2.)W.Horton ]r.,D.S.Powars,G.S.Gohn,Eds.,U.S.Geol. 17.D.S.Powars,U.S.GeoL Surv.Prof.Pap.1622 (2000) 33.The International Continental Scientific Drilling Program, Surv.Prof.Pap.1688 (2005). 18.C.W.Poag.The Chesopeake Invoder (Princeton Univ. the U.S.Geological Survey,and the NASA Science Mission 3.C.Koeberl,C.W.Poag,W.U.Reimold,D.Brandt, Press,Princeton,N].1999). Directorate provided funding for the drilling project. Science271,1263(1996). 19.E.R.McFarland,T.S.Bruce,U.S.Geol.Surv.Prof.Pop NSF and the Austrian Science Foundation provided 4.A.Deutsch.C.Koeberl.Meteorit.Plonet.Sai.41.689 (2006). 1688-K(2005). supplementary funding for the drill-site operations 5.G.S.Collins,K.Winnemann,Geology 33,925 (2005). 20.W.E.Sanford et ol.,Eos 85,369 (2004). DOSECC Inc.conducted the administrative and 6.G.S.Gohn et al.,Eos 87,349 (2006). 21.W.E.Sanford,J.Geochem.Explor.78-79,243(2003). operational management of the deep drilling project. 7.)W.Horton Jr.et al.,Geol.Soc.Am.Abstr.Prog.39. 22.W.E.Sanford,Geofluids 5,185 (2005). We thank the Buyrn family for use of their land as a 451,abstract167-5(2007). 23.See supporting material on Science Online. drilling site,the scientific and technical staff of the 8.)W.Horton Jr.,)N.Aleinikoff,M )Kunk,C.W.Naeser, 24.L Radford,L B.Cobb,R.L McCoy,U.S.DOE Rep. Chesapeake Bay Impact Structure Drilling Project for N.D.Naeser,U.S.GeoL Surv.Prof.Pap.1688 (2005). D0E/ET28373-1(1980). their many contributions,A.Gronstal (Open University. 9.P.B.Robertson,R.A.F.Grieve,in Impact and Explosion 25.F.T.Manheim,M.K.Horn,Southeast.Geol 9,215 UK)for the microbe enumeration data,L Edwards Crotering:Planetary and Terrestrial Implicotions, (1968). (USGS)for discussions of the post-impact geology,and D.)Roddy,R O.Pepin,R.B.Merrill,Eds.(Pergamon, 26.A.L.Gronstal et al.,Geol.Soc.Am.Abstr.Prog.39,316 G.Collins(Imperial College,UK)and K.Wunnemann New York,1977,pp.687-702. abstract116-22(2007). (Humboldt-University Berlin)for making available the 10.A.Wittmann,T.Kenkmann,L.Hecht,D.Stoffler, 27.R.]Parkes et al.,Noture 371,410 (1994). results of their numerical modeting. GeoL Soc.Am.Bull.119,1151 (2007). 28.5.D'Hondt et al.,Science 306,2216 (2004) 11.D.Stoffler et ol.,Meteorit.Planet.Sci.39,1035(2004). 29.R.]Parkes,B.A.Cragg,P.Wellsbury.Hydrogeol J.8, 12.T.Hayden et al.,Geology 36,327 (2008). 11(2000). Supporting Online Material 13.D.S.Powars,T.S.Bruce,U.5.Geol.Surv.Prof.Pap.1612 30.W.D.Grant,Philos.Trans.R.Soc.London Ser.B 359 www.sciencemag.org/cgi/content/full320/5884/1740/DC1 1249(2004). Materials and Methods 1999) 14.K.G.Miller,G.S.Mountain,Leg 150 Shipboard Party. 31.M.L.Malinconico,W.E.Sanford,]W.Horton Jr.. Figs.S1 to S10 Members of the New Jersey Coastal Plain Drilling Project, abstract for Conference on Large Meteorite Impacts and Table S1 Science271,1092(1996). References Planetary Evolution IV,Vredefort Dome,South Africa, 15.K.G.Miller et al.,Science 310,1293 (2005). abstract3068(2008). 4 April 2008:accepted 28 May 2008 16.]V.Browning et al,GeoL Soc.Am.Bull.118,657 (2006). 32.K.Kashefi,D.R.Lovley,Science 301,934 (2003). 10.1126/science.1158708 REPORT location glide paths and the investigation of vol- Dislocation Mean Free Paths umes that are representative of the bulk material (4).One can then derive a continuum formulation and Strain Hardening of Crystals on the basis of physically justified mechanisms and parameters,and this formulation is further integrated at the scale of a bulk crystal.Face- B.Devincre,1 T.Hoc,2 L.Kubin1* centered cubic (foc)crystals are taken as bench- mark materials because of their well-documented. Predicting the strain hardening properties of crystals constitutes a long-standing challenge for dislocation but rather complicated,stress/strain response. theory.The main difficulty resides in the integration of dislocation processes through a wide range of We started by considering the critical stresst time and length scales,up to macroscopic dimensions.In the present multiscale approach,dislocation for the activation of slip system i as a function of dynamics simulations are used to establish a dislocation-based continuum model incorporating discrete the dislocation densities p'stored (ie.,tempo- and intermittent aspects of plastic flow.This is performed through the modeling of a key quantity, rarily or permanently immobilized)in all slip sys- the mean free path of dislocations.The model is then integrated at the scale of bulk crystals,which temsj.This critical stress is given by a generalized allows for the detailed reproduction of the complex deformation curves of face-centered cubic crystals. Because of its predictive ability,the proposed framework has a large potential for further applications. Taylor relation ()of th omb where u is the shear modulus and b the modu- islocations are complex defects of crystal- translation of the crystal lattice.During plastic flow, lus of the Burgers vector.In fec crystals,the line materials,which have been investigated dislocations multiply and their mutual interactions symmetric tensor a contains six independent for more than 70 years.Their fundamental hinder their motion.As a consequence,a shear dimensionless coefficients,which account for the and economical importance arises from the number stress increase dt has to be imposed to produce a average strength of pair interactions between slip of properties they govem,such as the high strength shear strain increase dy.By definition,the ratio systems that result from short-and long-range of nanostructured materials,the reliability of semi- dildy is the strain hardening rate.Although dis- interactions.Their values were recently deter- conductor devices,the processing and service life location theory has successfully explained many as- mined from DD simulations(5). of structural materials,or the rheological proper- pects of the strength of crystalline solids,predicting For determining strain hardening,the key quan- ties of tectonic events in Earth's crust. strain hardening is "the most difficult remaining tity is the rate at which the critical stress evolves The irreversible,or plastic,deformation of crys- problem"(3).The present dislocation-based models with strain or,equivalently,the rate at which dis- tals results from the motion on crystallographic for strain hardening still have difficulties integrating locations accumulate under strain.For this purpose, planes of linear defects,the dislocations (/2). elementary dislocation properties into a continuum it is useful to define a dislocation mean free path L These defects carry an elementary amount of shear description of bulk crystals or polycrystals.As a which is the distance traveled by a dislocation seg- (the Burgers vector)that is usually the smallest consequence,current approaches cannot avoid ment of length before it is stored by interaction making use of extensive