Dynamic pathway model for the formation of C 60In-Ho Lee, Hanchul Kim, and Jooyoung Lee Citation: The Journal of Chemical Physics 120, 4672 (2004); doi: 10.1063/1.1645776 View online: http://dx.doi.org/10.1063/1.1645776 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/120/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Solvation of fullerene and fulleride ion in liquid ammonia: Structure and dynamics of the solvation shells J. Chem. Phys. 137, 134501 (2012); 10.1063/1.4754852 Bonding behavior and thermal stability of C 54 Si 6 : A first-principles molecular dynamics study J. Chem. Phys. 122, 084304 (2005); 10.1063/1.1844315 Molecular dynamics study of the Ag 6 cluster using an ab initio many-body model potential J. Chem. Phys. 109, 2176 (1998); 10.1063/1.476851 Detection and separation of radioactive fullerene families by radiochemical techniques AIP Conf. Proc. 416, 261 (1997); 10.1063/1.54558 A molecular dynamics study of impurity desorption from solid clusters of rigid C60 molecules J. Chem. Phys. 106, 6475 (1997); 10.1063/1.473613

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Dynamic pathway model for the formation of C 60

In-Ho Leea) and Hanchul Kimb)

Korea Research Institute of Standards and Science, Daejon 305-600, Korea

Jooyoung Leec)

Korea Institute for Advanced Study, Seoul 130-012, Korea

~Received 19 November 2003; accepted 12 December 2003!

We present a dynamic pathway model for the formation of C60 using the action-derived moleculardynamics simulations. We propose candidate precursors for dynamic pathway models in whichcarbons spontaneously aggregate due to favorable energetics and kinetics. Various planar polycyclicmodels are in a disadvantageous state where they cannot be trapped in the forward reaction due totheir high excess internal energies. Our simulation results show that precursors either in the shapeof tangled polycyclics or in the shape of open cages are kinetically favored over precursors in theshape of planar hexagonal graphite fragments. Calculated activation energies for the probableprecursor models are in good agreement with experiment. Existence of chains in the models oftangled polycyclics and open cages is beneficially for the formation of C60 molecule. Chainsattached to the precursor model are energetically favorable and display lithe movements along thedynamic pathway. ©2004 American Institute of Physics.@DOI: 10.1063/1.1645776#

The C60 molecule, nicknamed Buckyball, is a hollowsphere 0.71 nm in diameter made of 20 hexagons~single anddouble bonds! and 12 pentagons~single bonds!, arranged asa truncated icosahedron. The discovery of this third allotro-pic form of carbons uncovered a fundamentally differentstructure of closed carbon cages.1 This new family of non-planar carbon compound has induced extensive studies fromwide scientific communities. Currently, many C60 derivativeshave been found; multishell fullerenes,2 endohedralfullerenes,3 nanotubes,4 and nano-peapods.5

Mass production of the C60 molecule has a significantimpact on broad research areas with many applications. Tolist several of them, they are carbon chemistry in the galaxy,6

molecular-size ball bearings,7 superconductors,7

photoconductors,7,8 light-sensitive optical sensors,9 andatomic encapsulation cages.7 In addition, C60 molecules arepotential precursors in many plasma chemistry and chemicalvapor deposition processes.10 There are four recipes for thesynthesis of C60: graphite vaporization,11 resistive heating ofgraphite,12 graphite arching,13 and pyrolysis~decompositionby heating!.14 Among them, the pyrolysis method is uniquein that the starting material is not graphite, but hydrocarbons.

