Role of free volume in strain softening of as-cast and annealedbulk metallic glass
Byung-Gil YooDivision of Materials Science and Engineering, Hanyang University, Seoul 133-791, Korea
Kyoung-Won Park and Jae-Chul LeeDepartment of Materials Science and Engineering, Korea University, Seoul 136-701, Korea
U. RamamurtyDepartment of Materials Engineering, Indian Institute of Science, Bangalore 560012, India
Jae-il Janga)
Division of Materials Science and Engineering, Hanyang University, Seoul 133-791, Korea
(Received 22 September 2008; accepted 24 November 2008)
Plasticity in amorphous alloys is associated with strain softening, induced by the creationof additional free volume during deformation. In this paper, the role of free volume,which was a priori in the material, on work softening was investigated. For this, anas-cast Zr-based bulk metallic glass (BMG) was systematically annealed below its glasstransition temperature, so as to reduce the free volume content. The bonded-interfaceindentation technique is used to generate extensively deformed and well defined plasticzones. Nanoindentation was utilized to estimate the hardness of the deformed as well asundeformed regions. The results show that the structural relaxation annealing enhancesthe hardness and that both the subsurface shear band number density and the plastic zonesize decrease with annealing time. The serrations in the nanoindentation load-displacement curves become smoother with structural relaxation. Regardless of theannealing condition, the nanohardness of the deformed regions is �12–15% lower,implying that the prior free volume only changes the yield stress (or hardness) but not therelative flow stress (or the extent of strain softening). Statistical distributions of thenanohardness obtained from deformed and undeformed regions have no overlap,suggesting that shear band number density has no influence on the plastic characteristicsof the deformed region.
I. INTRODUCTION
Plastic deformation in metallic glasses is distinctlydifferent from that in crystalline metals. The fundamen-tal carriers of plasticity in amorphous alloys are sheartransformation zones (STZs), which occur through coop-erative shear displacements of clusters of atoms in re-sponse to the applied load.1 This deformation mode hastwo key attributes. (i) The matrix surrounding the STZhas to accommodate the dilation associated with the STZoperation. This results in pressure sensitivity of plasticflow, which in turn has many important consequences interms of mechanical properties.2–4 (ii) STZs occur, pref-erentially, in those regions that are readily amenable toplasticity by virtue of having lower than average atomicdensity. For amorphous materials, there are a number ofdifferent ways of describing this intrinsic state variableof atomic density, free volume being the most popular
one.5,6 “Liquid-like regions,”7 local shear modulus,8
fictive temperature,9 and atomic packing density10 areother descriptors.
Considerable attention is paid, in the recent literature,to understanding the influence of free volume on plasticdeformation of amorphous alloys. The driving force forsuch studies is the development of a detailed understand-ing of the origins of ductility and toughness in bulkmetallic glasses (BMGs), a new class of structural mate-rials with a unique and attractive set of properties. Oneway to vary the free volume systematically is to inducestructural relaxation in the metallic glasses throughannealing heat treatments below the glass transition tem-perature, Tg.
11–15 Structural relaxation, which does notlead to crystallization, reduces the free volume by anni-hilation and such free volume reduction is known tohave pronounced influence on the mechanical behaviorof BMGs. These include severe embrittlement, changein the ductile-to-brittle transition temperature, fatiguecharacteristics, and so on, of the as-cast material.16–20
In contrast, experiments on BMG samples that were
a)Address all correspondence to this author.e-mail: [email protected]
already deformed extensively by mechanical processessuch as rolling, show that post-deformation annealingenhances ductility.21
In addition to the free volume that is present in agiven amorphous alloy, the free volume model of Spae-pen5 suggests that a considerable amount of excess freevolume can be produced during plastic deformation. Thefree volume thus generated causes strain softening of theplastic flow, which in turn leads to localization of flowinto narrow regions known as shear bands.1,22,23 Theinstrumented indentation technique, especially nanoin-dentation, is widely used for investigating the shear bandplasticity in metallic glasses, which manifests as serra-tions in indentation load-displacement (P-h) curves.Examination of these serrations allows for detailedstudies on rate and temperature effects on plasticity.1,24
If the tests are performed under load control, discretedisplacement jumps or pop-ins are seen in the loadingpart of the P-h curve, whereas the displacement controltests show sudden load drops.
