Defense Technical Information Center Compilation Part Notice ADP012144

Transcription

1 UNCLASSIFIED Defense Technical Information Center Compilation Part Notice ADP TITLE: Designability Graphitic Cones DISTRIBUTION: Approved for public release, distribution unlimited This paper is part following report: TITLE: Materials Research Society Symposium Proceedings. Volume 675. Nanotubes, Fullerenes, Nanostructured and Disordered Carbon. Symposium Held April 17-20, 2001, San Francisco, California, U.S.A. To order complete compilation report, use: ADA component part is provided here allow users access individually authored sections f proceedings, annals, symposia, etc. However, component should be considered within [he context overall compilation report and not as a stand-alone technical report. following component part numbers comprise compilation report: ADP thru ADP UNCLASSIFIED

2 Mat. Res. Soc. Symp. Proc. Vol Materials Research Society Designability Graphitic Cones M. M. J. Treacy and J. Kilian NEC Research Institute, Inc. 4 Independence Way, Princen, NJ , USA ABSTRACT We show that, with pologically flexible seeds which are allowed explore different growth modes, graphitic cones are inherently more "designable" than flat graphitic disks. designability a structure is number seed pologies encoding that structure. We illustrate designability with a simple model, where graphite grows on C,, (5<n<30) ring seeds. For a wide range ring sizes, cones are most likely pological outcome. Results from model agree well with data from special cone-rich carbon black samples. concept designability allows entropy be incorporated in "pentagon road" model formation curved graphitic structures. INTRODUCTION Recently, Krishnan et al. (1997) [1] reported a new carbon black material, made by Kvaerner Engineering a.s. Norway in a proprietary industrial process that involved pyrolysis methane in a plasma rch. Transmission electron microscopy (TEM) showed that sample (designated KVR) contained high concentrations multilayer graphitic disks, cones and tubes. Cones had been sighted before in carbon black [2], but not in quantity and variety exhibited by KVR. Five distinct cone angles were observed, each corresponding inclusion a disclination order 15m<55. measured cone angles, 0, were found correspond closely predicted values 0=2sin-l(1-m/6). Although graphitic disks dominated sample, among cones medium-angle 600 (m=3) and (m=4) cones occurred most frequently. From an energetics viewpoint, small-angled cones (those with sharpest cone points and largest m value) should have highest elastic strain energy per carbon am, particularly near tip area where cone wall curvature is high. Furrmore, re is an additional energy penalty for forming tip structure, which must enclose one or more disclinations. Although pentagon inclusion probably costs least energy on forming core disclination, defect structures with dangling bonds and different ring sizes are also possible. From consideration enthalpy formation alone, disk pology with no strain and no pentagons, must be lowest energy form, and 19.2' (m=5) cone and cylinder (m=6) must be highest energy forms, and refore rarest. An enthalpy argument would predict, in terms 0 values, order likelihood be 1800 > > > 600 > > > 0'. TEM shows, however, that 180' disks, and 600 and cones dominate KVR sample. reason must lie with differences in Gibb's free energy, AG 0 j = AfHl - TASy. As usual, AH 0 j is enthalpy formation when a chemical system transforms from state i statej, and T is temperature at which transformation occurs. entropy change ASU is a measure number reaction pathways available system i that can lead productj. If a chemical system i is inundated with opportunities form a higher enthalpy productj, n entropy can dominate rate product formation. W2.6.1