To the best of our knowledge, there exists no micro-scopic understanding on the dynamic process of C60 forma-tion, nor any observation on the structure forming mecha-nism in experiments. Despite many experimental andtheoretical investigations on the energetics of carbon clus-ters, the kinetic mechanism of C60 formation requires betterunderstanding.7,15Moreover, in the conventional microscopickinetic models, it is difficult to observe the way carbon at-oms move on a complicated interatomic potential energy sur-

face. In particular, the C60 formation requires an astronomi-cally long time scale compared to the time for the vibrationof an individual carbon atom. Thermodynamic informationhas little to offer on the time evolution of the atomic posi-tions that would initiate the dome closure of the C60 mol-ecule. For these reasons, understanding the microscopicmechanism of the C60 formation demands intensive researchefforts. Direct information on the precursor models and tran-sition pathways associated with C60 formation can be used tomodel various activated processes in carbon-based nanoma-terials.

In this paper, we present globally convergent atomic tra-jectories for the formation of C60, providing a microscopickinetic model for the structural transformation. We optimizethe dynamic pathway from an initial to the final atomic con-figuration by minimizing the discretized classical action.16,17

This is a new computational method, derived from the leastaction principle, for investigating transition pathway con-structions of rare-event processes. We examine existing pre-cursor models in the literature15,18–21and propose candidatesfor dynamic pathway models in which carbons spontane-ously aggregate due to favorable energetics and kinetic fac-tors.

The energy of the carbon system is calculated employingthe tight-binding method that has been successfully appliedto carbon-based systems.22 The transferability of this poten-tial has been shown to be adequate to describe various C–Cinteractions, including the rehybridization of defect sites.22

The tight-binding method rather thanab initio method isdeliberately chosen: The required computational demand isquite intensive but manageable, and the errors in bond ener-gies from the tight-binding calculations are typically small.For example, the tight-binding method gives bond lengths of1.40 and 1.46 Å for the double and single bonds, respec-tively, in the fully relaxed C60 molecule. These results agree

a!Electronic mail: ihlee@kriss.re.krb!Electronic mail: hanchul@kriss.re.krc!Electronic mail: jlee@kias.re.kr

JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 10 8 MARCH 2004

46720021-9606/2004/120(10)/4672/5/$22.00 © 2004 American Institute of Physics

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very well with those from the first-principles local densityapproximation calculations of 1.39 and 1.45 Å for doubleand single bonds, respectively.

Recently, Passerone and Parrinello proposed the action-derived molecular dynamics~ADMD ! method that explicitlydetermines the dynamic trajectory of an atomic system forgiven initial and final configurations with a chosen simula-tion time t.16,17 The ADMD method is shown to be usefulespecially for the study of rare-events of atomic systems.Practical implementation of the algorithm requires the evalu-ation of the potential energy and the resulting atomic forcesas in ordinary molecular dynamics simulations. By carryingout the ADMD simulation we estimate the possibility ofspontaneous formation of C60 starting from various precursorconfigurations.

We consider classical atomic trajectories,$Rj% of anN-atom conservative system with given potential energy,V($R%). The time step index and the discretized time inter-val are represented byj (50,1,2,...,P) and D(5t/P),respectively. The trajectories start at the point$Rj 50%(5$R(0)%) at time t50 and finish at the point$Rj 5P%(5$R(t)%) at time t5t. A direct application of theleast action principle is limited in practice due to the fact thatthe nature of the stationary points depends on the choice ofD.16 Passerone and Parrinello proposed an action,Q($Rj%,E) to generate an atomic trajectory for given sets ofinitial, $R~0!%, and final, $R~t!%, configurations.16 The dy-namical trajectory,$Rj%, can be obtained by minimizing thefollowing discretized action, mimicking the least action prin-ciple in the classical mechanics,

Q~$Rj%,E!5D (j 50

P21 H (I 51

NMI

2D2 ~RjI2Rj 11

I !22V~$Rj%!J1m (

j 50

P21

~Ej2E!2. ~1!