Recently, Bhowmick et al.25 have used the bonded-interface technique to generate a well defined plasticflow zone underneath the indenter, which in turn wasprobed with nanoindentation to demonstrate that it isrelatively softer than the undeformed material far awayfrom the indenter tip. In this paper, we extend this workto examine the following issues related to plasticity inBMGs. While it is now well established that plastic flowcauses softening of the BMG, the role played by the freevolume content at the start is not known. For example,“will all amorphous alloys strain soften, irrespective ofthe prior free volume content?” is a question that has notbeen answered yet. A related question is, “if all amor-phous alloys strain soften, to what extent do they strainsoften, that is, will a BMG with higher amount of freevolume at the start of the plastic deformation strain soft-en less than an alloy with lower amount of free vol-ume?” Prior work by Ramamurty and coworkers17,26
has shown that reducing the free volume contentdecreases the shear band number density in the deformedzones. Consequently, “does the number density of shearbands determine the extent of softening?” is also animportant question to examine. Here we seek to addressthese issues, which have important bearing for betterunderstanding of plasticity in BMGs, through systematicexperiments on annealed BMG samples.
II. EXPERIMENTS
The bulk metallic glass that examined in this work is aZr-based BMG, Zr52.5Cu17.9Ni14.6Al10Ti5 (commerciallydesignated as Vit 105), which was obtained in the formof a rod having a diameter of �7 mm and a length of�70 mm. The sample, of which the glass-transitiontemperature (Tg) and the onset temperature of the
crystallization (Tx) are �673 and �720 K, respectively,was annealed at 630 K (0.93Tg) for two different times(20 and 90 min). To examine the possibility of induce-ment of crystallization during annealing, x-ray diffrac-tion (XRD) of the as-cast and annealed specimens wasperformed using D/MAX-2500 (Rigaku-Denki, Japan).To estimate the free volume change during annealing,differential scanning calorimetry (DSC) of all thespecimens was performed using a Perkin-Elmer DSC-7(PerkinElmer Inc., Waltham, MA) in a purified Ar atmo-sphere at a fixed heating rate of 20 K/min.The interface-bonded specimens for macroscopic
spherical indentation were prepared by cutting the rodfirst into two halves and then polishing them to a mirrorfinish prior to bonding them using a high-strength adhe-sive (Loctite, Henkel Ireland Ltd., Ireland). Subsequent-ly, the top surface of the bonded specimen was polishedto a mirror finish. On the bonded interface, sphericalindentations were performed using an instrumented in-dentation equipment, AIS-2100 (Frontics Inc., Seoul,Korea), with a WC ball indenter having a radius of500 mm. The applied indentation loads were 98 N and196 N, both with a fixed loading rate of 5 mm/s. Afterthis macroscopic spherical indentation, the bonded inter-face was opened by dissolving the adhesive in acetone,and then the subsurface deformation morphology wasexamined through optical microscopy. It should be notedthat the finite width of the complaint bond in the bondedinterface technique allows for the relaxation of theplastic constraint (that is otherwise present underneaththe indenter). This, in turn, allows for plastic flow intothe interface. For example, semicircular shear bandsmight be the result of the constraint release by the inter-face. Although the interface-bonded sample cannot gen-erate exactly the same fields of stress and strain duringindentation as those in a bulk sample without an inter-face, it is generally believed that one can gain insightfor better understanding the plasticity underneath theindenter by observing the subsurface deformation mor-phology of the samples. The soft adhesive effect mightbe avoided if the mechanical clamping method is appliedfor bonding the interface instead of using adhesive.However, clamping the sample in a vice can induce alarge additional stress in the specimen.Subsequent to the characterization of the subsurface
plastic zone, it was gently polished using alumina parti-cle of 0.3 mm or diamond paste of 0.5 mm to make it flatfor nanoindentation. To evaluate the hardness distribu-tion within the subsurface deformation region, a series ofnanoindentation experiments were performed on thepolished surface using a Nanoindenter-XP (Nano Instru-ments, Oak Ridge, TN) with two three-sided pyramidalindenters having a different centerline-to-face angle;65.3� (Berkovich indenter) and 35.3� (cube-corner in-denter). The nanoindentation hardness (often called
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nanohardness) was measured according to the Oliver-Pharr method,27,28 with the Berkovich indenter at a max-imum indentation load of 50 mN and the strain rate of0.5 s–1. To avoid possible artifacts, thermal drift wasmaintained below 0.05 nm/s. After nanoindentation, theprofiles of the indented surfaces were examined byatomic force microscopy (AFM) XE-100 (Park Systems,Suwon, Korea).
