Hall–Petch relationship in a nanotwinned nickel alloy
Leon L.Shaw,a,*Angel L.Ortiz b and Juan C.Villegas c
a
Department of Chemical,Materials and Biomolecular Engineering,University of Connecticut,Storrs,CT 06269,USA
b
Departamento de Ingenierı
´a Meca ´nica,Energe ´tica y de los Materiales Universidad de Extremadura,06071Badajoz,Spain c
Intel Corporation,Chandler,AZ 85226,USA
Received 5November 2007;revised 4January 2008;accepted 14January 2008
Available online 26January 2008
The Hall–Petch relationship in a nanotwinned alloy with absence of dislocation pile-ups is investigated for the first time.It is shown that,when the twin spacing is large (d >150nm),the hardness exhibits a d À1/2dependence.However,when the twin spacing is small (d <100nm),a d À1dependence results.These phenomena are interpreted based on dislocation-mediated mechanisms corroborated by the analysis of electron microscopy and X-ray diffractometry.Ó2008Acta Materialia Inc.Published by Elsevier Ltd.All rights reserved.
Keywords:Hardness;Plastic deformation;Twinning;Nanocrystalline microstructure;Hall–Petch relation集成灶品牌
Grain refinement has been a topic of intensive re-search for several decades.The driving force behind these enduring efforts is the enhancement of strength as the grain size decreases,as described by the empirical Hall–Petch (H–P)relationship [1,2]r y ¼r i þk y D À1=2
ð1Þ
where r y is the yield strength of a polycrystalline mate-rial,D is the average grain diameter,r i is the overall resistance of lattice to dislocation movement and k y is the H–P slope measuring the relative strengthening con-tribution of grain boundaries (GBs).Eq.(1)has been found to be applicable to a wide range of coarse-grained materials (D P $1l m),including applications to the flow stress at a give
n strain and the hardness of the material [3],the dislocation cells of heavily deformed materials [4],the materials with several levels of he co-presence of low-angle cell bound-aries and high-angle cell block boundaries [5],and materials with microsized twins [6,7].
Recently,the applicability of the H–P relation to ultrafine-grained materials (100nm <D <1l m)[8,9]and nanograined materials (D <100nm)[10–12]has been studied extensively because of their potentials to offer substantial improvements in mechanical properties over coarse-grained materials.Many of the studies in this area have been summarized in a recent review article
[13],where a comprehensive list of references can be found.However,studies on the H–P relation in nano-twinned materials are extremely scarce,with only one investigation on single-phase nanotwinned materials re-ported in the open literature [14].In that study,Eq.(1)is found to be valid for electrodeposited Cu with twin thickness as small as 13nm.Furthermore,the Hall–Petch slope is nearly the same as that determined from coarse-grained Cu,suggesting that the strengthening ef-fect of twin boundaries (TBs)is analogous to that of conventional GBs even in the nanometer scale [14].The applicability of the H–P relation to the nanotwin-ned Cu has been explained by dislocation pile-ups against TBs [14].However,it is well known that many materials do not exhibit dislocation pile-ups [15],and their H–P relations have been explained by the activa-tion of GB dislocation sources [1
6]or other alternative mechanisms [17–19].In this study we have investigated,for the first time,the dependence of the hardness of nanotwinned materials on the twin spacing with absence of dislocation pile-ups.The absence of dislocation pile-ups can be attained from low stacking fault energy materials when the twin thickness is too small to support dislocation pile-ups,as revealed in this study.
A nickel-base HASTELLOY C-2000Òalloy 1with a low stacking fault energy (1.22mJ m À2)[20]was chosen for this study in order to produce a gradient of
1359-6462/$-see front matter Ó2008Acta Materialia Inc.Published by Elsevier Ltd.All rights reserved.doi:10.1016/j.scriptamat.2008.01.025
*Corresponding author.Tel.:+18604862592;fax:+18604864745;e-mail:leon.shaw@uconn.edu
1
HASTELLOY and C-2000are registered trademarks of Haynes International,Inc.