Here the first term stands for a discretized classical actionS(5*0

tdt$( I 51N (MI /2) (RI)22V($R%)%) of an N-atom sys-

tem interacting by a given potentialV($R%). The initial andfinal configurations are fixed during the minimization of theaction Q. The number of intermediate configurations be-tween the two end points is set toP21, andMI is the atomicmass of atomI . The instantaneous total energy,Ej at timestep j is defined by16 Ej5( I 51

N (MI /2D2) (RjI2Rj 11

I )2

1V($Rj%). The second part of the actionQ is a penaltyterm, and the role of the parameterm(.0) is to control thevalue of the total energy of the system close to the targetenergyE along the path.16 In the ADMD simulation, it isworthy to check the quality of the trajectory based on the factthat the trajectory satisfies the Newton’s equation of motion,i.e., it follows closely the Verlet trajectory.16,23–25The qualityof a trajectory is often measured24 by the value of theOnsager–Machlup action,

O5(I 51

N

(j 51

P21 S 2RjI2Rj 21

I 2Rj 11I 2

D2

MI

]V~$Rj%!

]RjI D 2

.

~2!

The trajectory ofO50, i.e., all the arguments in the paren-theses become zero, is known as the Verlet trajectory.25 Thesmaller the value ofO is, the more closely the correspondingtrajectory follows the Verlet trajectory.

In the present ADMD simulation, we use an extendedaction17 that has an additional penalty term to Passerone andParrinello’s action.16 The resulting extended actionQ be-comes

Q~$Rj%,E;T!5Q~$Rj%,E!1n(I 51

N S ^KI&23kBT

2 D 2

, ~3!

where,KI5MI(RjI2Rj 11

I )2/(2D2) is the instantaneous ki-netic energy of the atomI at time stepj , ^KI& is its time-average value over the entire trajectory,T is the target tem-perature of the system which we set, andkB is theBoltzmann’s constant.17 The parametern(.0) controls thestrength of the kinetic energy enforcement to the value cor-responding to the target temperature. It should be noted thatthe target temperatureT is nothing but a computational pa-rameter to control the time-averaged kinetic energy inADMD simulations, and we do not intend to interpret it as aphysical quantity.17 The quality of atomic trajectories ob-tained by the extended ADMD method is significantly im-proved in terms of the smaller value of Onsager–Machlupaction compared to that from Passerone and Parrinello’sADMD method.17

Direct application of the conjugate gradient likemethod26 to a trajectory representation containing high fre-quency components is not an efficient approach for actionminimization since slowly varying low frequency compo-nents of the trial trajectory do not converge well. Therefore,when performing path relaxation, it is convenient to startwith a path containing low frequency components only andmove on by gradually adding higher frequency componentsto the path to accelerate the convergence.16,27

To start the atomic trajectory calculation in ADMD, wehave to choose both initial and final configurations. Since oursystem consists of 60 identical carbons, it is not trivial toassign each carbon atom of initial configuration to its appro-priate position in the final configuration. We use the least-squares superposition of two atomic coordinate sets viaquaternion method.28 The first coordinate set is fixed whilethe second one is translated and rotated to get the best super-position. In more complex cases, we introduce an atomicindex exchange process before carrying out the ADMDsimulation.17 To carry out systematic atomic index exchangeprocesses, we utilize the simulated annealing method.29

We have tested various precursor models, such as non-planar polycyclics, open cages, and planar polycyclic mod-els. For the latter, we considered monocyclic, dicyclic, tricy-clic, tetracyclic, pentacyclic, hexacyclic, and heptacyclicconfigurations as shown in Fig. 1. In experiments, in therange of 10–70 carbons, these types of clusters are dominantisomers.19,20 Furthermore, experimental evidence indicatesthat carbon cluster growth occur via coalescence of ionizedmedium-size carbon clusters.15,30 In order to construct afore-mentioned precursor models we first placed 60 carbons as anappropriate interaction range to form a designed ring-shape

4673J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 The formation of C60

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cluster and then we relaxed the carbons to settle into theirlocal-energy minima. By taking each precursor model as aninitial state~the final state being always the Buckyball con-figuration!, we performed ADMD simulations to investigatethe dynamic pathway of the C60 formation. The activationenergy barrier (Ea) is defined by the potential energy differ-ence between the highest potential energy along the pathwayand potential energy of the precursor model. Calculated en-ergy differences (DE) between precursors and the target C60

molecule, and the activation energy barrier (Ea) are shownin Table I. For each pathway model obtained through ADMD

simulation the corresponding potential energy evolution isalso shown in Fig. 2.