III. RESULTS
A. Structural characterization
Figure 1 shows the XRD scans of the as-cast andannealed samples. Because the annealing temperature,630 K, is below the Tg of the BMG, no perceptiblecrystalline peaks could be detected in the XRD spectra,suggesting that even the 90-min-annealed sample wasnot crystallized but only structurally relaxed.
To examine the change in structural characteristics inmore detail, DSC analysis was performed in a mannersimilar to that reported in literature.14,16,17 These studiesshow that the exothermic heat flow at the glass transition(DH) is directly associated with the degree of annihila-tion of excess free volume. Slipenyuk and Eckert14 ex-perimentally verified that there is a linear relationbetween DH and the reduced amount of free volume(Duf) during structural relaxation of a Zr-based BMG.Figure 2(a) shows the DSC thermograms recorded fromthe as-cast and annealed samples. The inset of the figure(showing a magnified view of the thermograms belowTg) reveals that the amount of exothermic heat flowdecreases as the annealing time increases. Estimatedamounts of DH, calculated by integrating the exothermicheat flow near the glass transition, are summarized asa function of annealing time in Fig. 2(b). It shows that
annealing reduces the DH, and in turn the amount of freevolume, significantly.
Structural relaxation in glasses can involve the simul-taneous operation of many processes, each associatedwith a characteristic timescale. The stretched exponen-tial function is a frequently used empirical relation todescribe the relaxation rates associated with any physicalproperty. In the context of the enthalpy change at theglass transition, it can be written as11,17,29
DHðtÞ ¼ DHð1Þ þ DHð0Þ � DHð1Þ½ �exp �t=tð Þb ;
ð1Þwhere DH(t), DH(0), and DH(1) are the values of DH attimes t, 0, and 1, respectively, t is the average relaxa-tion time, and b is the stretched exponent. b is indepen-dent of the annealing temperature in principle and can
FIG. 1. X-ray diffraction patterns of the as-cast and annealed
specimens.
FIG. 2. DSC analysis of structural change; (a) DSC scan of the
as-cast and annealed samples with inset showing subtle calorimetric
difference; (b) variation in the exothermic heat flow as a function of
annealing time.
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vary between 0 and 1. b � 1 implies that the systemis a strong glass former whereas b < 0.5 impliesthat the glass is fragile.30,31 A regression fit of Eq. (1)through the data in Fig. 2(b), with DH(0), DH(1), t, andb as variable parameters, yields values of 0.0935 J/g,2.78 J/g, 862 s, and 0.668, respectively, with a goodnessof fit R2 =1, suggesting excellent correlation. The lattertwo are consistent with the literature values.31
B. Subsurface deformation morphology
Figure 3 shows the optical micrographs of the defor-mation region underneath the spherical indentationimpressions. In the as-cast as well as the annealed sam-ples, the size of the plastic deformation zone, which hasmultiple shear bands, increases with the maximumload of spherical indentation, as expected. It is instructive
to note that, in Fig. 3(e), the initial formation (ornucleation) of shear bands occurs not just beneath theindented surface but at a distance from the surface. Thisis because the maximum shear stress during sphericalindentation is obtained at a distance of about 0.45 timesthe contact radius (a) below the specimen surface.32
Importantly, Fig. 3 illustrates the sub-Tg annealingeffect on the size of the deformed zone and the shearband number density at a given load, both of whichdecrease with increasing annealing time. For example,the variation in plastic zone size at a 196 N load (whichwas determined as the radial distance of the outermostshear band as shown in the inset of Fig. 4) as a functionof the annealing time is shown in Fig. 4, exhibiting thedecrease in the plastic zone size with the annealing time.This indicates that the indentation-induced plasticity islimited in the annealed samples compared to that in the
FIG. 3. Load-displacement (P-h) curves recorded during macroscopic spherical indentations on the bonded-interface at the maximum loads of
