How Grain Growth Stops: A Mechanism for Grain-Growth Stagnation in Pure Materials

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  • DOI: 10.1126/science.1187833, 1138 (2010);328 Science

    Elizabeth A. Holm and Stephen M. Foilesin Pure MaterialsHow Grain Growth Stops: A Mechanism for Grain-Growth Stagnation

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    CopyrightAmerican Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by theScience

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  • cidence angle. The calculated scattering spectraof a heptamer for different polarization angles areshown in Fig. 3C, where the cluster geometry isidentical to that used in Fig. 3A. These spectradisplay Fano minima at 1450 nm, with asym-metric line shapes that match the experimentalspectra. The nanoshell separationmodeled here issmaller than that used for the trimer calculationsto account for the strongly red-shifted Fano min-imum. This red shift is probably due to a combina-tion of at least three factors: (i) smaller nanoshellseparation due to inhomogeneous self-assembledmonolayer coverage, (ii) a higherrefractive-indexenvironment near the cluster due to excess poly-mer deposition, and (iii) increased capacitivecoupling between the nanoparticles due to nano-shell faceting.

    This cluster concept can be generalized toother functional 2D and 3D structures. One ex-ample is the tetrahedral cluster, which supportsisotropic electric and magnetic resonances inthree dimensions (28) and can be used as a build-ing block for isotropic metamaterials. Symmetrybreaking can be applied to engineer other typesof optical modes: Trimers comprising three dif-ferent particle types support magnetoelectricmodes, and tetrahedral clusters comprising fourdifferent particle types are chiral. Nonsphericalplasmonic particles can be used to construct moreelaborate structures, provided that their orienta-tions can be controlled during assembly. In allof these structures, resonances can be tuned byvarying individual particle geometries, interpar-ticle separation, and the dielectric environmentof the cluster. The assembly of clusters fromsolution is highly versatile: It can lead to liquidmetamaterials or metafluids (28), be integratedinto soft materials such as gels, or be encapsu-

    lated and dried onto surfaces of arbitrary curva-ture or patterning. Future work will focus on theseapplications and on achieving higher cluster yieldscomparable to those attained with lithographical-ly defined patterns (29), emulsion droplets (30),and DNA linking (31).

    References and Notes1. W. L. Barnes, A. Dereux, T. W. Ebbesen, Nature 424, 824

    (2003).2. V. M. Shalaev, Nat. Photonics 1, 41 (2007).3. V. M. Shalaev et al., Opt. Lett. 30, 3356 (2005).4. J. B. Pendry, D. Schurig, D. R. Smith, Science 312, 1780

    (2006); published online 25 May 2006 (10.1126/science.1125907).

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    10. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart,IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).

    11. T. J. Yen et al., Science 303, 1494 (2004).12. N. Verellen et al., Nano Lett. 9, 1663 (2009).13. N. Liu et al., Nat. Mater. 8, 758 (2009).14. K. J. Stebe, E. Lewandowski, M. Ghosh, Science 325, 159

    (2009).15. T. Ming et al., Angew. Chem. Int. Ed. 47, 9685

    (2008).16. J. H. Lee, Q. Wu, W. Park, Opt. Lett. 34, 443 (2009).17. M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin,

    M. Mojahedi, Phys. Rev. B 79, 073103 (2009).18. E. Prodan, C. Radloff, N. J. Halas, P. Nordlander,

    Science 302, 419 (2003).19. S. J. Oldenburg, R. D. Averitt, S. L. Westcott, N. J. Halas,

    Chem. Phys. Lett. 288, 243 (1998).20. J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo,

    G. M. Whitesides, Chem. Rev. 105, 1103 (2005).21. J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz,

    S. Schultz, J. Chem. Phys. 116, 6755 (2002).22. C. Enkrich et al., Phys. Rev. Lett. 95, 203901 (2005).23. Y. A. Urzhumov, G. Shvets, Solid State Commun. 146,

    208 (2008).

    24. D. W. Brandl, N. A. Mirin, P. Nordlander, J. Phys. Chem. B110, 12302 (2006).

    25. E. Plum et al., Phys. Rev. Lett. 102, 113902 (2009).26. Random symmetry breaking can be experimentally

    addressed by using smoother thick-shelled nanoparticles,assembling nanoparticles with atomically smooth facetssuch as crystalline nanocubes, or assembling clusters withlarger gap sizes, which would effectively reduce gapgeometry variation (but at the expense of total modestrength).

