Ideal shear strain of metals and ceramics

Transcription

1 PHYSICAL REVIEW B 70, (2004) Ideal shea stain of metals and ceamics Shigenobu Ogata, 1,2,3 Ju Li, 4 Naoto Hiosaki, 5 Yoji Shibutani, 2,3 and Sidney Yip 1,6, * 1 Depatment of Nuclea Engineeing, Massachusetts Institute of Technology, Cambidge, Massachusetts 02139, USA 2 Handai Fontie Reseach Cente, Osaka Univesity, Osaka , Japan 3 Depatment of Mechanical Engineeing and Systems, Osaka Univesity, Osaka , Japan 4 Depatment of Mateials Science and Engineeing, Ohio State Univesity, Columbus, Ohio 43210, USA 5 Advanced Mateials Laboatoy, National Institute fo Mateials Science, Ibaaki, , Japan 6 Depatment of Mateials Science and Engineeing, Massachusetts Institute of Technology, Cambidge, Massachusetts 02139, USA (Received 29 Mach 2004; evised manuscipt eceived 18 May 2004; published 15 Septembe 2004) Using density functional theoy we analyze the stess-stain esponses of 22 simple metals and ceamics to detemine the maximum shea stain a homogeneous cystal can withstand, a popety fo which we suggest the name sheaability. A sheaability gap is found between metals and covalent ceamics. Sheaability of metals futhe coelates with the degee of valence chage localization and diectional bonding. Depending on the defomation constaints, ionic solids may possess even lage sheaability than covalent solids. The Fenkel model of ideal shea stength woks well fo both metals and ceamics when sheaability is used in the scaling. DOI: /PhysRevB PACS numbe(s): x, Jj The ductility of solids is contolled by the enegy needed to beak a bond by shea compaed to that by tension. 1 7 It is chaacteistic of ceamics to have a lage atio of shea to bulk moduli; howeve, little is known about the ange of shea defomation in solids with diffeent types of bonding. The maximum shea and tensile distotions that chemical bonding can withstand ae paticulaly impotant fo defects, e.g., dislocation coes and cack tips. 6,7 A fist step towad bette undestanding begins with two aspects of affine defomation of pefect cystals. One is the elastic constant descibing the linea esponse of the lattice to small stain, and the othe is a fundamental chaacteization of the lage-stain nonlinea esponse While use of the fome in scaling elations is almost univesal in defect mechanics, the question of whethe the latte also factos into micostuctuecontolling quantities such as the intinsic stacking fault enegy has been examined only ecently. 12 Hee we apply density functional theoy (DFT) to compute the sheaability and tensibility of simple metals and ceamics, defined by the maximum shea and tensile stains at which a pefect cystal unde affine defomation becomes unstable, to bing out the fundamental connection between citical mechanical esponse of a solid and the undelying electonic stuctue such as edistibution of valence chage density. To be pecise, we define sheaability as s m ag max s, whee s is the esolved shea stess and s is the engineeing shea stain in a specified slip system. Similaly, tensibility is taken to be t m ag max P 1+t V 0, whee P V is the pessuevolume elation and V 0 is the equilibium volume, P V 0 =0. We have studied the following metals and ceamics using the Vienna Ab-Initio Simulation Package 13 : FCC Ag,Cu,Au, feomagnetic (FM) and paamagnetic NM Ni,Al; BCC W,Mo,Fe FM ; HCP Mg,Ti,Zn; L1 0 TiAl, D0 19 Ti 3 Al; diamond cubic C,Si; SiC,, Si 3 N 4 ; B1 NaCl, MgO, KB, CaO. The exchange-coelation density functionals adopted ae Pedew-Wang genealized gadient appoximation (GGA) fo metals except Au and Ag, and Cepeley-Alde local density appoximation (LDA) fo the othes. Ultasoft (US) pseudopotential is used in most cases, but fo difficult systems we switch to the pojecto augmented-wave (PAW) method. 