98 N and 196 N.
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as-cast sample, which is consistent with the results ofRamamurty and coworkers.17,26 The shear bands in theas-cast sample [Figs. 3(a) and 3(b)] can be broadly cate-gorized into two types; semicircular and radial shearbands. For the as-cast sample, only semicircular shearbands were observed underneath the indentation made at98 N [Fig. 3(a)]. At 196 N, a few secondary radial shearbands appear in addition to the semicircular shear bands[Fig. 3(b)]. For the annealed samples, however, the sec-ondary radial bands are rarely observed within the sub-surface deformation region. This relatively limitedplasticity in the annealed samples is a consequence ofthe reduction in free volume due to structural relaxation;as macroscopic plasticity in BMG is controlled by shearbanding behavior, the decreased probability of shearband nucleation due to the reduction in free volume canresult in the dramatic decay in shear band activity andthus plastic deformation.16,17,26
The spacing of semicircular shear bands was measuredalong the centerline from the indented surface [see theinset of Fig. 5(a)], and plotted as a function of thedistance from the tip in Fig. 5(a) and of the annealing
time in Fig. 5(b). Note that the data in Fig. 5(b) weretaken from the regime for the distance of 50–150 mm inFig. 5(a). For the as-cast sample, the interband spacing isin the range of 5–20 mm, similar to the values for a Zr-Cu-Ti-Ni-Be BMG previously reported.25 Further, it isalmost independent of the maximum load applied, de-spite statistical fluctuations. The spacing is much higherin the annealed samples and decreases with increasingload from 98 N to 196 N. This implies that, for theannealed samples, as the plastic strain increases, a newshear band nucleates between the preexisting shear bandsand hence the line density of shear bands increases. Con-versely, in the case of the as-cast sample, there is nodramatic change in the shear band density with increas-ing indentation strain.
Another clue for relatively limited plasticity in theannealed sample vis-a-vis the as-cast sample can befound in Fig. 6, which shows the P-h curves of nano-indentation made with a cube-corner indenter. Theannealed sample exhibits a smaller indentation depth atany given indentation load than the as-cast sample. Thisobservation implies a significant increase in hardness dueto structural relaxation. This observation is consistentwith the previous results on the same material byDmowski et al.15 who demonstrated a rapid increase inmicro-hardness after annealing for a short time. The P-hcurves show continuous serrations (serial pop-ins) thatare associated with shear band nucleation and/or pro-pagation.33–38 Using a cube-corner indenter instead of atypical Berkovich indenter is based on the fact that thesharper cube-corner indenter produces much higher stres-ses and strains than the Berkovich indenter,39,40 makingit possible to clearly observe serrations.41–43 In Fig. 6, theserrated flows seem to strongly depend on the annealingdegree. As the annealing time increases, the discreteevents gradually weaken. Since the serrations could arisefrom the inhomogeneous deformation by shear bandingactivity,33–38 this trend is evidence for the decreasedshear banding activity in the annealed samples.
Estimating the contribution of the serrations (pop-ins)to the total plastic deformation in the BMG can provide
FIG. 4. Change in plastic zone size at 196 N load (which was deter-
mined as the radial distance of the outermost shear band) with the
annealing time.
FIG. 5. Variation in the interband spacing as a function of (a) the distance from the tip of the hardness impression, and (b) the annealing time.
The data in (b) were taken from the regime for the distance of 50–150 mm in (a).