Available online at www.sciencedirect
Scripta Materialia 58(2008)
951–954
www.elsevier/locate/scriptamat
high-density nanotwins in the sample.The C-2000alloy is a corrosion-resistance alloy with a single-phase,face-centered-cubic (fcc)structure and a nominal chemical composition (in wt.%)of 23Cr,16Mo,1.6Cu,0.01C,0.08Si and balance Ni.The as-received C-2000plates were in an annealed condition.These annealed plates were subjected to a surface severe plastic deformation (S 2PD)treatment to create a surface region containing nanograins followed by a gradient of high-density nanotwins with a continuous increase in the twin thick-ness as the position moves away from the treated surface of the plate [21].The S 2PD method used in this study,similar to the surface mechanical attrition treatment [22,23],entails impacting the surface of the plate with high-energy balls (e.g.WC/Co balls)repeatedly under an argon atmosphere for 30min [24,25].The detail of other experimental conditions for S 2PD can be found elsewhere [21].
The microscopy analysis (Fig.1)reveals that a micro-structural gradient has been produced via the S 2PD pro-cess with a gradual decrease in the deformation twin density as the position moves away from the impacted surface.Parallel nanotwins (3–8nm thick)with and without 60°unit dislocations in b
etween are present near the impacted surface (Fig.1b),whereas twin–twin inter-sections with high-density dislocation entanglement in between are present below the nanotwinned surface re-gion (Fig.1c).Further away from the impacted surface is the deformation region,with small plastic strains exhibiting dislocation emission from twin boundaries (Fig.1d).Based on the peak broadening analysis of X-ray diffraction (XRD)patterns,the crystallite size and dislocation density as functions of the position measured from the impacted surface have been quantified [21].Be-cause of the absence of dislocation cells in the S 2PD-pro-
萧蔷 走光cessed C-2000alloy,the crystallite size measured via XRD reflects the size of the substructures mainly sepa-rated by twin boundaries with only a small proportion of fault and grain boundaries,and thus can be approx-imately regarded as the twin spacing [21].Furthermore,it is found that the twin spacing decreases as the position becomes closer to the impacted surface.Figure 2shows the dislocation density and the average number of dislo-cations within each twinned (or untwinned)region as a function of the twin spacing.Note that although the dis-location density increases as the twin spacing becomes smaller (i.e.as the position becomes closer to the im-pacted surface),the number of dislocations per twinned (or untwinned)region actually decreases.This is in excellent agreement with the transmission electron microscopy (TEM)analysis,as revealed by comparing Figure 1b and c.
Figure 3presents the H–P relationship between the Vickers hardness,H V ,and the twin spacing,d ,revealing that the H V Àd À1/2relationship is not linear,but is concave towards the d À1/2axis.Furthermore,the trend shown by the nanoindentation hardness is identical to that exhibited by the Vickers hardness.However,the nanohardness data have allowed the valid measurement of the hardness at locations 10l m away from the im-pacted surface [25],and thus extended the evaluation of the effect of the twin spacing down to 30nm.We pro-pose that the non-linear H Àd À1/2relationship is due to the change in the deformation mechanism as the
twin
Figure 1.Microstructures of the C-2000alloy after S 2PD processing.(a)An overall view of the cross-section from the impacted surface (indicated)to the nearly undeformed interior,showing a gradual increase in the deformation twin density (as indicated by the deformation marking density)when the position becomes closer to the impacted surface.(b)A Fourier-filtered lattice image near the impacted surface,showing the presence of parallel nanotwins with and without dislocations in between.(c)TEM bright-field image at a location $100l m from the impacted surface,showing the twin–twin intersection with dislocation entanglement in between.(d)TEM bright-field image from a location near the undeformed interior,showing emission of dislocations from a twin
boundary.
Figure 2.The dislocation density and the average number of disloca-tions per twinned (or untwinned)region as a function of the twin spacing.Note that when the twin spacing decreases from 750to 34nm,the position measured from the impacted surface changes from 500to 10l m,
respectively.
Figure 3.The hardness,H ,as a function of the inverse of the square root of the twin spacing,d À1/2.Note that the H Àd À1/2relationship is not linear,but is concave towards the d À1/2axis.