3 1 =0 ni= 3 A owi A3' m= 4 m=2 it.; 6 V. rb ýb n1 0 A---''- A i 2m4n I ",41 0 Figure 1: Illustration showing = how construct five graphite cones sheet. and a Multiples cylinder from 60 a segments hexagonal are systematically connected. removed, Each cone and can dangling be characterized bonds are re- by is number also known m as 60 segments disclination that are number. removed. m=i1,2,3,4,5 m correspond sheet, and true m=6 cones. cylinder m~=0 are considered planar here be end-members family cones. In this paper we present in more detail an entropy argument, first outlined in Krishnan et al. (1997) [ 1], that explains observed cone distribution. For simplicity, we focus discussion on monocyclic hydrocarbon rings as model nuclei for graphitic growth. We count number ways in which a carbon C,, ring can be embedded in a conical structure so as circumscribe cone tip. This count is "designability" structure [3]. We show that C,, rings (n>5, n#6) encode conical graphitic structures more prolifically than y do planar graphitic sheets. With pologically flexible seeds, cones are highly designable relative planar graphite. STRUCTURE OF GRAPHITE CONES To date, re have been several oretical conical papers shape discussing arises naturally cone pologies when [4, 600 segments 5]. severed. are cut edges out a planar ~ segment ~ _-;, graphite are sheet, rejoined. and A maximum ". Figure 1 shows how six this 60 process segments works. can be removed. removal physical one connection five 60 segments, reference points followed A by A', cone, and B tips B', results which in are five girdled distinct by types a continuous curved removal hexagonal six framework. 600 segments Topologically, does not equivalent necessarily lead a chain removal ams, which all when ams; connected it is identical rings end generates end, produce a cylinder. a ring. cylinder stacking a seven-membered and disk are family pological cones. end-members removal introduces (or addition) defects known 600 segments as disclinations. in a planar Cone lattice pologies number are 600 differentiated segments by that referring are removed. ams This is be disclination 3-connected, number cone tips m. must For all incorporate carbon ring introduce sizes or curvature than 6-rings. in a framework, Such rings sufficiently resulting in isolated rounded so cone that, tips. locally, Pentagons each pentagon can be Pentagons appears cannot be be a part arbitrarily an n=l1 (112.90) close each cone. or. applies isolated structures pentagon formed entirely rule (or from TPR), 5- which and 6-rings, stipulates that each pentagon must be W2.6.2

4 (a) (b) (c) (d) (e) () Figure 2: six pologically distinct open-ring C 1 8 circuits that occur in m=0 graphite. y can be represented by cyclic binary sequences, with 1 representing a bond "inwards" and 0 a bond "outwards". (c) and (d) are mirror images. re are 18 cyclic permutations each ring type, giving a tal 108 permutations. Cross-linked modes are excluded in this example. completely surrounded by hexagons. This rule can be justified by simple chemical arguments - extent sp 3 re-hybridization (in or words, non-coplanarity central am with its three bonds) must be minimized at each carbon am. Pentagons are generally considered be preferred "defect" ring size in hexagonal sp 2 - hybridized networks, as evidenced by ir abundance in Fullerenes. disclination parameter m is n equal number pentagons required form cone tip. CARBON n-ring MODES Consider a closed Cis circuit in a graphite sheet. Six distinct open-ring pologies are possible (Fig. 2), presenting a tal 6 l 8=108 cyclic configurations. A convenient method describing se circuits in planar graphite is label one free bond at each carbon that extends from ring. An outward-facing bond is assigned label "0", and an inward-facing bond label "1". se sequences can be permuted cyclically, anti-cyclically or mirrored without changing pology (except for handedness - for example Figs 2c and 2d). Upon removing se 18-carbon circuits from ir graphite matrix, "1" and "0" bonds become dangling bonds and are free move. In fact, ring is free change shape. Chemically, such a ring is free adopt one a huge number distinct chemical structures based on permutations single, double or triple bonds. We ignore such distinctions, since pologically rings are identical once y are embedded in graphite. For a cone or cylinder, distinction between a "0" and a "1" becomes clear by executing a clockwise path along circuit, beginning at any am. An am is assigned "0" when sense bond rotation in ring is clockwise, and "1" when rotation is counterclockwise. rules for growing a closed annulus graphite on perimeter a carbon ring (or circuit) are easily derived. For planar graphite formation, outer circuit graphitic annulus contains 12 additional ams. In general, for a disclination order m, relationship between size nk inner, and size nk 1 outer, circuits graphitic annulus is nk+l-nk=1 2-2 m. number inward-pointing bonds ik is related number outwardpointing bonds ok by ok - ik = 6 - m, with nk = ok + ik. re are 2" binary representations a C, ring. This number is greatly reduced by constraints on ok - ik. By examining binary form all integers between 0 and 2"-1, m-values can be obtained from m= 6 +ik-ok. A furr constraint is that ams may not collide, disallowing modes with more than five consecutive l's or O's. This last constraint greatly complicates problem enumerating valid rings. Here, we are interested only in 0:m:56 (i.e. cones), although or interesting m values also occur, such as saddles (m<0), closed shells such as buckyballs and nanotubes (m=12), and helicoids m>6. W2.6.3