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interesting clues for the governing deformation mecha-nism. For the purpose of this estimation, the followingdetailed analysis of the discrete displacements was con-ducted. The total displacement (Dhtot) at a given loadduring the loading part of nanoindentation can be parti-tioned, approximately, into three components: (i) dis-placement associated with the elastic deformation of thesolid, Dhe, (ii) displacement associated with homo-geneous plasticity, Dhcon, and (iii) displacement asso-ciated with discrete plastic deformation, Dhdis. Since theelastic portion of deformation can be expected to berecovered during unloading, Dhe can be estimated usingthe unloading part of the P-h curve [see Fig. 7(a)]. Thecontribution of discrete plasticity to the total plasticity(Dhtot � Dhe) can be assessed by examining the discreteplasticity ratio, defined as f =Dhdis/(Dhcon + Dhdis).
41
Note that f = 1 implies that plasticity is exclusivelyinhomogeneous (through shear bands), whereas f = 0means it is truly homogeneous (as in indentation ofmetallic polycrystalline materials). Accordingly, as theratio f increases, the portion of shear-band-mediatedplasticity increases and that of STZ-mediated plasticitydecreases.
Variation of f with h is plotted in Fig. 7(b) for theas-cast and annealed samples. For the as-cast sample, fappears to be independent of h and only the pop-in sizeincreases with displacement. This implies that the con-tributions of homogeneous and heterogeneous plasticity(STZ- and shear band-mediated plasticity, respectively)remain approximately constant, with the heterogeneousplasticity contribution being dominant (�80%). For theannealed samples, however, not only the pop-in size butalso the discrete plasticity ratio increases with the inden-tation depth. For h below �1200 nm, the f values of theannealed samples are smaller, whereas above a 1200 nmdepth of penetration, both the as-cast and annealed
samples exhibit similar f values. This implies that STZsand shear bands contribute equally at the initial stages ofplasticity in annealed samples, and only at later stages ofdeformation do shear bands start to dominate the overallplasticity.
C. Hardness of subsurface deformation zone
In order to analyze the mechanical response of thedeformed region in Fig. 3, nanoindentation experi-ments were performed with a Berkovich indenter onthe deformed area after gentle polishing. The P-hcurves obtained from the undeformed and deformedregions in the as-cast and annealed samples were com-pared in Fig. 8. For all the samples, there is a cleardifference in P-h curves and thus in hardness value.Note that the nanoindentation hardness values werecalculated according to the Oliver-Pharr method.27,28
Since this method cannot take the pileup (typicallyobserved around the hardness impression of BMGs)into consideration, the hardness value reported here isan overestimate.The nanohardness data obtained from the deformed
region at two different maximum loads (98 N and196 N) and those obtained from the undeformed regions(i.e., far away from the subsurface shear banded regions)
FIG. 7. Calculation of discret plasticity ratio. (a) How to measure a
discrete portion of plastic deformation (Dhdis), a continuous portion of
plastic deformation (Dhcon), and an elastic portion of deformation
(Dhe). (b) Plot of discrete plasticity ratio, f, versus depth of penetra-
tion, h.
FIG. 6. Typical load-displacement curves obtained from nanoinden-
tation with a cube-corner indenter at a constant loading rate of
0.1 mN/s.
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are plotted as cumulative probability distributions inFig. 9. Here, Figs. 9(a), 9(b), and 9(c) correspond to theas-cast, 20 min annealed, and 90 min annealed samples,respectively. The cumulative probability of the ithranked (in ascending order) data point is calculated as(i � 0.5)/N where N is the size of the data set.
The following points are noteworthy. (i) For all thethree different conditions, the hardness values of thedeformed region are smaller than the corresponding un-deformed hardness data. The softening tendency is ingood agreement with recent experimental work on thesame BMG as used here44,45 which report that the hard-ness of the compressed samples decreases with increas-ing compressive plastic strain. ”(ii) The hardnessdistributions, obtained from the deformed region, areindependent of the maximum load applied (98 N or196 N) and thus are also independent of the inden-tation strain to generate the plastic zone. (iii) Withannealing, the mean hardness of both deformed and
undeformed regions increase. The last point is illustratedin Fig. 10, wherein the average hardness values areplotted as a function of the annealing time. Although itappears that the undeformed and deformed hard-ness data diverge with annealing time, the relative dif-ference [(Hundeformed � Hdeformed)/Hundeformed] actuallydecreases, but only marginally (from �15.5% to 12.5%).