952L.L.Shaw et al./Scripta Materialia 58(2008)951–954
spacing becomes smaller.For large twin
d>$150nm),the hardness is mainly controlled by the increase in the internal stress due to the additional dislocation density which results from the presence of twin boundaries.For small twin d<$100nm),the hardness is dictated by the energy re-quired to spread and expand dislocations across the glide planes within the twinned or untwinned regions. The transition of the deformation mechanism occurs in the intermediate size of the twin $100nm<d<$150nm).
These proposed mechanisms are consistent with microstructural examinations.As shown in Figures1 and2,there are a relatively large number of dislocations within twinned and untwinned regions when the twin spacing is large.As such,dislocation glide across large twinned or untwinned regions would have to overcome the resistance resulting from interactions with disloca-tions.Since twin boundaries
can behave as a source for emitting dislocations,as revealed in this study (Fig.1d)and predicted theoretically recently[26],the dislocation density,q,is thus proportional to the reci-procal of the twin spacing,dÀ1[16].With this deforma-tion mechanism,the yield strength of the material will follow the Taylor relationship as well as the H–P rela-tionship[16],i.e.
r y¼r iþa MGb ffiffiffiq p¼r
i
þk y dÀ1=2ð2Þ
where a is the geometrical constant,M is the Taylor fac-
tor,G is the shear modulus,b is the Burgers vector,q is the dislocation density,d is the twin spacing,and r i and
k y have been defined in Eq.(1).Based on the hardness
data in the range of large twin spacings shown in Figure 3,the maximum H–P slope for the twinned C-2000with
d>$150nm,k H(max),is found to be0.696MPa m1/2,
which is larger than the H–P slope of coarse-grained pure Ni,k H=0.474MPa m1/2[4].Here k H is the H–P
slope referring to the Vickers hardness and H V is as-
sumed to follow the Tabor’s H V%3r y) [27,28]and thus k H=3k y.The higher H–P slope is attributed to solid solution strengthening in the C-2000
alloy.Thus,the similar H–P slopes indicate that the
strengthening effect of TBs is analogous to that of con-ventional GBs when the twin spacing is larger than 150nm for the C-2000alloy.The coefficient of a MGb in the Taylor relation can also be determined from the experimental data of H V vs.
ffiffiffiq p(Fig.4),and is found to be1.79Â10À8GPa m for the same range of the large twin spacings(d P150nm)used to calculate k H(max). This number is in good agreement with the theoretical prediction,which leads to a MGb=1.63Â10À8GPa m if one has chosen a=0.33[8],M=3.0[8], G=64.5GPa[20]and b=0.255Â10À9m[20].
The fact that the hardness data of the twinned C-2000
with the twin spacing larger than150nm can be de-
手机下载音乐scribed by both H–P and Taylor relations suggests that dislocations play a critical role in this twinned alloy when the twin spacing is larger than150nm.Further-more,the role of TBs can be regarded as‘‘increasing the dislocation density”since the Taylor relation can fully describe the hardness data without resorting to the twin spacing.However,such a situation is no longer true when the twin spacing is smaller than100nm, where a dÀ1dependence is present(Fig.5)and the coef-ficient of a MGb in the Taylor relation obtained from the experiment becomes much smaller than that predicted from the dislocation theory(Fig.4).
When the twin spacing is d<$100nm), the number of dislocations within each twinned and untwinned region is small(Figs.1b and2).As such, the resistance to plastic deformation and thus the hard-ness are not controlled by the resistance resulting from interactions with dislocations,but by the energy re-quired to spread and expand dislocations across the glide planes within the twinned and untwinned region. With this deformation mechanism,the yield strength and hardness should exhibit a linear relationship vs. the reciprocal of the dislocation cell size,as demon-strated by Langford and Co
hen[29,30],as well as re-viewed in a recent article[13].For the twinned C-2000 with a small twin spacing,the hardness should thus ex-hibit a dÀ1dependence.This is indeed the case observed in this study.As shown in Figure5,a dÀ1dependence is present for both the Vickers hardness and nanoindenta-tion hardness when the twin spacing is smaller than 100nm.Furthermore,the Taylor relation should not be present for the twinned C-2000with a small twin spacing because the hardness is not controlled by
the Figure4.The hardness,H,of the C-2000alloy with S2PD processing as a function of the square root of the dislocation density,
ffiffiffiq p.The slope of thefitting line for thefirst three data points(as indicated)is 5.37Â10À8GPa m,which leads to a MGb=1.79Â10À8GPa m.The twin spacing at the same location that gives the square root of the dislocation density shown is indicated on the top of the graph along the x
-axis.