    27. N. A. Mirin, K. Bao, P. Nordlander, J. Phys. Chem. A 113,4028 (2009).

    28. Y. A. Urzhumov et al., Opt. Express 15, 14129 (2007).29. Y. D. Yin, Y. Lu, B. Gates, Y. N. Xia, J. Am. Chem. Soc.

    123, 8718 (2001).30. V. N. Manoharan, M. T. Elsesser, D. J. Pine, Science 301,

    483 (2003).31. C. J. Loweth, W. B. Caldwell, X. G. Peng, A. P. Alivisatos,

    P. G. Schultz, Angew. Chem. Int. Ed. 38, 1808 (1999).32. J.A.F., F.C., C.W., and G.S. acknowledge funding by the

    NSF Nanoscale Interdisciplinary Research Team undergrant no. ECCS-0709323; G.S. and C.W. acknowledgefunding by Air Force Office of Scientific Research (AFOSR)Multidisciplinary University Research Initiative grantsFA9550-06-1-0279 and FA9550-08-1-0394; R.B, N.J.H.,and P.N. acknowledge support from the U.S. Departmentof Defense National Security Science and EngineeringFaculty Fellowship program, the Robert A. WelchFoundation (C-1220 and C-1222), AFOSR grantF49620-03-C-0068, the SUG@R (Shared UniversityGrid at Rice) team, and the Center for Advanced SolarPhotophysics, a U.S. Department of Energy EnergyFrontier Research Center. Electron microscopy wasperformed at the Center for Nanoscale Science at HarvardUniversity, a member of the National NanotechnologyInfrastructure Network. J.A.F. acknowledges R. Guerra forhelpful discussions and D. Bell for EM support.

    Supporting Online Materialwww.sciencemag.org/cgi/content/full/328/5982/1135/DC1Materials and MethodsSOM TextFigs. S1 to S6References

    4 February 2010; accepted 22 April 201010.1126/science.1187949

    How Grain Growth Stops: AMechanism for Grain-GrowthStagnation in Pure MaterialsElizabeth A. Holm* and Stephen M. Foiles

    The thermodynamic equilibrium state of crystalline materials is a single crystal; however,polycrystalline grain growth almost always stops before this state is reached. Although typicallyattributed to solute drag, grain-growth stagnation occurs, even in high-purity materials. Recentstudies indicate that grain boundaries undergo thermal roughening associated with an abruptmobility change, so that at typical annealing temperatures, polycrystals will contain both smooth(slow) and rough (fast) boundaries. Mesoscale grain-growth models, validated by large-scalepolycrystalline molecular dynamics simulations, show that even small fractions of smooth, slowboundaries can stop grain growth. We conclude that grain-boundary roughening provides analternate stagnation mechanism that applies even to high-purity materials.

    Most metals and ceramics are polycrys-talline: They are made up of manyindividual crystallites, called grains,separated by internal interfaces or grain bounda-ries. When polycrystalline materials are annealed

    at sufficiently high temperatures, grain bounda-ries move and rearrange so as to increase theaverage grain size and decrease the grain-boundary area per unit volume. However, evenat very high temperatures, grain growth only

    rarely proceeds to the equilibrium single-crystalstate. Instead, grain growth usually stops, thoughsubstantial internal interface remains. In fact,grain stagnation is so pervasive that most grain-growth models presume a finite maximum grainsize based purely on empirical observations (1).

    Understanding and controlling grain growthis important to nearly every engineered material.For materials that rely on strength, toughness orformability, including most nanocrystallinematerials, a stable, fine grain size is desirable.However, there are also important systems, suchas superalloy turbine blades and silicon photo-voltaics, in which a large (or even single-crystal)grain size is preferred.

    There have been many grain-growth stagna-tion mechanisms proposed, each valid in certainregimes. Well-known processes that reduce thedriving force for grain growth sufficiently to

    ComputationalMaterials Science and EngineeringDepartment,Sandia National Laboratories, Albuquerque, NM 871851411,USA.