5 S, * " e ", o o (a) (b) (d) Figure 3: a) A flexible C 18 H 36 hydrocarbon ring. b) and c) are two growth modes graphite on outside ring, each producing an m=2 disclination. Pentagons have yet appear, yet disclination is essentially fixed. d) Cross-linking in a C, 8 ring. Internal linkages "1" can not create an odd number pentagons. Outside linkages "0" can. Figures 3b and 3c show two 16,812 possible ways that 6-rings can be added a C 18 ring form a graphitic annulus corresponding an m=n2 cone. In each instance, tip structure is not yet resolved, and pentagons have not yet been formed. However, enclosed disclination is now fixed because stiffened annulus has already lost many its conformational degrees freedom. stiffened seed is now committed a particular value m. Tip closure, with pentagon creation, will occur early in growth stage, but after nucleation disclination. Table I lists number unique degrees freedom available an open n-ring (5<n<30) for each seven disclinations, 0<m56. We assume that a minimum ring size is required for each disclination. 6, 5, 12, 15, 18, 19 and 18 are minimum circuit sizes that can circumscribe m=o 6 unconnected pentagons in a manner consistent with isolated pentagon rule. Table shows that for smallest open ring sizes, n, small rn-values are favored. In this n n , , , , , , , , , , , , , , , , , , , , , , , ,173, , , ,618, ,464, , ,012, ,790, ,090, ,225, ,260, ,925, , ,595, ,230, ,142, ,381, ,068, ,476, , ,563, ,431, ,902,092 Table 1: number modes an open carbon ring n ams that correspond a disclination order 0<m<6. n-rings are not necessarily enclosing pentagons at this stage. Tip restructuring can occur after disclination is committed during growth phase. W2.6.4

6 open ring model, even-numbered rings n can only nucleate even-numbered disclinations m. Conversely, odd-numbered rings can only nucleate odd-numbered disclinations. With increasing n (n< 17), degrees freedom are maximum for m = 1, 2, 3 disclination values. For n>l 8, distribution peaks for m>3. Table also shows that large, pologically flexible, carbon rings favor nucleation cones rar than sheets. Cones with higher enthalpies formation are favored because Gibbs free energy is reduced by increase in entropy associated with large number growth modes that encode cones. More sophisticated ring-counting techniques allow us count cross-linked modes more efficiently. re is no room here describe our algorithm or results. most significant result is that cross-linking "0" bonds permits even n-rings create odd-m disclinations, and viceversa (e.g. Fig 3d). inclusion cross-linked seeds does not change significantly relative distribution modes given in Table 1. Data on KVR sample [1] reveal a bimodal distribution cones, with a broad peak in m=2,3,4 region, and a dominant peak at m=o, planar graphitic disks. Cylinders (m=6) occur relatively infrequently. This distribution could be explained by presence a relatively dilute concentration C. rings in an abundant supply graphitic carbon building units (i.e. C monomers, C 2 dimers etc). Self-assembly smaller carbon building units will inevitably favor formation graphite, which is lowest energy state for sp 2 carbon. This growth mode should produce prolific quantities graphitic sheets. A secondary growth mode involves accretion smaller carbon building units on C, rings. This latter growth mode clearly favors cones when n<30. re is a third possible growth mode where rings collide with rings. This would be equivalent ring-stacking model Wakabayashi and Achiba (1992) [6]. However, if monocyclic rings are dilute, this growth mode will not dominate. spheroidal particles commonly found in carbon black are suppressed in KVR sample. This may be because ir seeds are rapidly consumed by cone production. Ebbesen [7] has pointed out that growth phase is probably much more rapid than nucleation phase, since seed formation is a slow and difficult process. A high purity and yield cones can occur if carbon vapor/plasma is presented with ready-made flexible seeds or nuclei that possess many pological degrees freedom. Unfortunately, KVR synsis conditions are unknown. Clearly, re may be more than one nucleation mechanism present. 5-rings, as well as 6- rings and larger n-rings, are likely be present in plasma/vapor. 5-rings, when present at high concentration, could accrete form cup-like structures. KVR sample does show some m=2 cones in which pentagons are separated by distances more than a micron, implying that two m=l cones have merged. However, accretion alone can not so clearly explain observed distribution angles in cones with relatively sharp tips (i.e. pentagons closely spaced). Figure 3 shows distribution in m-values for Gaussian distributions seed ring sizes, 40 <n> =12 <n>=18. <n> 24 = 30 e 20 ~10 1. m m Figure 4: Distributions m values resulting when a graphite-forming system is seeding with Gaussian distributions monocyclic carbon rings, peaked at C 12, C, 8 and C 24 rings respectively. standard deviation is o, = 3 in each case. W2.6.5