A key observation that can be made from the hardnessdistributions of Figs. 9 and 10 is the following. For allthe three different material conditions, the highest hard-ness that is measured from the deformed region is lowerthan the lowest hardness measured in the undeformedregion. This is significant for the following reason. Byvirtue of shear banding, which is inhomogeneous in na-ture, the deformed region is inhomogeneous. The area ofthe nanoindent is 10 � 10 mm2, whereas the interband
FIG. 8. Representative P-h curves and hardness obtained from the
undeformed and the deformed zone underneath the indent; (a) as-cast,
(b) 20-min-annealed, and (c) 90-min-annealed specimen.
FIG. 9. Cumulative distribution of hardness measured at undeformed
and deformed region; (a) as-cast, (b) 20-min-annealed, and (c) 90-
min-annealed specimen.
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spacing is several tens of mm, especially for the 98 Nload case and in the annealed samples. (Note that wecould not intentionally make a nanoindentation on theregion between the shear bands or on top of a shearband, because the shear band zone was gently polishedbefore nanoindentation.) This being the case, the likeli-hood of an indent falling in between two shear bands ishigh. For such indents that do not sample a shear band,the hardness should be at least equal to the lowest hard-ness measured on an undeformed sample. However, thisis not so, with no overlap between the deformed andundeformed distributions. Further, the hardness distribu-tions obtained from the deformed regions are as tight asthose obtained from the undeformed regions. (This canbe gauged by examining the variance, which is definedas the ratio of the standard deviation to the average valueof the distribution. See Table I for data.)
D. Pileup around the nanoindents
The characteristics of the material pileup aroundthe nanoindents were evaluated using the AFM tounderstand the differences in the deformation behavior,if any, of the BMG in different states. At the continuumlevel, the plasticity in BMGs is volume conserving (forall practical purposes, as the dilatational component issmall) and hence can be treated as incompressible. Dueto the incompressibility of the material, the materialremoved from the indented volume can pile up aroundthe indentation.46 Figure 11 shows the representativeAFM images obtained from the undeformed as well asthe deformed regions. In these images, the high heightcontrast around the impression is due to the materialpileup. In the as-cast as well as annealed conditions,
AFM images obtained from the undeformed region ex-hibit a relatively larger amount of pileup than from thedeformed region. Tang et al.,47 who made nanoindenta-tions on the free surface around spherical indentationimpression, make a similar observation. It is noteworthythat, in Berkovich indentation, the shear bands are most-ly captured underneath the indenter and not extendedonto the surface, which is different from the case ofsharper cube-corner indentation. Thus, the observedpileup after Berkovich indentation is not the shear bandsthemselves.42,48
The influence of the sub-Tg annealing on the pileupphenomenon is clearly seen in Fig. 11; for both thesoftened and the undeformed region, a larger amount ofpileup is observed in the longer-annealed sample. Onemight imagine that the reduced amount of free volumeplays an important role in the increase in pileup in theannealed sample. Since the structurally relaxed samplehas a smaller amount of free volume than the as-castsample, the reduced free volume can reduce the abilityto accommodate plastic deformation induced by nano-indentation. This might result in a higher pileup in theannealed sample than in the as-cast sample.Based on the above results, it is instructive to clarify
that the development of material pileup has an oppositetrend on the activity of shear banding behavior, which isoften misunderstood. In Berkovich indentation of BMG,the shear bands (which occur mostly underneath theindenter)42,48 accommodate the inhomogeneous subsur-face deformation. Accordingly, the higher activity of theshear banding (arising from the larger amount of freevolume) can lead to a larger plastic deformation and thusa smaller amount of pileup. A comparison of pileupbehavior and serrated flow behavior (known to be asso-ciated with shear banding) observed in this work sup-ports this hypothesis. From Figs. 6 and 11, it is obviousthat less pronounced serrations in the P-h curve (possi-bly for the lower activity of shear banding) correspond tothe larger amount of pileup.The analogy between the AFM results and the obser-
vation by Jiang et al.,21 who conducted nanoinden-tation experiments on a Al-based metallic glass ribbonsample produced by melt-spinning, are noteworthy.They reported that the rolled (i.e., deformed andthus possibly softened) sample shows smaller pileup
FIG. 10. Change in hardness and its relative difference between de-
formed and undeformed region as a function of annealing time.
