APPLIED PHYSICS LETTERS 91, 161913 共2007兲 Sample size effect and microcompression of Mg65Cu25Gd10 metallic glass C. J. Lee and J. C. Huanga兲 Institute of Materials Science and Engineering, Center for Nanoscience and Nanotechnology, National Sun Yat-Sen University, Kaohsiung, Taiwan 804, Republic of China T. G. Nieh Department of Materials Science and Engineering, the University of Tennessee, Knoxville, Tennessee 37996-2200, USA 共Received 18 July 2007; accepted 27 September 2007; published online 19 October 2007兲 Micropillars with diameters of 1 and 3.8 ␮m were fabricated from Mg-based metallic glasses using focus ion beam, and then tested in compression at strain rates ranging from 6 ⫻ 10−5 to 6 ⫻ 10−1 s−1. The apparent yield strength of the micropillars is 1342– 1580 MPa, or 60%–100% increment over the bulk specimens. This strength increase can be rationalized using the Weibull statistics for brittle materials, and the Weibull modulus of the Mg-based metallic glasses is estimated to be about 35. Preliminary results indicated that the number of shear bands increased with the sample size and strain rates. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2800313兴 The strengths of face-center-cubic single crystals are known to be strongly dependent on the sample dimensions, especially when the dimension is close to micron or submicron scales.1–5 For example, the strengths of Ni or Au microsized single crystal specimens are several times higher than that of the bulk specimens, and the strength can be even further increased by more than one order of magnitude in submicron pillar specimens. This dramatic effect occurs when the size of samples is smaller than the characteristic length for multiplication, which results in the dislocation starvation. In contrast, bulk metallic glasses 共BMGs兲 do not have a long range ordered structure. The plastic deformation of BMGs at room temperature proceeds in the form of highly localized shear bands.6,7 Yielding is induced by shear band nucleation and propagation. BMGs are not expected to work harden because localized shear bands do not exert long range interactive force like dislocations do. Thus, once yielding starts shear band propagates very rapidly and leads to catastrophic failure. It has been recently shown that the strength of Mg-based BMGs is sensitive to the specimen size.8 Specifically, the strength is higher in a smaller size sample. There is yet another sample size effect in BMGs. Schuh et al.9 recently employed nanoindentation to study the shear band dynamics and identified a region 共homogeneous region II兲, which was characterized by extensive emission of multiple shear bands. Gao et al.10 also developed a free-volume-based, thermoviscoplastic constitutive law to describe the inhomogeneous deformation in amorphous alloys and concluded that homogeneous region II actually corresponded to a region with fine shear bands, and the shear band spacing decreases with increasing strain rate. The question then arises as to what would happen when the specimen size is reduced to a range where the specimen can no longer support even one single shear band. A preliminary result10 indicated that the size scale is probably around 1 ␮m or less. In the current paper, we fabricated microscale pillar specimens and subsequently tested them in compression to study their mechanical response as a function of specimen size and strain rate. We selected a Mg65Cu25Gd10 共in at. %兲 amorphous alloy for this study. Compression samples were prepared using the dual focus ion beam system 共FIB兲 of Seiko, SMI3050 SE, following the method developed previously.5 Firstly, the 30 keV and 7 – 12 nA currents Ga beam in FIB was faced to the surface of the BMG disk to mill a crater with a bigger size island being located in the center. Secondly, the same voltage and smaller currents of 3 down to 0.2 nA were used to refine the preserved island in the center to a desired diameter and height of pillars. Two kinds of micropillars with diameters of 3.8 and 1 ␮m, and the respective height-to-diameter