parameter fitting. with the microstructure.When the line moves by a Laboratoire d'Etude des Microstructures,Unite Mixte de Re- The present work takes advantage of three- distance d,it sweeps an area ldr and produces a cherche (UMR)104 CNRS,CNRS-Office National d'Etudes et dimensional dislocation dynamics (DD)simula- shear strain dy=bldd/V,where Vis the volume of de Recherches Aerospatiales (ONERA),20 Avenue de la Division tions (4-8)for averaging dislocation properties at the crystal.The stored density has then statistically Lederc,BP 72,92322 Chatillon Cedex,France.Laboratoire MSSMat,UMR 8579 CNRS,Ecole Centrale Paris,Grande Voie the intermediate scale of slip systems,which are increased by dp =(dL)V and the incremental des Vignes,92295 Chatenay-Malabry Cedex,France. ensembles of dislocations having the same Burgers storage rate is dp/dy=1/bL.This definition is only *To whom correspondence should be addressed.E-mail: vector and slip plane.The use of periodic boundary valid in differential form.as dislocation lines mul- ladislas.kubin@onera.fr conditions allows for the tayloring of large dis- tiply when they move.Following Kocks et al.and www.sciencemag.org SCIENCE VOL 320 27 JUNE 2008 1745
2. J. W. Horton Jr., D. S. Powars, G. S. Gohn, Eds., U.S. Geol. Surv. Prof. Pap. 1688 (2005). 3. C. Koeberl, C. W. Poag, W. U. Reimold, D. Brandt, Science 271, 1263 (1996). 4. A. Deutsch, C. Koeberl, Meteorit. Planet. Sci. 41, 689 (2006). 5. G. S. Collins, K. Wünnemann, Geology 33, 925 (2005). 6. G. S. Gohn et al., Eos 87, 349 (2006). 7. J. W. Horton Jr. et al., Geol. Soc. Am. Abstr. Prog. 39, 451, abstract 167-5 (2007). 8. J. W. Horton Jr., J. N. Aleinikoff, M. J. Kunk, C. W. Naeser, N. D. Naeser, U.S. Geol. Surv. Prof. Pap. 1688 (2005). 9. P. B. Robertson, R. A. F. Grieve, in Impact and Explosion Cratering: Planetary and Terrestrial Implications, D. J. Roddy, R. O. Pepin, R. B. Merrill, Eds. (Pergamon, New York, 1977), pp. 687–702. 10. A. Wittmann, T. Kenkmann, L. Hecht, D. Stöffler, Geol. Soc. Am. Bull. 119, 1151 (2007). 11. D. Stöffler et al., Meteorit. Planet. Sci. 39, 1035 (2004). 12. T. Hayden et al., Geology 36, 327 (2008). 13. D. S. Powars, T. S. Bruce, U.S. Geol. Surv. Prof. Pap. 1612 (1999). 14. K. G. Miller, G. S. Mountain, Leg 150 Shipboard Party, Members of the New Jersey Coastal Plain Drilling Project, Science 271, 1092 (1996). 15. K. G. Miller et al., Science 310, 1293 (2005). 16. J. V. Browning et al., Geol. Soc. Am. Bull. 118, 657 (2006). 17. D. S. Powars, U.S. Geol. Surv. Prof. Pap. 1622 (2000). 18. C. W. Poag, The Chesapeake Invader (Princeton Univ. Press, Princeton, NJ, 1999). 19. E. R. McFarland, T. S. Bruce, U.S. Geol. Surv. Prof. Pap. 1688-K (2005). 20. W. E. Sanford et al., Eos 85, 369 (2004). 21. W. E. Sanford, J. Geochem. Explor. 78–79, 243 (2003). 22. W. E. Sanford, Geofluids 5, 185 (2005). 23. See supporting material on Science Online. 24. L. Radford, L. B. Cobb, R. L. McCoy, U.S. DOE Rep. DOE/ET/28373-1 (1980). 25. F. T. Manheim, M. K. Horn, Southeast. Geol. 9, 215 (1968). 26. A. L. Gronstal et al., Geol. Soc. Am. Abstr. Prog. 39, 316, abstract 116-22 (2007). 27. R. J. Parkes et al., Nature 371, 410 (1994). 28. S. D’Hondt et al., Science 306, 2216 (2004). 29. R. J. Parkes, B. A. Cragg, P. Wellsbury, Hydrogeol. J. 8, 11 (2000). 30. W. D. Grant, Philos. Trans. R. Soc. London Ser. B 359, 1249 (2004). 31. M. L. Malinconico, W. E. Sanford, J. W. Horton Jr., abstract for Conference on Large Meteorite Impacts and Planetary Evolution IV, Vredefort Dome, South Africa, abstract 3068 (2008). 32. K. Kashefi, D. R. Lovley, Science 301, 934 (2003). 33. The International Continental Scientific Drilling Program, the U.S. Geological Survey, and the NASA Science Mission Directorate provided funding for the drilling project. NSF and the Austrian Science Foundation provided supplementary funding for the drill-site operations. DOSECC Inc. conducted the administrative and operational management of the deep drilling project. We thank the Buyrn family for use of their land as a drilling site, the scientific and technical staff of the Chesapeake Bay Impact Structure Drilling Project for their many contributions, A. Gronstal (Open University, UK) for the microbe enumeration data, L. Edwards (USGS) for discussions of the post-impact geology, and G. Collins (Imperial College, UK) and K. Wünnemann (Humboldt-University Berlin) for making available the results of their numerical modeling. Supporting Online Material www.sciencemag.org/cgi/content/full/320/5884/1740/DC1 Materials and Methods Figs. S1 to S10 Table S1 References 4 April 2008; accepted 28 May 2008 10.1126/science.1158708 REPORTS Dislocation Mean Free Paths and Strain Hardening of Crystals B. Devincre,1 T. Hoc,2 L. Kubin1 * Predicting the strain hardening properties of crystals constitutes a long-standing challenge for dislocation theory. The main difficulty resides in the integration of dislocation processes through a wide range of time and length scales, up to macroscopic dimensions. In the present multiscale approach, dislocation dynamics simulations are used to establish a dislocation-based continuum model incorporating discrete and intermittent aspects of plastic flow. This is performed through the modeling of a key quantity, the mean free path of dislocations. The model is then integrated at the scale of bulk crystals, which allows for the detailed reproduction of the complex deformation curves of face-centered cubic crystals. Because of its predictive ability, the proposed framework has a large potential for further applications. Dislocations are complex defects of crystalline materials, which have been investigated for more than 70 years. Their fundamental and economical importance arises from the number of properties they govern, such as the high strength of nanostructured materials, the reliability of semiconductor devices, the processing and service life of structural materials, or the rheological properties of tectonic events in Earth’s crust. The irreversible, or plastic, deformation of crystals results from the motion on crystallographic planes of linear defects, the dislocations (1, 2). These defects carry an elementary amount of shear (the Burgers vector) that is usually the smallest translation of the crystal lattice. During plastic flow, dislocations multiply and their mutual interactions hinder their motion. As a consequence, a shear stress increase dt has to be imposed to produce a shear strain increase dg. By definition, the ratio dt/dg is the strain hardening rate. Although dislocation theory has successfully explained many aspects of the strength of crystalline solids, predicting strain hardening is “the most difficult remaining problem” (3). The present dislocation-based models for strain hardening still have difficulties integrating elementary dislocation properties