TABLE I. Summary of the nanoindentation test results performed on
deformed and undeformed regions of various samples.
Condition
Undeformed region
hardness (GPa)
Deformed region
hardness (GPa)
Relative
difference (%)
As-cast 7.71 � 0.36 (4.1%) 6.50� 0.27 (4.7%) 15.7
20 min
annealed
9.22 � 0.26 (4.2%) 8.02� 0.34 (2.8%) 13.0
90 min
annealed
10.60� 0.44 (2.2%) 9.25� 0.20 (4.1%) 12.7
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around the hardness impression than in the as-spun(i.e., undeformed) sample and the rolled/annealed sam-ple exhibits a much higher pileup than in the as-spunsample and the rolled sample. Though they did not pro-vide the AFM analysis for the annealed/rolled sample(which can be analogous to the softened region in theannealed sample in this work), it is obvious from theirwork21 that the annealing induces a significant pileup.
IV. DISCUSSION
Now we return to the questions that we posed in theintroduction and discuss the experimental results of thisstudy in their context. In this study, the BMG samplesexperience free volume changes due to two reasons; oneis due to the structural relaxation by sub-Tg annealing ofthe as-cast sample that leads to free volume reductionthrough annihilation, and the other is the increased freevolume due to plastic deformation. While the formerresults in enhanced resistance to plastic flow, the latterresults in flow softening. Spaepen5 suggested an equa-tion for the amount of free volume created during defor-mation:
Dþu ¼ gu
uf
2kT
Scosh
tO2kT
!� 1
" #Nn
� exp �DGm
kT
!exp � gu
uf
!;
ð2Þ
where g is a geometric factor between 1 and 1/2, u* isthe atomic volume, uf is the average free volume of an
atom, k is Boltzmann’s constant, T is the absolute tem-perature, S is the elastic distortion energy, t is the shearstress, O is the atomic volume, N is the total number ofthe atoms, n is the frequency, and DGm is the activationenergy of motion. In this equation, if u*/uf (which isalways smaller than unity for an atom to be on a poten-tial jump site) decreases, (gu*/uf) � exp(�gu*/uf) getsalgebraically smaller. As per this relation, relativelymore free volume can be created during deformation ina sample that has a smaller initial free volume uf to startwith. In the present work, due to the reduction in freevolume by annealing, the sample annealed for 90 minhas the least amount of initial free volume among thethree samples [see Fig. 2(b)]. Therefore, one might ex-pect that the relative amount of the excess free volumeduring spherical indentation is the largest for the 90-min-annealed sample, which can result in more pronouncedstrain softening. However, the results obtained in thepresent work (Fig. 10) suggest that the degree of strainsoftening (i.e., the difference in relative hardness be-tween the deformed region and the undeformed region)is almost independent of the initial free volume (�15.5%for the as-cast sample and �12.5% for the 90-min-annealed sample).
Very recently, Xie and George45 analyzed the degree ofstrain softening measured by compression test on the as-cast and annealed BMG having the same composition asexamined here. Calculations using the Vickers hardnessdata in Fig. 2(b) of their work45 yield the relative hard-ness difference [(Hundeformed � Hdeformed (at strain = 0.3)/Hundeformed] of �4.5 and �1% for the as-cast sample and
FIG. 11. AFM images of hardness impression produced during Berkovich indentation at Pmax = 50 mN and (dh/dt)h-1 = 0.5/s: [(a)–(c)] the
softened and the [(d)–(f)] undeformed zone; (a), (d) as-cast sample, (b), (e), the sample annealed for 20 min, and (c), (f) for 90 min.