    *To whom correspondence should be addressed. E-mail:eaholm@sandia.gov

    28 MAY 2010 VOL 328 SCIENCE www.sciencemag.org1138

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  • produce a metastable polycrystal include pinningby dispersed particles (2), pinning by thermalgrooves (3), and the film-thickness effect (4).These occur only in films or in systems contain-ing an immobile second phase. In other mate-rials, grain-growth stagnation is often attributedto the segregation of solute species to the grainboundary. Solutes inhibit grain growth by requir-ing the boundary tomove in concert with its solutecloud (59) or by directly changing boundary-migration mechanisms (10, 11). For most solutesand temperatures of interest, this substantiallyslows grain-boundary motion and, hence, alsoslows grain growth. However, grain-growth stag-nation is also observed in systems in which solutesshould not limit growth, including high-puritymaterials (12, 13), nanocrystalline materials thatshould be in the solute breakaway regime (14, 15),and materials at high homologous temperatureswhere solute diffusivity should be high (15). It isreasonable to suppose that there might be anotheractive grain-growth stagnation mechanism in thesesystems.

    We recently calculated grain-boundary mo-bility, which scales how fast a boundary canmove in response to a driving force, for a catalogof 388 grain boundaries in Ni using a syntheticdriving force molecular dynamics (MD) method(16, 17). At a given temperature, mobility fallsinto two ranges: (i) high-mobility (fast) bounda-ries and (ii) low-mobility (slow) boundaries,whose mobility is too small for us to measurevia MD. Typical high-mobility boundaries movecontinuously with well-defined activation ener-gies and atomically rough-boundary structures.On the other hand, low-mobility boundariesmove in a jerky, stepwise manner with differentactivation energies and atomically smooth bound-ary structures (18). The boundary structurechanges from smooth/low-mobility to rough/high-mobility at a characteristic temperature, Tr

    (the roughening temperature), which can vary byhundreds of degrees from boundary to boundary,as discussed byOlmsted et al. (17). Figure 1 showsthe distribution of roughening temperatures for ourcatalog of Ni boundaries: At low temperature,most boundaries are smooth and slow, but evenat 0.9 Tm (where Tm is the melting temperature),~10% of the boundaries remain smooth and slow.

    It is important to note that the rougheningtransition is not related to faceting, in whichcertain crystallographically favored grain bound-aries decompose into flat facets that lie alonglow-energy inclinations (1921). Because facetingresults in both high and low grain-boundarymobilities (22, 23), it is not generally implicatedin grain-growth stagnation (24). In contrast, theroughening transition occurs in all boundaries, notjust those that are thermodynamically inclined tofacet, and the structure of smooth boundaries doesnot include crystallographic facets (18). Smoothboundaries generally have very lowmobility,whichcould inhibit grain growth. Because faceting androughening are distinct phenomena, it is possiblethat a single grain boundary could undergo bothtransitions at different temperatures; however, thisreport concerns only the microstructural effects ofgrain-boundary roughening and its associatedmobility transition.

    To determine the effect of boundary rougheningon grain growth in a polycrystal, we incorporatethe boundary mobility and roughening data in

    Fig. 1 into a mesoscale simulation of polycrystal-line microstructural evolution (2528). When allboundaries are rough/high-mobility, we find thatgrain growth proceeds to the equilibrium single-crystal state. In contrast, in simulations that in-clude smooth/low-mobility boundaries, graingrowth slows dramatically at a finite grain sizethat decreases as the fraction of smooth bounda-ries, f0, increases, as shown in Fig. 2A. It is notsurprising that immobile boundaries can slow orstop grain growth; however, it is unexpected thatthe grain structure stagnates at such small smoothboundary fractions, because themotion ofmobileboundaries can sweep out immobile boundaries.Our results show that in a polycrystal, grainboundaries form an interconnected network thatmay be pinned, even when many individualboundaries remain unpinned. This observationapplies, not only to pinning by smooth bounda-ries, but also to other stagnation mechanisms. Forexample, particle pinning theories disagree onwhether all or merely a fraction of boundariesmust be pinned by intersecting particles for thegrain structure to be stabilized (29, 30). Thepresent results indicate that pinning a fraction ofboundaries is sufficient.

    The stagnant grain size relative to the initialgrain size (that is, Dp/D0) decreases as a powerlaw in f0, as shown in Fig. 2B. By combiningFig. 1 and Fig. 2B, we can determine the stagnantgrain size as a function of temperature, as shown

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    Fig. 1. Distribution of grain-boundary roughening-transition temperatures for several hundred grainboundaries in Ni calculated by MD simulations (17).The width of the bars corresponds to the temper-ature bin w...