14 Billouin zone (BZ) k-point sampling is pefomed using the Monkhost-Pack algoithm. Fo metals, BZ integation follows the Methfessel- Paxton scheme 15 with the smeaing width chosen so the TS tem is less than 0.5 mev/atom. Fo nonmetallic systems, the tetahedon method with Blöchl coections 16 is used. Incemental affine shea stains ae imposed on each cystal along expeimentally detemined common slip systems to obtain the coesponding unelaxed and elaxed enegies and stesses, defined espectively by the conditions, ij =0 except s x/d 0 with d 0 being the inteplana sepaation and x taken along the Buges vecto, and ij =0 except fo the esolved shea stess. Fo, Si 3 N 4, the common slip systems ae unknown expeimentally. We theefoe calculate six systems fo each phase, and take the one that has the lowest ideal shea stength. 17 In Table I, the equilibium lattice constants a 0 c 0 obtained fom enegy minimization, with attention to enegy cutoff E cut and k-sampling convegence, ae compaed with expeimental esults. In Table II, the calculated elaxed and unelaxed shea moduli G,G u in the specified slip systems ae compaed with analytical values computed fom expeimental elastic constants. The esolved moduli ae calculated using fine meshes s=0.5% 1% along the shea path, wheeas coase meshes s=1% 5% ae used to intepolate the s cuves. Affine stess components ae elaxed to within a convegence toleance 0.05 GPa, and in cystals with intenal degees of feedom, the foce on each atom is elaxed to less than 0.01 ev/å. The elaxed ideal shea stess m nomalized by G and the sheaability s m fo diffeent mateials ae plotted togethe in Fig. 1(a). Fo simplicity we display only esults in the expeimentally detemined pimay slip system, except fo /2004/70(10)/104104(7)/$ The Ameican Physical Society

2 OGATA et al. PHYSICAL REVIEW B 70, (2004) TABLE I. Equilibium popeties: Calculation vs expeiment. a 0, c 0 Å Expt. Elastic const GPa Mateial # atoms Method # k-points E cut ev Calc. Expt. C 13 C 33 C 11 C 12 C 44 C 2 US-LDA d e Si 2 US-LDA e e -SiC 2 US-LDA e k -Si 3 N 4 28 US-LDA , ,5.591 f -Si 3 N 4 14 US-LDA , , g l NaCl 8 US-LDA e e KB 8 US-LDA e e MgO 8 US-LDA e e CaO 8 US-LDA e e Mo 1 US-GGA e e W 1 US-GGA e e Fe a 1 PAW-GGA h e Ti c 2 PAW-GGA , , e m Mg 2 PAW-GGA , , e e Zn 2 PAW-GGA , , e e TiAl 4 US-GGA , ,4.068 i i Ti 3 Al 8 US-GGA , ,4.649 j j Al 6 US-GGA e e Ni a 1 US-GGA e e Ni b 1 US-GGA Ag 1 US-LDA e e Au 1 US-LDA e e Cu 6 US-GGA e e a Feomagnetic. b Paamagnetic. c p-valence. d Refeence 23. e Refeence 24. f Refeence 27. g Refeence 28. h Refeence 30. i Refeence 31. j Refeence 32. k Refeence 25. l Refeence 29. m Refeence 26. BCC metals whee all thee slip systems ae equally likely, 8,18 in which case we use the one that gives the minimum m. Fo Ni we plot only the FM case. The coesponding unelaxed esults ae shown in Fig. 1(b). Note that elaxation has a paticulaly ponounced effect in ionic ceamics. Fom these esults we see gaps in the distibutions of s m and s u m between the metals and the covalent solids. Such gaps also may be seen by compaing esults of pevious woks fo elastic shea instability of metals 8,12,18 and covalent solids. 17,19 Moeove, among the metals the noble metals Au, Ag, Cu and the moe diectionally bonded Al and BCC Mo,W,Fe FM ae at opposite sides of the distibutions. This suggests that diectional bonding allows fo longe-ange shea distotion of the bonds befoe peak esistance is attained, which one can ationalize by obseving that the geate the covalency, the moe valence chage will concentate in non-nuclea-centeed egions, 20,21 e.g., bond centes and othe high-symmety intestices, as can be veified by an examination of the chage density isosuface plots (see Fig. 2, and also Supplementay Mateial 22 ). These localized chage pockets would equie cetain spatial aangement among them fo the total enegy to be well-minimized. In contast, if the valence chage density in the intestices is completely delocalized, then thee would be no such constaints and the enegy baie to shea would come mainly fom a misfit-volume effect. In FCC Cu and Ag one can see that when the local intestice volumes completely ecove thei equilibium values at the intinsic stacking fault, the enegy penalities ae vey low, despite the wong bond angles. 