Figure5.The hardness,H,as a function of the reciprocal of the twin spacing,dÀ1.Note that a linear relationship is present for both the Vickers hardness and nanoindentation hardness when d<100nm.
杨幂的421事件是一男还是多人L.L.Shaw et al./Scripta Materialia58(2008)951–954953
resistance resulting from interactions with dislocations. This trend is also confirmed by the experimental data, as shown in Figure4,where the Taylor relation works well for the twin spacing larger than150nm,but fails for the twin spacing smaller than100nm.Based on these phenomena,it can be concluded that when the twin spacing is smaller than100nm,the hardness is dic-tated by the energy required to spread and expand dislo-cations across the glide planes within the twinned and untwinned region.Furthermore,it is this dÀ1depen-dence that causes the bending of the linear HÀdÀ1/2 relation towards the dÀ1/2axis.
It should be mentioned that the phenomena disclosed above were obtained from the C-2000alloy with nanotwins generated via S2PD.Thus,there are three po-tential sources of errors that can alter the hardness in the surface region of the alloy.First,some impurities from the ball can diffuse into the sample.Second,there may be ultrafine-scale pores and minorflaws due to the se-vere impacting of th
e surface,a trace amount of which could drop the hardness[31].Third,some re-distribu-tion of solutes in the icrosegregation to grain boundaries)may be possible during S2PD.How-ever,the detailed analysis indicates that all of these potential sources of errors are unlikely to affect the result of the present study.Our previous XRD investiga-tion[21]indeed revealed the WC contamination of the impacted surface due to material transfer between balls and the C-2000plate.Nevertheless,the contamination becomes minute at locations10l m below the impacted surface and almost imperceptible at20l m below the im-pacted surface[21].Since the dÀ1dependence is observed from20to190l m below the impacted surface for the Vickers hardness and from10to190l m below the im-pacted surface for the nanoindentation hardness,it can rule out the possibility of the WC contamination leading to a dÀ1dependence.Similarly,extensive SEM examina-tion of the cross-sections like the one shown in Figure1a has never revealed the presence of ultrafine cracks and pores at locations5l m below the impacted surface [21,25,32],and therefore the second source of errors is also very unlikely.Finally,if diffusion of contaminants and microsegregation occurred at depths20l m below the impacted surface during S2PD processing,the inte-grated intensities of XRD peaks at these locations would have changed because of the alternation of the structure factors for all of the crystallographic planes, and thus the ratio of the integrated intensities of differ-ent XRD I111/I200,where I111and I200are the integrated intensities of{111}and{200}planes, respecti
vely)would also have changed at these locations, which is not observed in the XRD analysis[21].Thus, the third source of errors is also unlikely.
In short,the present study demonstrates that the strengthening effect of TBs is similar to that of conven-tional GBs.Furthermore,we have presented thefirst evidence that,when the twin spacing is
d>150nm),the resistance to plastic deformation is dictated by interactions with dislocations which results from the presence of TBs,and this deformation mecha-nism leads to a dÀ1/2dependence of the hardness.When the twin spacing is small(d<100nm),the resistance to plastic deformation is controlled by the energy required to spread and expand dislocations across the glide planes within the twinned and untwinned region,and this deformation mechanism leads to a dÀ1dependence of the hardness.The transition of the deformation mechanism occurs in the intermediate size of the twin spacing(100nm<d<150nm).
The authors acknowledge thefinancial support by National Science Foundation through Grant No. DMR-0207729.
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