First, we consider the planar hexagonal sheet shown inFig. 1~a! as a precursor of the C60,18 which can serve as areference configuration to curl the cage structure of fullerene.This model is found to require a high activation energy bar-rier of 15.1 eV. The creation of the twelve pentagons fromthe planar hexagonal lattice requires a significant amount ofkinetic energy in order to overcome the energy barrier be-tween the two structural configurations. In fact, this precur-sor model contradicts the experimental observation that C60

molecules are formed starting from a few carbon fragments.The isotope (12C and 13C) scrambling experiments clearlysupport the gradual addition of carbon atoms.31 In addition,the graphitic fullerene precursors have not been observedfrom the condensation of carbon clusters.32

Much experimental evidence indicates that fullerenescan be produced by the coalescence of carbon rings ratherthan the gradual addition of small-size clusters such as C2

and C3.15,30 Based on this experimental evidence, Jarroldand co-workers20 proposed that the main mechanism of thecarbon molecule growth comes from the reaction betweenmidsize rings. The reaction between planar rings creates newplanar polycyclic compounds. Thus, it is worthwhile investi-gating the activation energy of the planar type precursormodels. We performed ADMD simulations starting from thisdicyclic precursor@Fig. 1~b!# and found that its activationenergy barrier is 8.2 eV. A similar simulation gives 12.7 eVfor the activation energy barrier of a monocyclic structure@Fig. 1~c!#. The rate of the C60 formation via thermal activa-tion can be estimated bynvib exp(2Ea /kBT), whereEa is thecalculated activation energy barrier andnvib51012 s21 is theattempt rate. For the barrier of 8 eV, the estimated rate is toolow for significant C60 molecule formation atT;2000 K. Asshown in Table I, the planar type of tricyclic33 or tetracyclicconfiguration requires activation energy barriers in the rangeof 8.1–9.4 eV along the calculated dynamic pathways in theC60 formation. Thus, our ADMD simulations show thataforementioned precursor models are kinetically well sepa-rated from the C60 molecule. Especially for the monocyclicand dicyclic precursor models, as noted earlier, our activa-tion energy barrier estimations are consistent with the experi-mental fact that the monocyclic and dicyclic rings do notappear to convert into the fullerene.15,20 Through the experi-mental study of annealing and dissociation of carbon rings,Jarrold and co-workers20 showed that the conversion of themonocyclic ring to the fullerene is very inefficient comparedto the process of the monocyclic ring dissociation. A secondfragmentation of the monocyclic ring with an energy injec-tion is a probable route for the fullerene formation.20 In ad-dition, it has been shown that the more rings the initial con-formation has, the smaller the activation barrier is totransform to a fullerene.15 This experimental finding is alsoin broad agreement with the present activation energy esti-mations as shown in Table I.

In order to see the variation of activation energy barriersin the model of the coalescence of carbon rings, it is useful toinvestigate other types of precursor models. Pentacyclic,hexacyclic, and heptacyclic planar precursor models in Figs.

FIG. 1. Twelve precursor models described in the text are shown:~a! hex-agonal sheet,~b! dicyclic, ~c! monocyclic,~d! tricyclic, ~e! tetracyclic,~f!pentacyclic, ~g! hexacyclic, ~h! heptacyclic, ~i! tangled polycyclic I,~j!tangled polycyclic II~k! open cage I, and~l! open cage II. The energy ofeach precursor structure is locally minimized. The energy of the system iscalculated using the tight-binding method~Ref. 22!.