ratio of 2–2.5 and 2.5–3 were fabricated. These pillars, as shown in Fig. 1, are tapered because the divergence of ion beam, and the convergence angle is about 2.5°. Microcompression tests were performed with an MTS nanoindenter XP under the continuous stiffness measurement mode using a flat punch, which was machined out of a standard Berkovich indenter also by FIB. The projected area of the tip of the punch is an equilateral triangle of 13.5 ␮m. A prescribed displacement rate of 0.25– 500 nm/ s, which corresponds to the initial strain rates of 6 ⫻ 10−5 – 6 ⫻ 10−2 s−1, was used to deform the pillars. The load-displacement data are presented in Fig. 2. Traditionally, the curves are readily converted into the engineering stress and strain curves, with the assumption that the sample is uniformly deformed. However, the assumption is obviously violated since a bulk metallic glass is deformed by the emission of highly localized shear bands, as will be shown later. Thus, we choose to present the load-displacement data for the convenience of discussion. Author to whom correspondence should be addressed. Tel.: ⫹886-7-5252000 Ext 4063. Electronic mail: [email protected] FIG. 1. Mg-BMG micropillars fabricated by focus ion beam technique: 共a兲 1 ␮m in diameter and 共b兲 3.8 ␮m in diameter. a兲 0003-6951/2007/91共16兲/161913/3/$23.00 91, 161913-1 © 2007 American Institute of Physics Downloaded 20 Oct 2007 to 140.117.53.150. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 161913-2 Appl. Phys. Lett. 91, 161913 共2007兲 Lee, Huang, and Nieh TABLE I. Summary of the compressive stress 共in unit of MPa兲 of Mg65Cu25Gd10 glass at different strain rates. 1 ␮m ⌽ 3.8 ␮m ⌽ Bulk 3 mm ⌽ FIG. 2. 共Color online兲 Compression load-displacement curves of Mg-BMG at different strain rates: 共a兲 1 ␮m and 共b兲 3.8 ␮m. Since the pillars are uniformly tapered, the total displacement ⌬h is ⌬h = 1 h0 冕 h0 ⌬u共x兲dx, 共1兲 0 where h0 is the height of the pillar ⌬u共x兲 is the local displacement. In the elastic range, with ␴ = E␧ and ␴共x兲 = P / A共x兲, where P and A are instantaneous load and sample cross section area, respectively, Eq. 共1兲 is deduced to ⌬h = 冕 h0 0 P dx, ␲共d0 + x sin ␪兲2E 共2兲 where d0 is the radius of the pillar and ␪ = 2.5° is the taper angle. As sin ␪ ⬇ ␪, we obtain 冉冊冤 P E= ⌬h 冉 2␪h0 d0 ␲ d 02 ␪ ln 1 + 冊冥 . 共3兲 Substitute d0 = 0.5 ␮m and h0 = 2.5 ␮m, we finally have E = 2.64 ⫻ 1012 冉冊 P , ⌬h 共4兲 where P and ⌬h are in the units of Newton and meter, respectively. Thus, the elastic modulus is the slope of loaddisplacement curve multiplied by a constant of 2.64⫻ 1012, which is about 41 GPa. However, the above calculation was based on a rigid substrate. In our case, the substrate is also the same Mg-BMG. Under such a condition, according to a finite element analysis of microcompression,11 the actual Young’s modulus should be corrected by a factor of about 1.25. This results in a value of 51.3 GPa, which is in excellent agreement with the literature value of 50.6 GPa measured from bulk Mg-BMG sample.12 Since the sample is tapered, yielding will, in principle, start from the top of the sample because this is the location with the minimum cross section. The departure load divided by the top area is then the “apparent” yield strength. Despite a slight variation, the apparent yield strengths of the 3.8 and 1 ␮m pillars range from 1342 to 1580 MPa, which are much higher than that for the bulk Mg65Cu25Gd10 samples, which is about 800 MPa.13 These data are summarized in Table I. The current result of improved strength caused by a decrease in sample size is in accord with a previous study on millimeter-scaled BMG.8 Also noted in Table I is the fact that the average strength of the 1 ␮m pillar 共⬃1500 MPa兲 is higher than that of the 3.8 ␮m pillar 共⬃1390 MPa兲. This demonstrates that the sample size effect can extend from the millimeter to micron range. The strength improvement from 10−4 s−1 10−3 s−1 10−2 s−1 6 ⫻ 10−2 s−1 1400 1342 800 1488 1362 1580 1384 1525 1464 millimeter-scaled sample to micron-scaled pillar is noted to be about 