12 Consequently a athe geneal intepetation of ou esults is that, so fa as the eaangement of chage density in esponse to mechanical defomation is concened, bondangle dependence bings about geometic constaints on the atomic configuations above and beyond the volumetic constaints. Fom the standpoint of enegy landscape, the stable attactive basin of the gound state is steepe and wide in the shea diection because many low-enegy metastable states ae eliminated o geatly elevated by the exta constaints in configuational space

3 IDEAL SHEAR STRAIN OF METALS AND CERAMICS PHYSICAL REVIEW B 70, (2004) TABLE II. Shea moduli G (elaxed), G u (unelaxed), ideal shea stains s m (unelaxed) in common slip systems. (elaxed), s m u (unelaxed) and stesses m (elaxed), m u Expt. GPa a G/B GPa a Calc. GPa Relaxed Unelaxed b Mateial Slip system G G u G /B G u /B G G u s m m GPa m /G u s m u m GPa m u /G u C Si SiC Si 3 N Si 3 N NaCl KB MgO CaO Mo Mo Mo W W W Fe c Fe c Fe c Ti e Mg Zn TiAl Ti 3 Al Al Ni c Ni d Ag Au Cu a Computed analytically fom expt. elastic constants of Table I. b Subsidiay stess components ae unelaxed, but intenal degees of feedom ae elaxed. c Feomagnetic. d Paamagnetic. e p-valence. In the elaxed s m, m distibutions [Fig. 1(a)], ionic ceamics lie midway between diectionally-bonded metals and covalent solids, while and Si 3 N 4, being moe ionic than SiC, ae also in this ange. 17 Howeve in unelaxed shea, these solids manifest abnomally lage ideal shea stains s u m and stesses u m, which ae attibuted to the bae Coulomb epulsion between like-chage ions as in a simple Madelung sum model. In an atomic envionment like a cack tip, the suounding medium would not allow fo eithe fully elaxed o fully unelaxed local shea. This implies that ionic mateials could be eithe less o much moe bittle than covalent mateials, depending on the subsidiay defomation constaints pesent, a situation that is analogous to the distinction of plane-stess vs plane-stain loading conditions in the factue of metals. Anothe notewothy featue of Fig. 1 is the appoximately univesal linea scaling between s m and m /G acoss a ange of cystal stuctues, natue of bonding, and slip systems. The oiginal Fenkel mode, 1 containing a single paamete G, is well-known and widely used 3 5,33 in an empiical fashion. It

4 OGATA et al. PHYSICAL REVIEW B 70, (2004) FIG. 1. (Colo) DFT calculation esults of 22 mateials (metallic: blue, ionic: geen, covalent: ed). (a) Relaxed, and (b) unelaxed ideal shea stesses (nomalized) and shea stains. The solid line indicates a unit slope, while the dashed line coesponds to a slope of 2/. The ionic solids ae fa out of ange in (b) and ae shown in the inset. has been pointed out, based on physical intuition, that a moe ealistic desciption is to teat the peak position of s as an adjustable paamete 6,7 athe than as a fixed value at b/4d 0. What we have shown hee, on the basis of ab initio esults, is that a two-paamete epesentation = 2Gs m s sin 0 s s m, m = 2Gs m 1 2s m, with the shea modulus G and the sheaability s m as fundamental mateials paametes, povides a satisfactoy desciption of simple metals and ceamics. As can be seen in Fig. 1 the slope of 2/, implied by ou poposed extension of the Fenkel model, Eq. (1), indeed epesents the data well. Looking at this coelation in anothe way, we suggest that the fundamental constitutive behavio fo shea defomation can be captued in a maste cuve in tems of nomalized stess /Gs m and stain s s/s m, as shown in Fig. 3. In this escaling all cuves have initial unit slope and each maximum at s =1. The behavio labeled as Fenkel (enomalized) eflects a univesal shea-softening esponse, fo s s m. It is woth emphasizing that this new, enomalized Fenkel model is the shea countepat to the Univesal Binding Enegy Relation 2 which also has two paametes and has been quantitatively checked against ab initio calculations