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the sample which was first deformed and thenannealed at 633 K for 120 min, respectively. Thissmall difference (�3.5% between the as-cast and theannealed sample) also indicates that the extent of soft-ening is not seriously affected by the initial free vol-ume. The reason for the difference in decreasingamount between this work and the work of Xie andGeorge45 (i.e., for as-cast sample, �15.5% versus�4.5%) might be related with different loading sys-tem, that is, the uniaxial stress state in compressionversus the complex triaxial stress state in indentation.Also, the different range of applied plastic strain couldconceivably be due to another reason. In the work ofXie and George,45 the minimum plastic strain of thedeformed sample is 0.3, which is much higher thanthat reached here during spherical indentation (ofwhich the representative plastic strain is often esti-mated as 0.2a/R where a and R are the contact radiusand the radius of the indenter tip)49; for example, inthe case of the as-cast sample, approximately 0.035and 0.049 for Pmax = 98 N and 196 N, respectively.In their work,45 no information was given for sucha small plastic strain.
Another important question that we posed earlier is“does the number density of shear bands determinethe extent of softening?” Observations made in thisstudy imply that the flow stress of a metallic glassthat has undergone plastic deformation is location in-dependent, that is, the shear band has the same flowstress as that in between the bands. The fact thatnanohardness data obtained at two different preloads(of 98 N and 196 N) fall on top of each supports thehypothesis. This is because the density of shear bandsin these two cases is significantly different. If shearband number density were important, the plastic re-gime generated using the 196 N load should be rela-tively softer vis-a-vis that generated using the 98 Nload, as the former has a higher number of bands.However, this was not the case. This observation ledus to a possible answer to the above question, that is,the degree of softening is not seriously affected by theshear band density.
V. SUMMARY
In this study, the role of free volume in inhomoge-neous plastic flow of the BMG was systematically ana-lyzed by performing macroscopic spherical indentationand nanoindentation on the surface and subsurface of theas-cast and annealed samples. The primary results of thisinvestigation are:(1) The annealing treatment applied in this work did
not induce crystallization but structural relaxation, and alarge amount of free volume was annihilated during thestructural relaxation process.
(2) While the structural relaxation annealingenhances hardness, both the subsurface shear bandnumber density and the plastic zone size decrease withannealing time.(3) The serrations in the nanoindentation load-
displacement curves become smoother with structuralrelaxation, which could be analyzed with the discreteplasticity ratio in terms of the BMG’s governing defor-mation mechanism.(4) Regardless of the annealing condition, the nano-
indentation hardness of the deformed regions is�12–15% lower than the undeformed region, implyingthat the prior free volume only changes the yield stress(or hardness) but not the relative flow stress (or theextent of strain softening).(5) Statistical distributions of the nanoindentation
hardness obtained from deformed and undeformedregions have no overlap, suggesting that shear bandnumber density has no influence on the plastic charac-teristics of the deformed region.(6) AFM images obtained from the undeformed re-
gion exhibit a relatively larger amount of material pileuparound the nanohardness impression than that fromthe deformed region. Also, for both the softened andthe undeformed region, a larger amount of pileup is ob-served in the longer-annealed sample. Both phenomenacould be analyzed in terms of free volume change and itsrole in plastic deformation.Collectively, the important conclusion that can be
drawn from this work is that the initial amount offree volume, while important in controlling plasticdeformation (or strength/hardness) of BMGs, doesnot influence the extent of strain softening duringdeformation.
ACKNOWLEDGMENTS
This research was supported by the Korea ResearchFoundation (KRF) grant funded by the Korean Govern-ment, MEST (Grant No. KRF-2006-331-D00273), andpartially by the Korea Science and Engineering Founda-tion (KOSEF) grant funded by MEST (Grant No. R01-2008-000-20778-0). One of the authors, U.R., acknowl-edges the financial support received for this work fromthe Department of Science and Technology, Governmentof India through a Swarna Jayanthi Fellowship. Theauthors thank Dr. H. Bei (at Oak Ridge National Labora-tory) for providing the samples, and Mr. B-W. Choi(at Hanyang University) for his assistance in calculatingthe discrete plasticity ratio.
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