60%–100%, which is quite remarkable. After yielding, the load remains essentially constant at different strain rates, as evident in Fig. 2. The corresponding engineering stress-strain curves, not shown here, have a similar shape. This particular shape of stress-strain curve usually, according to the conventional wisdom, leads to a conclusion that BMG behaves like a perfectly plastic material. However, the raw displacement-time data, as given in Fig. 3, seem to reveal a different message. Specifically, data in Fig. 3 indicate the displacement 共or strain兲 takes place almost instantly. In other words, strain does not occur in a gradual fashion but in the form of burst, even though the tests were conducted under strain rate control conditions. Every strain burst event, regardless of the strain rate, proceeds within about 1 s, suggesting that the strain rate during these bursts was at least 10−1 s−1. There is also a notable trend that, for both micropillars, a faster compression rate results in a larger strain burst. The instant strain burst is peculiar but is usually indicative of a sudden propagation of a localized shear band.9 The morphology of compressed pillar samples is shown in Fig. 4. It is well known that bulk Mg-BMGs generally fracture into fragments at the onset of yielding since they have a low Poisson’s ratio of ⬃0.32.14 However, the present micropillar samples do not exhibit such a catastrophic fracture mode even at a “macroscopic” strain of 10%–15%. In fact, all of them deformed by macroscopic shear, presumably resulted from localized shear band propagation. In principle, shear band is anticipated to initiate from the corner of contact between specimen and compression punch. This is the location where the sample experiences the maximum stress, not only because it has the minimum cross section as a result of sample taper, but also due to the large constraint caused by the friction between the test specimen and punch. Indeed, it is clearly shown in Fig. 4, in which all pillars are observed to deform by this shear mode after yielding. Furthermore, immediately following the yielding, the upper part of the specimen begins to slide along a major shear plane. However, the effective load-bearing cross-sectional area always remains constant, as the punch impresses onto the bottom part of the FIG. 3. 共Color online兲 Scanning electron microscopy 共SEM兲 micrographs showing the appearance of deformed pillars: 共a兲 3.8 ␮m, ⬃6 ⫻ 10−4 s−1; 共b兲 1 ␮m, ⬃2 ⫻ 10−4 s−1; 共c兲 3.8 ␮m, ⬃1.2⫻ 10−3 s−1; 共d兲 1 ␮m, ⬃1 ⫻ 10−3 s−1; 共e兲 3.8 ␮m, ⬃6 ⫻ 10−2 s−1; and 共f兲 1 ␮m, ⬃6 ⫻ 10−2 s−1. Downloaded 20 Oct 2007 to 140.117.53.150. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 161913-3 Appl. Phys. Lett. 91, 161913 共2007兲 Lee, Huang, and Nieh An increase in strength with decreasing sample size can be rationalized in the following. For brittle materials, the variability of their strengths is expected based on their flaw sensitivity and may be analyzed using the Weibull statistics. The Weibull equation describes the fracture probability P f as a function of a given uniaxial stress ␴ in the form of17 冋冉冊册 P f = 1 − exp − V ␴ − ␴u ␴0 m , 共5兲 where ␴0 is a scaling parameter, m is the Weibull modulus, and V is the volume of the tested sample. The parameter ␴u denotes the stress at which there is a zero failure probability, and is usually taken to be zero. Assume the characteristic flaw causing fracture in both millimeter and micron samples are the same, then, at a fixed fracture probability, i.e., P f = constant, the above equation reduces to V 冉冊 ␴ ␴0 m = const. 