5 IDEAL SHEAR STRAIN OF METALS AND CERAMICS PHYSICAL REVIEW B 70, (2004) TABLE III. Ideal volumetic tensile stain t m and stess P m. B is the bulk modulus computed fom DFT. Mateial B GPa t m P m GPa P m /B FIG. 2. (Colo) Valence chage density x isosuface plots of (a) FCC Ag, (b) FCC Al, (c) BCC Mo, (d) diamond cubic Si, at thei stess-fee states. 0 is the atomic volume. Taken togethe they allow mateials design and pefomance citeia to be fomulated in which tensile and shea dissipation modes compete. 5,7,33 35 Fo example, the bittleness paamete of Rice 5 that compaes the unstable stacking enegy us to the suface enegy s, may be vey cudely estimated as C Si SiC Si 3 N Si 3 N NaCl KB MgO CaO Mo W Fe a Ti b Mg Zn TiAl Ti 3 Al Al Ni a Ag Au Cu a Feomagnetic. b p-valence. = us Gs 2 m G 2 s s Bt m B 2 m / t 2 m 2 by scaling aguments. G/B (see Table II) is accessible expeimentally and has been used as a pefomance pedicto in FIG. 3. (Colo) Relaxed shea stess-stain cuves of 22 mateials, escaled such that all have unit slope initially and each maximum at 1. The enomalized Fenkel model Eq. (1) is shown fo compaison

6 OGATA et al. PHYSICAL REVIEW B 70, (2004) FIG. 4. (Colo) Ideal tensile stengths and tensibilities (volumetic) of 22 mateials. Note the naowe ange of data compaed to Fig. 1. alloy design, fo instance to pedict the best compomise between ductility and coosion esistance in stainless steels. 35 But s m and t m, while easy to obtain in ab initio calculations, 10 ae unavailable expeimentally and theefoe have neve been used in a pactical manne, despite having been theoetically established to be impotant. 6,7 Ou esults, along with othe DFT calculations, 8 12,17 19 indicate that a wide gap in exists between metals and ceamics because G/B and s m ae govened not only by the cystal stuctue, but also by the natue of bonding (e.g., s m =0.105 in FCC Au vs s m =0.200 in FCC Al) and the loading condition (e.g., s m =0.221 vs s m u =0.658 in B1 NaCl). On the othe hand, the elative vaiation of t m is less sensitive than s m, and it has no spectal gap (see Table III and Fig. 4). Besides contolling dislocation nucleation 5,7 via Eq. (2), s m also contol dislocation mobility. Based on the model of Foeman et al., 6 it can FIG. 5. (Colo) Schematic map of mateial ideal stengths, showing stability boundaies (dashed cuves) of the affine stain enegy landscapes fo metals and ceamics, beyond which bonds beak spontaneously in a pefect cystal (Ref. 34). s m and t m indicate the maximum stable engineeing shea and volumetic tensile stains of a pefect cystal, espectively. Ceamics tend to have lage sheaability s m than metals, with s m depending not only on the cystal stuctue (efeence value is b/4d 0 ), but also on the natue of bonding (e.g., in FCC Au vs in FCC Al), and the loading condition (e.g., in elaxed vs in unelaxed shea, in NaCl). In contast, the elative vaiation of tensibility t m of solids in moe limited ( in the 22 mateials studied). As in the Univesal Binding Enegy Relation (UBER) (Ref. 2) fo tension, a two-paamete model fo shea can be established by modifying the empiical fomula of Fenkel (Ref. 1). Based on Figs. 1, 3, and 4 and elastic moduli fo shea and tension, a simple scaling elation Eq. (2) suggests it is much moe difficult to beak bonds in ceamic solids in shea than in metals, elative to in tension. Electonic-stuctue featues such as valence chage localization and anisotopy ae found to coelate stongly with mechanical-esponse featues duing defomation [see Fig. 2 and Supplementay Mateial (Ref. 22)], such as the sheaability

