the size effect of different materials, as proposed by Korte and Clegg [18] and Lee and Nix [24]. Recent results in Mo micropillars [26], a bcc metal with a high lattice resistance, have also shown similar trends, with the size exponent increasing at temperatures close to 250 °C, the critical temperature for Mo at which the lattice resistance is expected to become negligible. However, the micropillars in that study were machined by FIB milling, and the uncertainty in the initial dislocation density precluded any estimation of the size-dependent and size-independent terms at each temperature.
5. Conclusions The influence of temperature on the compressive strength of [1 1 1] LiF single crystals was measured by means of micropillar compression tests. Micropillars of diameters in the range 1–5 lm were obtained by means of directional solidification and surface etching of eutectic compounds, leading to single crystals with a controlled initial dislocation density of 2.5 1013 m2. The micropillar flow strength was independent of the micropillar diameter at ambient temperature, but a strong size effect developed with temperature: micropillars 1 lm in diameter were twice as strong as those 5 lm in diameter at 250 °C. The different contributions to the flow stress of the micropillars, namely the lattice resistance, forest hardening and the sizedependent contribution as a result of operation of singlearm dislocation sources were rigorously accounted for as a function of both temperature and micropillar diameter. It was demonstrated that the size effect observed during micropillar compression comes about as a result of the relative weights of the size-independent (lattice resistance plus forest hardening) and size-dependent contributions to strength. At room temperature, the former contribution dominated, and no size effect was found, while at 250 °C, both contributions were of the same order for the micropillar diameters studied, leading to a strong size effect. Thus, the role of the lattice resistance on the size effect of micrometer-size single crystals was demonstrated unambiguously for the first time. This result rationalizes the different values of power-law exponent for the size effect found in the literature for fcc and bcc metals, as well as covalent and ionic solids. Acknowledgements This investigation was supported by the Spanish Ministry of Economy and Competitiveness through project MAT2012-31889. HYC acknowledges the support of a Juan del Cierva post-doctoral fellowship from the Spanish Ministry of Economy and Competitiveness. The authors are indebted to Prof. V.M. Orera and Dr. R.I. Merino from the Institute of Materials Science of Arago´n (CSIC), who grew and provided the eutectic crystals. References [1] Uchic MD, Dimiduk DM, Florando JN, Nix WD. Science 2004;305:986–9. [2] Greer JR, Oliver WC, Nix WD. Acta Mater 2005;53:1821–30. [3] Volkert CA, Lilleodden ET. Philos Mag 2006;86:5567–79. [4] Uchic MD, Shade PA, Dimiduk DM. Annu Rev Mater Res 2009;39:361–86. [5] Dehm G. Prog Mater Sci 2009;54:664–88. [6] Greer JR, De Hosson JTM. Prog Mater Sci 2011;56:654–724. [7] Parthasarathy TA, Rao SI, Dimiduk DM, Uchic MD, Trinkle DR. Scr Mater 2007;56:313–6. [8] Rao SI, Dimiduk DM, Parthasarathy TA, Uchic MD, Tang M, Woodward C. Acta Mater 2008;56:3245–59.
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