into a continuum description of bulk crystals or polycrystals. As a consequence, current approaches cannot avoid making use of extensive parameter fitting. The present work takes advantage of threedimensional dislocation dynamics (DD) simulations (4–8) for averaging dislocation properties at the intermediate scale of slip systems, which are ensembles of dislocations having the same Burgers vector and slip plane. The use of periodic boundary conditions allows for the tayloring of large dislocation glide paths and the investigation of volumes that are representative of the bulk material (4). One can then derive a continuum formulation on the basis of physically justified mechanisms and parameters, and this formulation is further integrated at the scale of a bulk crystal. Facecentered cubic (fcc) crystals are taken as benchmark materials because of their well-documented, but rather complicated, stress/strain response. We started by considering the critical stress ti c for the activation of slip system i as a function of the dislocation densities r j stored (i.e., temporarily or permanently immobilized) in all slip systems j. This critical stress is given by a generalized Taylor relation (9) of the form, ti c ¼ mb ffiffiffiffiffiffiffiffiffiffiffiffi ∑ j aijr j q , where m is the shear modulus and b the modulus of the Burgers vector. In fcc crystals, the symmetric tensor aij contains six independent dimensionless coefficients, which account for the average strength of pair interactions between slip systems that result from short- and long-range interactions. Their values were recently determined from DD simulations (5). For determining strain hardening, the key quantity is the rate at which the critical stress evolves with strain or, equivalently, the rate at which dislocations accumulate under strain. For this purpose, it is useful to define a dislocation mean free path L, which is the distance traveled by a dislocation segment of length l before it is stored by interaction with the microstructure. When the line moves by a distance dx, it sweeps an area ldx and produces a shear strain dg = bldx/V, where V is the volume of the crystal. The stored density has then statistically increased by dr = (dx/L)l/V, and the incremental storage rate is dr/dg = 1/bL. This definition is only valid in differential form, as dislocation lines multiply when they move. Following Kocks et al. and 1 Laboratoire d'Etude des Microstructures, Unité Mixte de Recherche (UMR) 104 CNRS, CNRS–Office National d'Etudes et de Recherches Aérospatiales (ONERA), 20 Avenue de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France. 2 Laboratoire MSSMat, UMR 8579 CNRS, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France. *To whom correspondence should be addressed. E-mail: ladislas.kubin@onera.fr www.sciencemag.org SCIENCE VOL 320 27 JUNE 2008 1745
REPORTS Teodosiu etal.(10,ID),the net storage rate in each unzipping of a single junction initiates several suc- tals.This suggests that a continuous description of slip system i is written as cessive bursts of dislocation motion and expansion. uniform storage events through a coarse-graining As a consequence,the stored dislocation density procedure that smoothens out intermittency can be dp'11 =元p increases in a discontinuous but progressive man- established.To ensure compatibility between the ner during straining(Fig.1B),thus inducing strain discrete and continuum approaches of storage,the hardening.Figure 1C shows the probability distri- parameters involved in the continuum approach The last term at the right-hand side.where y is bution functions of plastic strain-burst amplitudes, discussed below are estimated as averages over proportional to the critical annihilation distance for P(p),as obtained from three DD simulations the fluctuating output of DD simulations. screw dislocations.describes the effect of a mech- performed with different loading axes.For each The most important contribution to the mean anism called dynamic recovery (4).The mean free orientation,a power law P(Yp)-Yp is ob- free paths arises from the interactions of moving path L'also appears in Eq.1.Very little is presently tained in a bounded domain of amplitudes,with dislocations with "forest"dislocations (that is.dis- known about the way L'depends on dislocation a scaling exponent n1.6 in the range of previ- locations of other systems that pierce their slip interactions,stress,and specimen orientation. ously measured values (12,/3). plane)and their subsequent storage.We examined At a small scale.plastic flow is not continuous Because the shear strain is the area swept by conditions such that nslip systems are active,with but exhibits intermittency,as has been shown in dislocations divided by the volume of the deform- n>1,and we considered one active slip system i. several recent studies (/2,/3).Figure 1A.which is ing crystal,these dislocation avalanches cannot be We made a simplifying assumption by replacing extracted from a DD simulation,illustrates that the observed on the stress/strain curves of bulk crys- the interaction coefficients between slip systems 0.9 B 10 45 0.6 0.53 0.55 0.57 2(%) 如分 =1.6 o001) 4[111] 。[112 5 um 104 Fig.1.Strain bursts during large-scale DD simulations of tensile deformation in copper crystals.The elementary simulation cell has a size of 4.4-by-4.9-by-5.9-m,the imposed strain rate is 10 s and periodic boundary conditions were used (4).(A)Successive dislocation avalanches occurring in the slip systems =[011](111)of a deforming [001]crystal.Superimposed configurations taken at constant time 0102 intervals are shown in a thin film of thickness =0.25 um containing the active set of (111)extended slip C planes.The forest slip systems(short green lines)form junctions (red straight lines)with the active slip 10 10-6 10-5 104 system (blue lines).During its expansion,the unpinned segment(1)sweeps an area of-130 um2before 10-3 being stored again at dense forest tangles.(B)Corresponding evolutions of the resolved stress (t)and SYp dislocation density (p)in s versus the total shear strain ()Each jerk results from an avalanche,and arrows mark the dislocation configurations shown in(A). (C)Probability P(Yp)per bin size of the strain-burst amplitudes Yp for three simulations with high-symmetry orientations. 0.4 3.0 0.5 [112 B [112] [111 2.5 [111 0.4 0.3 [001] [001] 2.0 0.3 02 5 0.2 1.0 [112 0.1 1 0.5 111 001] 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 00 02 0.40.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 Yp(%) Yp(%) Yp(%) Fig.2.Measurement of the dimensionless constants Po(A),ko(B),and K(C)as a function of plastic shear strain by large-scale DD simulations and for three symmetrical orientations.Black lines represent the mean values.The elementary cell size is about (5 um),and the imposed strain rate isy=10 s-1. 1746 27 JUNE 2008 VOL 320 SCIENCE www.sciencemag.org
Teodosiu et al. (10, 11), the net storage rate in each slip system i is written as dri dgi ¼ 1 b 1 Li − yri ð1Þ The last term at the right-hand side, where y is proportional to the critical annihilation distance for screw dislocations, describes the effect of a mechanism called dynamic recovery (4). The mean free path Li also appears in Eq. 1. Very little is presently known about the way Li depends on dislocation interactions, stress, and specimen orientation. At a small scale, plastic flow is not continuous but exhibits intermittency, as has been shown in several recent studies (12, 13). Figure 1A, which is extracted from a DD simulation, illustrates that the unzipping of a single junction initiates several successive bursts of dislocation motion and expansion. As a consequence, the stored dislocation density increases in a discontinuous but progressive manner during straining (Fig. 1B), thus inducing strain hardening. Figure 1C shows the probability distribution functions of plastic strain-burst amplitudes, PðdgpÞ, as obtained from three DD simulations performed with different loading axes. For each orientation, a power law PðdgpÞ ∼ dg−h p is obtained in a bounded domain of amplitudes, with a scaling exponent h ≈ 1.6 in the range of previously measured values (12, 13). Because the shear strain is the area swept by dislocations divided by the volume of the deforming crystal, these dislocation avalanches cannot be observed on the stress/strain curves of bulk crystals. This suggests that a continuous description of uniform storage events through a coarse-graining procedure that smoothens out intermittency can be established. To ensure compatibility between the discrete and continuum approaches of storage, the parameters involved in the continuum approach discussed below are estimated as averages over the fluctuating output of DD simulations. The most important contribution to the mean free paths arises from the interactions of moving dislocations with “forest” dislocations (that is, dislocations of other systems that pierce their slip plane) and their subsequent storage. We examined conditions such that n slip systems are active, with n > 1, and we considered one active slip system i. We made a simplifying assumption by replacing the interaction coefficients between slip systems A 1 2 3 4 5 7 9 10 6 11 8 (MPa) 7.5 7.0 6.5 6.0 5.5 5.0 4.5 0.9 0.8 0.7 0.6 ρ (1012 m-2) γp B τ 0.53 0.55 0.57 δγp 10-6 10-3 10-4 10-5 101 105 104 103 102 [001] [111] [112] P (δγp) / bin size η C =1.6 (%) Fig. 1. Strain bursts during large-scale DD simulations of tensile deformation in copper crystals. The elementary simulation cell has a size of 4.4-by-4.9-by-5.9–mm3 , the imposed strain rate is 10 s–1 , and periodic boundary conditions were used (4). (A) Successive dislocation avalanches occurring in the slip system s = [011](111) of a deforming [001] crystal. Superimposed configurations taken at constant time intervals are shown in a thin film of thickness = 0.25 mm containing the active set of (111) extended slip planes. The forest slip systems (short green lines) form junctions (red straight lines) with the active slip system (blue lines). During its expansion, the unpinned segment (1) sweeps an area of ~130 mm2 before being stored again at dense forest tangles. (B) Corresponding evolutions of the resolved stress (t) and dislocation density (r) in s versus the total shear strain (gp). Each jerk results from an avalanche, and arrows mark the dislocation configurations shown in (A). (C) Probability P(dgp) per bin size of the strain-burst amplitudes dgp for three simulations with high-symmetry orientations. 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.4 0.3 0.2 0.1 0.0 C γ p (%) γ p (%) γ p (%) κ 0.0 0.2 0.4 0.6 0.8 1.0 3.0 2.5 2.0 1.5 1.0 0.5 0.0 B ko [112] [111] [001] 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.3 0.2 0.1 0.0 A po [112] [111] [001] [112] [111] [001] Fig. 2. Measurement of the dimensionless constants p0 (A), k0 (B), and k (C) as a function of plastic shear strain by large-scale DD simulations and for three symmetrical orientations. Black lines represent the mean values. The elementary cell size is about (5 mm)3 , and the imposed strain rate is g˙ = 10 s −1. 1746 27 JUNE 2008 VOL 320 SCIENCE www.sciencemag.org REPORTS
REPORTS a by their average value a.During a time interval Collecting all terms,one eventually obtains system,and (ii)large-scale simulations of tensile dt,mobile dislocations in i sweep an area dsand the storage rate per active slip system or,equiv- deformation tests along three symmetrical axes, produce a strain increment dy'=bds'per unit of alently,the inverse of the mean free path (Eq.1). [1121.[111],and [0011.Figure 2 illustrates the de- volume.The increase in stored density dp'is the The latter takes a relatively simple form in the termination of the constants po.ko.and K by large- product of the number of stable junctions formed case of loading along symmetrical orientations scale simulations.Table 1 gives average values during a time increment and the dislocation den- like [001],[111],or [112]when n active slip for the constants in Eq.2,as obtained from the sity stored per junction. systems (n=4,3,and 2,respectively)equally two sets of DD simulations.The values of the The number of stable junctions formed during contribute to the total strain.One then has mean free path coefficients,K,can then be dt is the product of two terms.The first one is the compared to the ones predicted by Eq.2. number of intersections,dNn of mobile dislo- with From the values of po and K,one can infer cations with the forest dislocations of i,in density L ubK that the fraction of attractive intersections that p(where frefers to the forest).This quantity is n(1+x)2 result in junction formation is ~25%,whereas proportional to p'and to the swept area;hence, K 2) the average density of junctions is~30%of the dNimt =pfdy/b.The second term incorporates Pokova(n-1-K) total density in symmetrical conditions.The sim- the fact that not all intersections produce stable ulated and calculated values for Ki2 and Ki are junctions (Fig.1A).Because the stability of a This mean free path exhibits interesting prop- in good agreement with one another.The differ- junction is proportional to the average strength of erties that are also present in more general con- ence between the values for Koo actually results the forest interactions (5).it is written in the form ditions.From Eq.2,one can see that it is from a particular dislocation mechanism that is pova,where po is a constant. inversely proportional to the Taylor stress.It is specific to the [001]orientation (/4).We used The density stored by each junction is given also proportional to an orientation-dependent the measured value of Koor to correct the pre- by e/V,where e'is the average length of the coefficient Kik,which depends on the three diction of the model in that case. stored segments.As usual in dislocation theory, dimensionless constants (po,ko,and K)and also Storage by forest interactions and Taylor hard- this length is inversely proportional to stress,and on the number n ofactive slip systems.In short, ening constitutes the two major building blocks one has e'=koub/te,where ko is a dimen- as n increases,the forest density seen by each for modeling strain hardening.Other building sionless constant.However.one has to account active slip system increases too.and the mean blocks are discussed in the supporting online ma- for the contribution of junctions to the average free path decreases.Because n depends on the terial text.They are essentially concemned with lengths.Although junction lines do not neces- orientation [h,]of the loading axis,an orien- self-interaction mechanisms,which govern strain sarily share the attributes of perfect dislocations tation dependence of the mean free path arises. hardening in single slip conditions (/4),and dy- they are redistributed into the densities stored in which was not consistently modeled to date. namic recovery,which is related to the thermally the active slip systems.This way,they can further The constants defining the mean free paths activated annihilation of screw dislocations by react with mobile segments to form second-order were evaluated from two sets of independent DD cross-slip.Because the annihilation distanceyin junctions (6).As a result,one defines a last con- simulations caried out with copper as a model ma- Eq.1 incorporates two poorly known factors,a stant parameter,the ratio of junction density to terial (4):(1)model simulations,in which a mobile reference value was estimated from an experi- total density in each slip system,K. slip system interacts with an immobile forest slip mental stress/strain curve. To allow for a comparison between the pre- Table 1.Average values of the dimensionless constants for fcc crystals involved in Eq.2 and their dicted and experimental mechanical responses of single crystals,a change in scale is performed from variance.The number of independent measurements is indicated in parentheses in the top row.The values of the mean free path coefficients K can be compared to the ones calculated from Eq.2 mesoscopic to macroscopic dimensions.For this (italic numbers in parentheses).For a resolved shear stress of 10 MPa in copper,these values are purpose,use is made of a crystal plasticity code almost identical to those of the mean free paths expressed in microns. (4,/5),which is a specific type of finite element code that takes into account the crystallographic Po(6) k(6) K(9) K112(3) K11(3) Koo1(3) nature of dislocation glide.deformation conditions. 0.117±0.012 1.08±0.005 0.291±0.015 10.42±0.4 7.29±1.6 4.57±0.3 and lattice rotations during plastic flow.We then 11.87八 (738) (6.21) integrated the set of dislocation-based equations on a meshed tensile specimen (fig.S1). 100 100 [111] [001] B PP 0.6 C 80 【1121 ps 80 化 60 0.4 log[p(m)] 60 [123) Cu 40 40 Ag 0.2 10 0 20 0 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.30.40.50.6 yt Yi Fig.3.Simulated mechanical response of fcc single crystals at room tem- active slip systems.The crossing of the [001]and [111]curves was exper- perature.t and total strains (y)were drawn using traditional conventions for imentally observed.This crossing occurs because of a competition between the plotting experimental results.(A)Copper.Stress/strain curves were resolved on orientation dependencies of stages ll and Ill.(B)Copper,[123]orientation. the primary slip system.Notice the strong orientation effect and the occurrence Densities (p)and resolved strains (y)on the primary (p)and secondary (s)slip of three stages for the low-symmetry [123]orientation.The strain hardening systems as a function of the total resolved strain,showing the transition between rate increases from dissymmetrical double slip along [123]to symmetrical slip easy glide in stage I and forest hardening in stage ll.(C)Resolved stress versus along [112],[111],and [001];i.e.,it increases with an increasing number of total shear strain curves for [123]Cu,Al,Ag,and Ni crystals at room temperature. www.sciencemag.org SCIENCE VOL 320 27 JUNE 2008 1747
aij by their average value a. During a time interval dt, mobile dislocations in i sweep an area dSi and produce a strain increment dgi = bdSi per unit of volume. The increase in stored density dri is the product of the number of stable junctions formed during a time increment and the dislocation density stored per junction. The number of stable junctions formed during dt is the product of two terms. The first one is the number of intersections, dNint, of mobile dislocations with the forest dislocations of i, in density ri f (where f refers to the forest). This quantity is proportional to ri f and to the swept area; hence, dNint ¼ ri f dg=b. The second term incorporates the fact that not all intersections produce stable junctions (Fig. 1A). Because the stability of a junction is proportional to the average strength of the forest interactions (5), it is written in the form p0 ffiffiffi a p , where p0 is a constant. The density stored by each junction is given by ‘ i =V, where ‘ i is the average length of the stored segments. As usual in dislocation theory, this length is inversely proportional to stress, and one has ‘ i ¼ k0mb=ti c, where k0 is a dimensionless constant. However, one has to account for the contribution of junctions to the average lengths. Although junction lines do not necessarily share the attributes of perfect dislocations, they are redistributed into the densities stored in the active slip systems. This way, they can further react with mobile segments to form second-order junctions (6). As a result, one defines a last constant parameter, the ratio of junction density to total density in each slip system, k. Collecting all terms, one eventually obtains the storage rate per active slip system or, equivalently, the inverse of the mean free path (Eq. 1). The latter takes a relatively simple form in the case of loading along symmetrical orientations like [001], [111], or [112] when n active slip systems (n = 4, 3, and 2, respectively) equally contribute to the total strain. One then has 1 Li ¼ ti c mbKhkl , with Khkl ¼ nð1 þ kÞ 3=2 p0k0 ffiffiffi a p ðn − 1 − kÞ " # ð2Þ This mean free path exhibits interesting properties that are also present in more general conditions. From Eq. 2, one can see that it is inversely proportional to the Taylor stress. It is also proportional to an orientation-dependent coefficient Khkl, which depends on the three dimensionless constants (p0, k0, and k) and also on the number n of active slip systems. In short, as n increases, the forest density seen by each active slip system increases too, and the mean free path decreases. Because n depends on the orientation [h,k,l] of the loading axis, an orientation dependence of the mean free path arises, which was not consistently modeled to date. The constants defining the mean free paths were evaluated from two sets of independent DD simulations carried out with copper as a model material (4): (i) model simulations, in which a mobile slip system interacts with an immobile forest slip system, and (ii) large-scale simulations of tensile deformation tests along three symmetrical axes, [112], [111], and [001]. Figure 2 illustrates the determination of the constants p0, k0, and k by largescale simulations. Table 1 gives average values for the constants in Eq. 2, as obtained from the two sets of DD simulations. The values of the mean free path coefficients, Khkl, can then be compared to the ones predicted by Eq. 2. From the values of p0 and k, one can infer that the fraction of attractive intersections that result in junction formation is ~25%, whereas the average density of junctions is ~30% of the total density in symmetrical conditions. The simulated and calculated values for K112 and K111 are in good agreement with one another. The difference between the values for K001 actually results from a particular dislocation mechanism that is specific to the [001] orientation (14). We used the measured value of K001 to correct the prediction of the model in that case. Storage by forest interactions and Taylor hardening constitutes the two major building blocks for modeling strain hardening. Other building blocks are discussed in the supporting online material text. They are essentially concerned with self-interaction mechanisms, which govern strain hardening in single slip conditions (14), and dynamic recovery, which is related to the thermally activated annihilation of screw dislocations by cross-slip. Because the annihilation distance y in Eq. 1 incorporates two poorly known factors, a reference value was estimated from an experimental stress/strain curve. To allow for a comparison between the predicted and experimental mechanical responses of single crystals, a change in scale is performed from mesoscopic to macroscopic dimensions. For this purpose, use is made of a crystal plasticity code (4, 15), which is a specific type of finite element code that takes into account the crystallographic nature of dislocation glide, deformation conditions, and lattice rotations during plastic flow. We then integrated the set of dislocation-based equations on a meshed tensile specimen (fig. S1). Table 1. Average values of the dimensionless constants for fcc crystals involved in Eq. 2 and their variance. The number of independent measurements is indicated in parentheses in the top row. The values of the mean free path coefficients Khkl can be compared to the ones calculated from Eq. 2 (italic numbers in parentheses). For a resolved shear stress of 10 MPa in copper, these values are almost identical to those of the mean free paths expressed in microns. p0 (6) k0 (6) k (9) K112 (3) K111 (3) K001 (3) 0.117 ± 0.012 1.08 ± 0.005 0.291 ± 0.015 10.42 ± 0.4 (11.87) 7.29 ± 1.6 (7.38) 4.57 ± 0.3 (6.21) 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 [001] [111] [112] [123] I II III A γ t γ t γ t (MPa) τ (MPa) τ 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 γ log[ (m-2)] 8 10 12 14 ρp ρs γ p γ s B ρ 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 C Ni Cu Ag Al Fig. 3. Simulated mechanical response of fcc single crystals at room temperature. t and total strains (gt ) were drawn using traditional conventions for plotting experimental results. (A) Copper. Stress/strain curves were resolved on the primary slip system. Notice the strong orientation effect and the occurrence of three stages for the low-symmetry [123] orientation. The strain hardening rate increases from dissymmetrical double slip along [123] to symmetrical slip along [112], [111], and [001]; i.e., it increases with an increasing number of active slip systems. The crossing of the [001] and [111] curves was experimentally observed. This crossing occurs because of a competition between the orientation dependencies of stages II and III. (B) Copper, [123] orientation. Densities (r) and resolved strains (g) on the primary (p) and secondary (s) slip systems as a function of the total resolved strain, showing the transition between easy glide in stage I and forest hardening in stage II. (C) Resolved stress versus total shear strain curves for [123] Cu, Al, Ag, and Ni crystals at room temperature. www.sciencemag.org SCIENCE VOL 320 27 JUNE 2008 1747 REPORTS
REPORTS A selection of results is presented in Fig.3 ery.To address this,we tentatively used a scaling 5.B.Devincre,L Kubin,T.Hoc,Scr.Mater.54,741 (2006) for tensile deformation at 300 K.with emphasis law derived from Escaig's model for cross-slip (4). 6.V.V.Bulatov et al.,Nature 440,1174 (2006). on copper crystals.Although the model does The present results indicate that,paradoxi- 7.V.Bulatov,F.Abraham,L.Kubin,B.Devincre,S.Yip, Nature391,669(1998) not incorporate ad hoc switches,the resolved cally,realistic strain hardening properties in uni- 8.R.Madec.B.Devincre,L Kubin,T.Hoc,D.Rodney, stress/strain curves (Fig.3A)exhibit the tradi- axial deformation are obtained without accounting Science301,18792003). tional stages that characterize fcc single crystals for dislocation pattering (18,19);that is,for 9.P.Franciosi,M.Berveiller,A.Zaoui,Acta Metall.28,273 (16).The low-hardening stage I,during which a the emergence of non-uniform microstructures (1980). 10.U.F.Kocks,H.Mecking,Prog.Mater.Sci.48,171 (2003). single slip system is activated,appears for low- during plastic flow.A possible reason is that the 11.C.Teodosiu,).-L.Raphanel,L Tabourot,in Large Plastic symmetry orientations like [123].The linear wavelength of dislocation patters and the mean Deformations,C.Teodosiu,)L.Raphanel,F.Sidoroff,Eds. stage II is due to forest hardening,and its slope free path values follow the same scaling rela- (A.A Balkema,Rotterdam,Netherlands,1993),p.153. increases with the number of active slip systems. tion,in tension or compression. 12.M.-Carmen Miguel,A.Vespignani,S.Zapperi,]Weiss, 1.-R.Grasso,Nature410,667(2001). The subsequent decrease in strain hardening rate This study shows that the mean free path 13.F.F.Csikor,C.Motz,D.Weygand,M.Zaiser,S.Zapperi, is also orientation-dependent and stems from of dislocations is the missing link connecting Science318,251(2007). dynamic recovery.All of these features are in discrete dislocation interactions and avalanche 14.B.Devincre,L Kubin,T.Hoc,Scr.Moter.57,905 (2007) excellent agreement with published experimen- processes to strain hardening properties in the 15.T.Hoc,C.Rey,1.-L Raphanel,Acto Mater.49.1835 (2001). 16.T.E.Mitchell,Prog.AppL Mater.Res.6,117 (1964). tal results (16,/7).Figure 3B shows the evo- bulk.The present multiscale methodology should 17.T.Takeuchi,Trans.JIM 16,629 (1975) lution of the shear strains and densities on the apply to several areas of practical importance, 18.L Kubin,Science 312,864 (2006). primary and secondary slip systems during a such as the mechanical response of polycrystal- 19.L Kubin,B.Devincre,T.Hoc,Mater.Sci.Eng.A simulated [123]test.It is representative of the line materials or size effects in small dimensions. 483-484,19(2008). 20.The authors acknowledge funding by their host wealth of detailed information that can be ob- institutions:CNRS,ONERA and Ecole Centrale Paris tained at the scale of slip systems.Finally,Fig.3C illustrates a broader aspect of this type of mod- References and Notes 1.]Friedel,Dislocotions (Pergamon,Oxford,1967). Supporting Online Material eling by presenting [123]stress/strain curves for 2.)Hirth,]Lothe,Theory of Dislocations (Krieger www.sciencemag.org/cgi/content/full/320/5884/1745/DC1 several foc crystals at room temperature.In addi- Malabar,FL 1992). SOM Text tion to a rescaling of lattice parameters and elastic 3.A.H.Cottrell,in Dislocations in Solids,vol.11, Fig.S1 constants,shifting from one fcc material to the F.R.N.Nabarro,M.S.Duesbery,Eds.(Elsevier, References Amsterdam,2002),p.vii. other implies changes in the annihilation proper- 4.Simulation methods and additional information are 5 February 2008;accepted 2 May 2008 ties of screw dislocations during dynamic recov- avallable as supporting material on Science Online. 10.1126/cience.1156101 Despite this progress,the conversion of metal- Ordered Mesoporous Materials polymer hybrids into mesoporous materials with ordered and large pores (10 nm)has not been from Metal Nanoparticle-Block accomplished,in part because of the low volume fraction of metals in most hybrids and the wide- Copolymer Self-Assembly spread use of gold,which has a high diffusion coefficient and therefore retains its mesostructure only at low temperatures(7-9).Although a protec- Scott C.Warren,1.2 Lauren C.Messina,2 Liane S.Slaughter,2 Marleen Kamperman,1 Qin Zhou,2 tive organic layer can be added to metal NPs to Sol M.Gruner,3 Francis ]DiSalvo,2 Ulrich Wiesner* prevent uncontrolled aggregation,even a thin or- ganic layer represents a considerable volume of The synthesis of ordered mesoporous metal composites and ordered mesoporous metals is a the overall material:For example,a 1-nm-diameter challenge because metals have high surface energies that favor low surface areas.We metal NP with a relatively thin 1-nm organic shell present results from the self-assembly of block copolymers with ligand-stabilized platinum is just 4%metal by volume.As a result,the typica nanoparticles,leading to lamellar CCM-Pt-4 and inverse hexagonal(CCM-Pt-6)hybrid metal content in most block copolymer-metal NP mesostructures with high nanoparticle loadings.Pyrolysis of the CCM-Pt-6 hybrid produces an hybrids is only a few volume%,and the prospects ordered mesoporous platinum-carbon nanocomposite with open and large pores for converting the hybrid into an ordered meso- (>10 nanometers).Removal of the carbon leads to ordered porous platinum mesostructures. porous material,in which the metal would have a The platinum-carbon nanocomposite has very high electrical conductivity (400 siemens per volume fraction between 60 and 75%for an in- centimeter)for an ordered mesoporous material fabricated from block copolymer self-assembly. verse hexagonal structure,are poor. Mesoporous metals have been synthesized at a espite considerable progress in the field tion of Raney nickel and other metals(3).Dealloy- smaller length scale,with 2-to 4-nm pores,through of porous solids,major challenges re- ing processes provide limited control over structural the coassembly of metal ions with small-molecule main in the synthesis of ordered meso- parameters such as pore geometry and order.In surfactants followed by reduction (10-13).The porous materials with high metal content from contrast,block copolymer self-assembly or tem- small pore size,however,limits the flow of liquids the coassembly of macromolecular surfactants plating with metal species provides access to highly through the material,which is essential for many and inorganic species.Controlling the structure of ordered structures.Synthetic routes to such struc- applications (14,15).Metals have also been de- metals at the mesoscale (2 to 50 nm)is crucial for tures have included adsorbing and then reducing posited onto (/6)or into (/7)thin films of block the development of improved fuel cell electrodes metal ions within a preassembled block copolymer and may also assist in the miniaturization of scaffold (4)and coassembling ligand-stabilized Department of Materials Science and Engineering,Cornell optical and electronic materials for data transmis- nanoparticles (NPs)with block copolymers (5). University,Ithaca,NY 14853,USA.Department of Chemistry sion,storage,and computation (/2). More recently,polymer-coated NPs that behave and Chemical Biology,Cornell University,Ithaca,NY 14853,USA.Department of Physics,Comell University. An early route to preparing mesoporous metals like surfactants have been isolated at the interface Ithaca,NY 14853.USA. involves the dealloying of a less noble metal from a of block copolymer domains,which can create a *To whom correspondence should be addressed.E-mail: bimetallic alloy;this has been used for the prepara- bicontinuous morphology at higher loadings (6). ubw1@cornell.edu 1748 27 JUNE 2008 VOL 320 SCIENCE www.sciencemag.org
A selection of results is presented in Fig. 3 for tensile deformation at 300 K, with emphasis on copper crystals. Although the model does not incorporate ad hoc switches, the resolved stress/strain curves (Fig. 3A) exhibit the traditional stages that characterize fcc single crystals (16). The low-hardening stage I, during which a single slip system is activated, appears for lowsymmetry orientations like [123]. The linear stage II is due to forest hardening, and its slope increases with the number of active slip systems. The subsequent decrease in strain hardening rate is also orientation-dependent and stems from dynamic recovery. All of these features are in excellent agreement with published experimental results (16, 17). Figure 3B shows the evolution of the shear strains and densities on the primary and secondary slip systems during a simulated ½123 test. It is representative of the wealth of detailed information that can be obtained at the scale of slip systems. Finally, Fig. 3C illustrates a broader aspect of this type of modeling by presenting [123] stress/strain curves for several fcc crystals at room temperature. In addition to a rescaling of lattice parameters and elastic constants, shifting from one fcc material to the other implies changes in the annihilation properties of screw dislocations during dynamic recovery. To address this, we tentatively used a scaling law derived from Escaig’s model for cross-slip (4). The present results indicate that, paradoxically, realistic strain hardening properties in uniaxial deformation are obtained without accounting for dislocation patterning (18, 19); that is, for the emergence of non-uniform microstructures during plastic flow. A possible reason is that the wavelength of dislocation patterns and the mean free path values follow the same scaling relation, in tension or compression. This study shows that the mean free path of dislocations is the missing link connecting discrete dislocation interactions and avalanche processes to strain hardening properties in the bulk. The present multiscale methodology should apply to several areas of practical importance, such as the mechanical response of polycrystalline materials or size effects in small dimensions. References and Notes 1. J. Friedel, Dislocations (Pergamon, Oxford, 1967). 2. J. Hirth, J. Lothe, Theory of Dislocations (Krieger, Malabar, FL, 1992). 3. A. H. Cottrell, in Dislocations in Solids, vol. 11, F. R. N. Nabarro, M. S. Duesbery, Eds. (Elsevier, Amsterdam, 2002), p. vii. 4. Simulation methods and additional information are available as supporting material on Science Online. 5. B. Devincre, L. Kubin, T. Hoc, Scr. Mater. 54, 741 (2006). 6. V. V. Bulatov et al., Nature 440, 1174 (2006). 7. V. Bulatov, F. Abraham, L. Kubin, B. Devincre, S. Yip, Nature 391, 669 (1998). 8. R. Madec, B. Devincre, L. Kubin, T. Hoc, D. Rodney, Science 301, 1879 (2003). 9. P. Franciosi, M. Berveiller, A. Zaoui, Acta Metall. 28, 273 (1980). 10. U. F. Kocks, H. Mecking, Prog. Mater. Sci. 48, 171 (2003). 11. C. Teodosiu, J.-L. Raphanel, L. Tabourot, in Large Plastic Deformations, C. Teodosiu, J. L. Raphanel, F. Sidoroff, Eds. (A. A. Balkema, Rotterdam, Netherlands, 1993), p. 153. 12. M.-Carmen Miguel, A. Vespignani, S. Zapperi, J. Weiss, J.-R. Grasso, Nature 410, 667 (2001). 13. F. F. Csikor, C. Motz, D. Weygand, M. Zaiser, S. Zapperi, Science 318, 251 (2007). 14. B. Devincre, L. Kubin, T. Hoc, Scr. Mater. 57, 905 (2007). 15. T. Hoc, C. Rey, J.-L. Raphanel, Acta Mater. 49, 1835 (2001). 16. T. E. Mitchell, Prog. Appl. Mater. Res. 6, 117 (1964). 17. T. Takeuchi, Trans. JIM 16, 629 (1975). 18. L. Kubin, Science 312, 864 (2006). 19. L. Kubin, B. Devincre, T. Hoc, Mater. Sci. Eng. A 483–484, 19 (2008). 20. The authors acknowledge funding by their host institutions: CNRS, ONERA, and Ecole Centrale Paris. Supporting Online Material www.sciencemag.org/cgi/content/full/320/5884/1745/DC1 SOM Text Fig. S1 References 5 February 2008; accepted 2 May 2008 10.1126/science.1156101 Ordered Mesoporous Materials from Metal Nanoparticle–Block Copolymer Self-Assembly Scott C. Warren,1,2 Lauren C. Messina,2 Liane S. Slaughter,2 Marleen Kamperman,1 Qin Zhou,2 Sol M. Gruner,3 Francis J. DiSalvo,2 Ulrich Wiesner1 * The synthesis of ordered mesoporous metal composites and ordered mesoporous metals is a challenge because metals have high surface energies that favor low surface areas. We present results from the self-assembly of block copolymers with ligand-stabilized platinum nanoparticles, leading to lamellar CCM-Pt-4 and inverse hexagonal (CCM-Pt-6) hybrid mesostructures with high nanoparticle loadings. Pyrolysis of the CCM-Pt-6 hybrid produces an ordered mesoporous platinum-carbon nanocomposite with open and large pores (≥10 nanometers). Removal of the carbon leads to ordered porous platinum mesostructures. The platinum-carbon nanocomposite has very high electrical conductivity (400 siemens per centimeter) for an ordered mesoporous material fabricated from block copolymer self-assembly. Despite considerable progress in the field of porous solids, major challenges remain in the synthesis of ordered mesoporous materials with high metal content from the coassembly of macromolecular surfactants and inorganic species. Controlling the structure of metals at the mesoscale (2 to 50 nm) is crucial for the development of improved fuel cell electrodes and may also assist in the miniaturization of optical and electronic materials for data transmission, storage, and computation (1, 2). An early route to preparing mesoporous metals involves the dealloying of a less noble metal from a bimetallic alloy; this has been used for the preparation of Raney nickel and other metals (3). Dealloying processes provide limited control over structural parameters such as pore geometry and order. In contrast, block copolymer self-assembly or templating with metal species provides access to highly ordered structures. Synthetic routes to such structures have included adsorbing and then reducing metal ions within a preassembled block copolymer scaffold (4) and coassembling ligand-stabilized nanoparticles (NPs) with block copolymers (5). More recently, polymer-coated NPs that behave like surfactants have been isolated at the interface of block copolymer domains, which can create a bicontinuous morphology at higher loadings (6). Despite this progress, the conversion of metalpolymer hybrids into mesoporous materials with ordered and large pores (≥10 nm) has not been accomplished, in part because of the low volume fraction of metals in most hybrids and the widespread use of gold, which has a high diffusion coefficient and therefore retains its mesostructure only at low temperatures (7–9). Although a protective organic layer can be added to metal NPs to prevent uncontrolled aggregation, even a thin organic layer represents a considerable volume of the overall material: For example, a 1-nm-diameter metal NP with a relatively thin 1-nm organic shell is just 4% metal by volume. As a result, the typical metal content in most block copolymer–metal NP hybrids is only a few volume %, and the prospects for converting the hybrid into an ordered mesoporous material, in which the metal would have a volume fraction between 60 and 75% for an inverse hexagonal structure, are poor. Mesoporous metals have been synthesized at a smaller length scale, with 2- to 4-nm pores, through the coassembly of metal ions with small-molecule surfactants followed by reduction (10–13). The small pore size, however, limits the flow of liquids through the material, which is essential for many applications (14, 15). Metals have also been deposited onto (16) or into (17) thin films of block 1 Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853, USA. 2 Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 14853, USA. 3 Department of Physics, Cornell University, Ithaca, NY 14853, USA. *To whom correspondence should be addressed. E-mail: ubw1@cornell.edu 1748 27 JUNE 2008 VOL 320 SCIENCE www.sciencemag.org REPORTS