7 <n>=12, 18 and 24, with e, = 3 in each case. Each distribution strongly favors cones, 1< m <5, with peak shifting higher m as <n> increases. A hisgram for <n>=20, o7,, = 3 (not shown), which peaks at m=3, is qualitatively similar that observed for KVR sample. re is negligible production m=o disks in se hisgrams. As explained above, disks can be formed in a separate nucleation process, with no large rings necessarily involved. validity designability argument does not rely on assumption that monocyclic carbon rings are seeds. Any flexible seed that is allowed explore its pological degrees freedom may suffice. We note that similar "designability" arguments, applied peptide chains, have been used explain rmodynamic stability certain folded protein structures [3]. m<o hyperboloidal structures, with negative Gaussian curvature, are not observed in KVR sample, even though y are abundantly present as potential C, ring modes. This appears argue against carbon rings as seeds - y are o flexible. However, it is possible that rmal vibrations will destroy hyperboloidal structures, once seeded. In cones, rmal vibrations will resemble bell modes with highest stresses near open cone rim. cone tip will not be subjected large stresses. However, in saddle-shaped hyperboloids, dominant vibration modes will be "clapping", or "butterfly wing", modes. Such vibrations may subject high curvature cores disclinations large tensile stresses, ripping such hyperboloids apart. In addition, more steeply inclined -65m<-4 structures may cross-link at ir open rims (literally clapped shut), leaving an unrecognizable mass. m>o cones are expected be relatively stable rmal vibrations, which may explain ir survival in KVR sample. A consequence our designability model is that pentagon creation becomes a structural tidying-up step that happens after disclination is committed. In "pentagon road" model [8], pentagons are formed in seed structures in order eliminate high-energy dangling bonds, and as an annealing mechanism reduce overall energy structure. In that model, creation pentagon implies creation disclination. designability argument and pentagon road model are not in conflict. designability mechanism implicitly refers an earlier instant in seed's existence, when it still possesses pological degrees freedom. Disclinations are inherently present as conformational modes, but are not yet committed. Pentagons have yet be formed. In a pologically flexible seed pentagons appear "along road", rar than at beginning. number pentagons in later seed structure is predetermined by mode committed by seed at an earlier stage growth. pentagon road model provides a mechanism for minimizing enthalpy seed structures. By incorporating concept designability, pentagon road model can be extended include entropy, which drives emergence preferred graphitic structures. ACKNOWLEDGMENTS We wish thank Thomas Ebbesen and Ned Wingreen for stimulating discussions. 1. A. Krishnan, E. Dujardin, M.M.J. Treacy, J. Hugdahl, S. Lynum, and T.W. Ebbesen, Nature 388, 451 (1997). 2. M. Ge and K. Sattler, Chem. Phys. Letts. 220, 192 (1994). 3. H. Li, R. Helling, C. Tang, and N. Wingreen, Science 273, 666 (1996). 4. A.T. Balaban, D.J. Klein, and X. Liu, Carbon 32, 357 (1994). 5. S. Ihara, S. Ih, K. Akagi, R. Tamura, and M. Tsukuda, Phys. Rev. B 54, (1996). 6. T. Wakabayashi and Y. Achiba, Chem. Phys. Letts. 190, 465 (1992). 7. T.W. Ebbesen, Personal communication (1997). 8. R.E. Smalley, Acc. Chem. Res. 25, 98 (1992). W2.6.6