7 IDEAL SHEAR STRAIN OF METALS AND CERAMICS PHYSICAL REVIEW B 70, (2004) be shown that inceasing the skewness s m / b/4d 0 of s while keeping G fixed leads to shaply inceased Peiels stesses. 3,4 Ou finding is summaized in Fig. 5. We believe thee is sufficient basis, in tems of theoetical fomulation 1,5 7 and ab initio popety data, 8 12,17 19 to popose that the sheaability s m of a pefect cystal is an impotant chaacte of the mateial. Like the elastic constants, it is a mateial-specific popety that can be detemined by fist-pinciples calculations and used in mateials design and selection at the macoscopic level. Unlike the elastic constants which petain to the solid at equilibium, it eflects quantitatively the electonic and atomic esponse of the solid at the point of bond beaking. S.O. thanks M. Kohyama, H. Kitagawa, Y. Koizumi, and M. Nishiwaki and acknowledges suppot by the Hattoi- Houkoukai fellowship. J.L. acknowledges suppot by Honda R&D Co., Ltd. and the OSU Tanspotation Reseach Endowment Pogam. S.Y. acknowledges suppot by Honda R&D, AFOSR, DARPA, NSF, and LLNL. *Electonic addess: syip@mit.edu 1 J. Fenkel, Z. Phys. 37, 572 (1926). 2 J. H. Rose, J. R. Smith, and J. Feante, Phys. Rev. B 28, 1835 (1983). 3 R. Peiels, Poc. Phys. Soc. London 52, 34(1940). 4 F. R. N. Nabao, Poc. Phys. Soc. London 59, 256 (1947). 5 J. R. Rice, J. Mech. Phys. Solids 40, 239 (1992). 6 A. J. Foeman, M. A. Jaswon, and J. K. Wood, Poc. Phys. Soc., London, Sect. A 64, 156 (1951). 7 G. Xu, A. S. Agon, and M. Otiz, Philos. Mag. A 72, 415 (1995). 8 C. R. Kenn, D. Roundy, J. W. Mois, J., and M. L. Cohen, Mate. Sci. Eng., A , 111 (2001). 9 J. W. Mois, J., D. M. Clattebuck, D. C. Chzan, C. R. Kenn, W. Luo, and M. L. Cohen, Mate. Sci. Foum 426, 4429 (2003). 10 M. Čeny, J. Pokluda, M. Šob, M. Fiák, and P. Šandea, Phys. Rev. B 67, (2003). 11 D. M. Clattebuck, C. R. Kenn, M. L. Cohen, and J. W. Mois, J., Phys. Rev. Lett. 91, (2003). 12 S. Ogata, J. Li, and S. Yip, Science 298, 807 (2002). 13 G. Kesse and J. Futhmülle, Phys. Rev. B 54, (1996). 14 P. R. Blöchl, Phys. Rev. B 50, (1994). 15 M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 (1989). 16 P. E. Blöchl, O. Jepsen, and O. K. Andesen, Phys. Rev. B 49, (1994). 17 S. Ogata, N. Hiosaki, C. Koce, and Y. Shibutani, J. Mate. Res. 18, 1168 (2003). 18 D. Roundy, C. R. Kenn, M. L. Cohen, and J. W. Mois, J., Philos. Mag. A 81, 1725 (2001). 19 D. Roundy and M. L. Cohen, Phys. Rev. B 64, (2001). 20 B. Silvi and C. Gatti, J. Phys. Chem. A 104, 947 (2000). 21 P. Moi-Sanchez, A. M. Pendas, and V. Luana, J. Am. Chem. Soc. 124, (2002) J. Donohue, The Stuctues of the Elements (Wiley, New Yok, 1974). 24 Landolt-Bönstein III/7a, 7b1, 14a, 29a (Spinge-Velag, Belin, ). 25 W. R. L. Lambecht, B. Segall, M. Mathfessel, and M. van Schilfgaade, Phys. Rev. B 44, 3685 (1991). 26 E. S. Fishe and C. J. Renken, Phys. Rev. 135, A482 (1964). 27 K. Kato, Z. Inoue, K. Kijima, I. Kawada, H. Tanaka, and T. Yamane, J. Am. Ceam. Soc. 58, 90(1975). 28 R. Gün, Acta Cystallog., Sect. B: Stuct. Cystallog. Cyst. Chem. B35, 800 (1979). 29 R. Vogelgesang, M. Gimsditch, and J. S. Wallace, Appl. Phys. Lett. 76, 982 (2000). 30 M. Acet, H. Zähes, E. F. Wassemann, and W. Peppehoff, Phys. Rev. B 49, 6012 (1994). 31 K. Tanaka, T. Ichitsubo, H. Inui, M. Yamaguchi, and M. Koiwa, Philos. Mag. Lett. 73, 71(1996). 32 K. Tanaka, K. Okamoto, H. Inui, Y. Minonishi, M. Yamaguchi, and M. Koiwa, Philos. Mag. A 73, 1475 (1996). 33 J. W. Kysa, J. Mech. Phys. Solids 51, 795 (2003). 34 J. Li, K. J. Van Vliet, T. Zhu, S. Yip, and S. Suesh, Natue (London) 418, 307 (2002). 35 L. Vitos, P. A. Kozhavyi, and B. Johansson, Nat. Mate. 2, 25 (2003)