共6兲 Since V ⬀ d3, where d is the diameter of the compression sample, we obtain FIG. 4. Displacement as a function of loading time during microcompression for the 共a兲 1 ␮m and 共b兲 3.8 ␮m pillars. sample. This explains the observation of “lips” on the top of deformed samples and the constancy of load in Fig. 2. Additional remarks must be made from the examination of Fig. 4. First, the recorded displacement 共or strain burst兲 comes from a sudden macroscopic sliding, which is triggered by microscopic shear band propagation in the specimen, not from a uniform plastic deformation per se. It is, in fact, equivalent to the situation of rigid-body sliding or bicrystal sliding.15,16 The measured strain should not be interpreted as general plasticity, at least in the conventional sense, since neither the two “crystals” deform. Furthermore, the constancy of load 共or engineering stress兲 is not indicative of a perfectly plastic material, or has any thing to do with work hardening or softening. Rather, it is a measure of friction of sliding between two rigid bodies. In the current case, it represents the self frictional force of the Mg-BMG. Gao et al.10 predicted a finer shear band spacing at a faster strain rate. Although the strain rate differs by only two orders of magnitude in the present experiments, there is a general tendency of having more shear bands in pillars deformed at a higher strain rate; this is clearly demonstrated in Fig. 4. It is also noted that there are very limited number of shear bands in these micropillar samples, especially in the 1 ␮m pillar sample. In fact, at the slowest strain rate of 2 ⫻ 10−4 s−1, only one shear band is present. This suggests that 1 ␮m is close to the scale limit of shear band spacing. Should it be the case, then, the measured apparent yield strength of the 1 ␮m pillar 共1580 MPa兲 would represent the strength for a material which does not contain any volumetric defect with size greater than the critical value that can lead to catastrophic fracture. The critical strength of the present Mg BMG is about G / 12, where G is the shear modulus of 19.3 GPa. Although there is no existing theory to predict the theoretical strength of amorphous metals, it is interesting to note that G / 共10– 12兲 is also coincidentally the predicted value for polycrystalline metals. d3␴m = const. 共7兲 As listed in Table I, the apparent strengths of the 3 mm and 1 ␮m Mg-BMG samples are 800 and 1580 MPa, respectively. Insert these values into Eq. 共7兲, the Weibull modulus is calculated to be about 35, which is within the range of the m values recently reported for Zr48Cu45Al7 共m = 73.4兲 and 共Zr48Cu45Al7兲98Y2 共m = 25.5兲.18 The above analysis supports the observation that the compression stress increases with decreasing specimen size. The authors gratefully acknowledge the sponsorship from the National Science Council of Taiwan, Republic of China, under the Project No. NSC 96-2218-E-110-001. This work was also partially supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy under Contract No. DEFG02-06ER46338 with the University of Tennessee. 1 M. D. Uchick, D. M. Dimiduk, J. N. Florando, and W. D. Nix, Science 305, 986 共2004兲. 2 D. M. Dimiduk, M. D. Uchic, and T. A. Parthasarathy, Acta Mater. 53, 4065 共2005兲. 3 J. R. Greer, W. C. Oliver, and W. D. Nix, Acta Mater. 53, 1821 共2005兲. 4 C. A. Volkert and E. T. Lilleodden, Philos. Mag. 86, 5567 共2006兲. 5 M. D. Uchick and D. M. Dimiduk, Mater. Sci. Eng., A 400-401, 268 共2005兲. 6 F. Spaepen, Acta Metall. 25, 407 共1976兲. 7 A. S. Argon, Acta Metall. 27, 47 共1979兲. 8 Q. Zheng, S. Cheng, J. H. Strader, E. Ma, and J. Xu, Scr. Mater. 56, 161 共2007兲. 9 C. A. Schuh, A. C. Lund, and T. G. Nieh, Acta Mater. 52, 5879 共2004兲. 10 Y. F. Gao, B. Yang, and T. G. Nieh, Acta Mater. 55, 2319 共2004兲. 11 H. Zhang, B. E. Schuster, Q. Wei, and K. T. Ramesh, Scr. Mater. 54, 181 共2006兲. 12 X. K. Xi, R. J. Wang, D. Q. Zhao, M. X. Pan, and W. H. Wang, J. Non-Cryst. Solids 344, 105 共2004兲. 13 H. M. Chen, Y. C. Chang, T. H. Hung, X. H. Du, J. C. Huang, J. S. C. Jang, and P. K. Liaw, Mater. Trans. 48, 1802 共2007兲. 14 J. Schroers and W. L. Johnson, Phys. Rev. Lett. 93, 255506 共2004兲. 15 T. Watanabe, M. Yamada, S. Shima, and S. Karashina, Philos. Mag. A 40, 667 共1979兲. 16 H. Miura, K. Hamashima, and K. T. Aust, Acta Metall. 28, 1591 共1980兲. 17 W. J. Weibull, J. Appl. Mech. 18, 293 共1951兲. 18 W. F. Wu, Y. Li, and C. A. Schuh, 共to be published兲. Downloaded 20 Oct 2007 to 140.117.53.150. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp