LETTERS NATUREVol 440127 April 2006 nodes and links.A node is where three or more lines merge together,and a link is 5.Hirth,J.P.Lothe,J.Theory of Dislocations 2nd edn,9-13 (Wiley,New York, any line segment connecting two nodes of the network.For the present analysis, 1982). we label any 3-node that bounds a regular binary junction as a 'normal'or 6 Saada,G.Sur le durcissement do a recombinaison des dislocations.Acta N-node.Likewise,any 4-node formed by two dislocations crossing each other Metoll..8.841-852(1960). is also labelled an N-node.All other nodes are regarded as multi-nodes or 心 Franciosi,P.Berveiller,M.Zaoui,A.Latent hardening in copper and aluminium single crystals.Acto Metall.28,273-283 (1980). M-nodes,including the symmetric 4-nodes shown in Fig.le as well as 3-nodes 甲 Guiu,F.Pratt,P.L.The effect of orientation on the yielding and flow of produced by dissociation of the symmetric 4-nodes.Three types of links can now molybdenum single crystals.Phys.Status Solidi 15,539-552 (1966). be defined with respect to the types of nodes they connect:NN links,NM links 9 Hirth,J.P.Lothe,J.Theory of Dislocations 2nd edn,269-270 (Wiley,New and MM links.Here,notation NM is used for any link that connects an N-node York.1982). to an M-node.To compute the fraction of lines involved in the multi-junctions 10.Basinski,S.J.Basinski,Z.S.in Dislocations in Solids Vol.4 (ed.Nabarro, shown in Fig.3d,we summed the lengths of all MM links with half the lengths of F.R.N.)261-362,(North Holland,Amsterdam,1979). all NM links and divided this sum by the total length of all links.The colour Nabarro,F.R.N.,Basinski,Z.S.Holt,D.B.The plasticity of pure single scheme used in Fig.4 is as follows:all MM links are shown in white,whereas the crystals.Adk.Phy5.50,193-233(1964). 12.Madec.S.R..Devincre.B.Kubin.L.P.From dislocation junctions to forest colour of NM links is graded from white at the M-nodes to green at the N-nodes. hardening.Phys.Rev.Lett.89,255508 (2002). All NN links,including binary junctions,are shown in green. Atomistic simulation.The simulation volume was a small cube-shaped block of 13. Hahner,P.Zaiser,M.Dislocation dynamics and work hardening of fractal dislocation cell structures.Mater.Sci.Eng.A 272,443-454 (1999). a perfect b.c.c.single crystal,17 nm on each side.The initial geometry contained 14. Abraham,F.F.et al.Simulating materials failure by using up to one billion three dislocations with Burgers vectors 1/2[111],1/2[111],and 1/2[111]inter- atoms and the world's fastest computer:Work hardening.Proc.Natl Acad.Sci. secting at the block centre.The atom positions inside the block were then relaxed U5A99.5783-5787(2002). to mechanical equilibrium using the conjugate gradient method and an 15. Frank,F.C.Read,W.T.Multiplication processes for slow moving interatomic interaction function for molybdenum2.The atoms on the block dislocations.Phys.Rev.79,722-723 (1950). surfaces were fixed throughout the simulation.To visualize crystal defects,only 16. Bulatov,V.V.et al.Scalable line dynamics in ParaDiS.Supercomputing (http:// the atoms inside the block with energies exceeding the ideal bulk value by www.sc-conference.org/sc2004/schedule/pdfs/pap206.pdf)(2004). 0.095eV are shown. 17. Mughrabi,H.Dislocation wall and cell structures and long-range internal Experiment.The experiments involved three steps:(1)compression of a single- stresses in deformed metal crystals.Acta Metall.31,1367-1379 (1983). 18. Kubin,L P.Canova,G.R.The modelling of dislocation patterns.Scr.Metall. crystal molybdenum specimen to 1%total strain along the [001]axis,(2)cutting Mater..27,957-962(1992) and thinning the deformed specimen along the (101)plane to obtain electron 19. Ghoniem,N.M,Huang J.M.Wang,Z.Q.Affine covariant-contravariant transparent foils,and (3)TEM observations using a set of reflection vectors g vector forms for the elastic field of parametric dislocations in isotropic crystals that can reveal multi-junctions.In the view shown in Fig.2b the zone axis Phl.Mag.Let.82.55-63(2002). [101]and the diffraction vector g=[020],making all four dislocations 20.Finnis,M.W.Sinclair,J.5.A simple empirical N-body potential for transition entering the 4-node visible.The views in Fig.2c and d were obtained using metals.Phil.Mag.A 50,45-55 (1984). g=[121]and g=[121]which made lines b=1/2[111]and bs=1/2[111] invisible owing to the g-b =0 condition.To access additional diffraction vectors, Supplementary Information is linked to the online version of the paper at the specimen was tilted to a new zone axis (=[201])making it possible to www.nature.com/nature. identify the Burgers vectors of two remaining dislocations in a similar manner: Acknowledgements This work was supported by the US DOE Office of Basic b2 1/2[111]and ba 1/2[111],respectively. Energy Sciences and the NNSA ASC program.We thank E.Chandler and C.Mailhiot for their encouragement and unwavering support of the ParaDiS Received 14 November 2005;accepted 14 February 2006. code development effort,D.Lassila for providing the purified molybdenum crystals,M.LeBlanc for performing the mechanical tests,R.Cook for graphic 1.Orowan,E.Zur kristallplastizitat.Z.Phys.89,605-659 (1934). design and F.Abraham and G.Campbell for critical reading and editorial 2 Taylor,G.The mechanism of plastic deformation in crystal.Part I.Theoretical. suggestions for the manuscript.This work was performed under the auspices of Pr0c.R5oc.A145.363-404(1934). the US DOE by the Lawrence Livermore National Laboratory. 3.Polanyi,M.Uber eine Art Gutterstorung,die einen kristall plastich machen konnte.Z.Phys.89,660-664(1934). Author Information Reprints and permissions information is available at 4.Hirsch,P.B.,Homne,R.W.Whelan,M.J.Direct observations of the npg.nature.com/reprintsandpermissions.The authors declare no competing arrangement and motion of dislocations in aluminium.Phil.Mag.1,677-684 financial interests.Correspondence and requests for materials should be (1956). addressed to V.V.B.(bulatov1@llnl.gov). 1178 2006 Nature Publishing Group© 2006 Nature Publishing Group nodes and links. A node is where three or more lines merge together, and a link is any line segment connecting two nodes of the network. For the present analysis, we label any 3-node that bounds a regular binary junction as a ‘normal’ or N-node. Likewise, any 4-node formed by two dislocations crossing each other is also labelled an N-node. All other nodes are regarded as multi-nodes or M-nodes, including the symmetric 4-nodes shown in Fig. 1e as well as 3-nodes produced by dissociation of the symmetric 4-nodes. Three types of links can now be defined with respect to the types of nodes they connect: NN links, NM links and MM links. Here, notation NM is used for any link that connects an N-node to an M-node. To compute the fraction of lines involved in the multi-junctions shown in Fig. 3d, we summed the lengths of all MM links with half the lengths of all NM links and divided this sum by the total length of all links. The colour scheme used in Fig. 4 is as follows: all MM links are shown in white, whereas the colour of NM links is graded from white at the M-nodes to green at the N-nodes. All NN links, including binary junctions, are shown in green. Atomistic simulation. The simulation volume was a small cube-shaped block of a perfect b.c.c. single crystal, 17 nm on each side. The initial geometry contained three dislocations with Burgers vectors 1/2[111], 1/2[111], and 1/2[111] inter￾secting at the block centre. The atom positions inside the block were then relaxed to mechanical equilibrium using the conjugate gradient method and an interatomic interaction function for molybdenum20. The atoms on the block surfaces were fixed throughout the simulation. To visualize crystal defects, only the atoms inside the block with energies exceeding the ideal bulk value by 0.095 eV are shown. Experiment. The experiments involved three steps: (1) compression of a single￾crystal molybdenum specimen to 1% total strain along the [001] axis, (2) cutting and thinning the deformed specimen along the (101) plane to obtain electron transparent foils, and (3) TEM observations using a set of reflection vectors g that can reveal multi-junctions. In the view shown in Fig. 2b the zone axis <[101] and the diffraction vector g ¼ [020], making all four dislocations entering the 4-node visible. The views in Fig. 2c and d were obtained using g ¼ [121] and g ¼ [121] which made lines b1 ¼ 1/2[111] and b3 ¼ 1/2[111] invisible owing to the gzb ¼ 0 condition. To access additional diffraction vectors, the specimen was tilted to a new zone axis (<[201]) making it possible to identify the Burgers vectors of two remaining dislocations in a similar manner: b2 ¼ 1/2[111] and b4 ¼ 1/2[111], respectively. Received 14 November 2005; accepted 14 February 2006. 1. Orowan, E. Zur kristallplastizitat. Z. Phys. 89, 605–-659 (1934). 2. Taylor, G. The mechanism of plastic deformation in crystal. Part I. Theoretical. Proc. R. Soc. A 145, 363–-404 (1934). 3. Polanyi, M. Uber eine Art Gutterstorung, die einen kristall plastich machen konnte. Z. Phys. 89, 660–-664 (1934). 4. Hirsch, P. B., Horne, R. W. & Whelan, M. J. Direct observations of the arrangement and motion of dislocations in aluminium. Phil. Mag. 1, 677–-684 (1956). 5. Hirth, J. P. & Lothe, J. Theory of Dislocations 2nd edn, 9–-13 (Wiley, New York, 1982). 6. Saada, G. Sur le durcissement duˆ a` recombinaison des dislocations. Acta Metall. 8, 841–-852 (1960). 7. Franciosi, P., Berveiller, M. & Zaoui, A. Latent hardening in copper and aluminium single crystals. Acta Metall. 28, 273–-283 (1980). 8. Guiu, F. & Pratt, P. L. The effect of orientation on the yielding and flow of molybdenum single crystals. Phys. Status Solidi 15, 539–-552 (1966). 9. Hirth, J. P. & Lothe, J. Theory of Dislocations 2nd edn, 269–-270 (Wiley, New York, 1982). 10. Basinski, S. J. & Basinski, Z. S. in Dislocations in Solids Vol. 4 (ed. Nabarro, F. R. N.) 261–-362, (North Holland, Amsterdam, 1979). 11. Nabarro, F. R. N., Basinski, Z. S. & Holt, D. B. The plasticity of pure single crystals. Adv. Phys. 50, 193–-233 (1964). 12. Madec, S. R., Devincre, B. & Kubin, L. P. From dislocation junctions to forest hardening. Phys. Rev. Lett. 89, 255508 (2002). 13. Ha¨hner, P. & Zaiser, M. Dislocation dynamics and work hardening of fractal dislocation cell structures. Mater. Sci. Eng. A 272, 443–-454 (1999). 14. Abraham, F. F. et al. Simulating materials failure by using up to one billion atoms and the world’s fastest computer: Work hardening. Proc. Natl. Acad. Sci. USA 99, 5783–-5787 (2002). 15. Frank, F. C. & Read, W. T. Multiplication processes for slow moving dislocations. Phys. Rev. 79, 722–-723 (1950). 16. Bulatov, V. V. et al. Scalable line dynamics in ParaDiS. Supercomputing khttp:// www.sc-conference.org/sc2004/schedule/pdfs/pap206.pdfl (2004). 17. Mughrabi, H. Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals. Acta Metall. 31, 1367–-1379 (1983). 18. Kubin, L. P. & Canova, G. R. The modelling of dislocation patterns. Scr. Metall. Mater. 27, 957–-962 (1992). 19. Ghoniem, N. M., Huang, J. M. & Wang, Z. Q. Affine covariant-contravariant vector forms for the elastic field of parametric dislocations in isotropic crystals. Phil. Mag. Lett. 82, 55–-63 (2002). 20. Finnis, M. W. & Sinclair, J. S. A simple empirical N-body potential for transition metals. Phil. Mag. A 50, 45–-55 (1984). Supplementary Information is linked to the online version of the paper at www.nature.com/nature. Acknowledgements This work was supported by the US DOE Office of Basic Energy Sciences and the NNSA ASC program. We thank E. Chandler and C. Mailhiot for their encouragement and unwavering support of the ParaDiS code development effort, D. Lassila for providing the purified molybdenum crystals, M. LeBlanc for performing the mechanical tests, R. Cook for graphic design and F. Abraham and G. Campbell for critical reading and editorial suggestions for the manuscript. This work was performed under the auspices of the US DOE by the Lawrence Livermore National Laboratory. Author Information Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to V.V.B. (bulatov1@llnl.gov). LETTERS NATURE|Vol 440|27 April 2006 1178