TABLE I. Calculated energy differences (DE) between the target C60 mol-ecule and precursor molecules, and values of the activation energy barrier(Ea) associated with the corresponding formation processes are shown forvarious precursor models. The energy of the system is calculated using thetight-binding method~Ref. 22!. Each nonplanar precursor model was pre-pared by a preliminary simulated annealing run~Refs. 29 and 35! startingfrom a random configuration. The energy of each precursor structure islocally minimized. Four nonplanar precursor models shown in Figs. 1~i!–1~l!, were prepared by incomplete simulated annealing procedures of therandomly chosen structures on condition ofDE,46 eV. The estimated for-mation rates at temperature in the rangeT51000– 2000 K.

Precursor model DE ~eV! Ea ~eV! Formation rate (s21)

Planar formSheet 30.64 15.1Monocyclic 49.25 12.7Dicyclic 50.39 8.2Tricyclic 51.95 8.1Tetracyclic 49.91 9.4Pentacyclic 54.79 4.6Hexacyclic 57.81 3.7Heptacyclic 62.51 0.3

Nonplanar formTangled polycyclic I 45.72 3.3 231025– 53103

Tangled polycyclic II 41.63 1.0 93106– 33109

Open cage I 41.94 2.0 83101– 93106

Open cage II 39.55 0.4 93109– 931010

4674 J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 Lee, Kim, and Lee

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1~f!, 1~g!, and 1~h! represent locally stable cluster structures,and consist of five, six, and seven ring-shape carbon clusters,respectively. These precursors or their derivatives are wellwithin the vicinity of C60 formation in that they require rela-tively low activation energies~4.6, 3.7, and 0.3 eV, respec-tively!. However, these models are not adequate, since theexcess energy to be released from the exothermic reactionwill lead to immediate disintegration of C60.

Using molecular dynamics simulations, Kim and To-manek have shown that the shape of the C60 molecule isstable34 below temperatureT;3000 K. At T53000 K, thecluster system has an additional internal energy of about 46eV/molecule over the pristine C60 molecule. Thus, precursormodels with the energy difference over;46 eV/moleculewould result in immediate melting and/or evaporation ofC60. This factor should be adequately taken into account formodeling of precursor models. Pentacyclic, hexacyclic, andheptacyclic precursor models have excess energies of 54.79,57.81, and 62.51 eV with respect to the target C60 molecule,respectively. Thus, as a realistic precursor model of the C60,pentacyclic and hexacyclic precursor models should have ap-propriate energy-loss processes. In the experimental setting,this is realized by the heat transfer process from the collisionbetween carbon clusters and cold inert gas. Ring-shape pla-nar precursor structures with more than seven rings are notconsidered since total energies of these systems are so highthat a series of high energy flights of carbon atoms is foundin the dynamic pathways.

In general, the annealing process will generate low en-ergy configurations. If we take sufficiently long annealingtime, we expect the final system would take the form of giantfullerenes or soot particles, rather than a collection of C60

molecules. This is based on the fact that carbons are morestable in the form of giant fullerenes or soot particles than inthe form of many C60 molecules at normal condition. Thus, arealistic kinetic model for the C60 molecule formation shouldtake into account the importance of partial energy lowering

processes so as to form the metastable C60 molecules in ashort period of the molecule formation.

To simulate such a partial energy lowering process, weemploy a stochastic energy optimization method, the simu-lated annealing.29,35 Four nonplanar precursor models shownin Figs. 1~i!–1~l!, were prepared by incomplete simulatedannealing procedures of the randomly chosen structures. Theresulting precursor models have the excess energies of 45.72,41.63, 41.94, and 39.55 eV with respect to the target C60

molecule. Each of the four nonplanar precursor models sat-isfy the condition ofDE,46 eV. Two tangled polycyclicmodels~I and II! have activation energy barriers of 3.3 and1.0 eV, respectively. The precursors in the shape of tangledpolycyclics have tolerably smooth energy fluctuations alongthe dynamical pathways as shown in Fig. 2. The precursorsin the shape of open cages~I and II! also have smooth energyfluctuations and low activation energy barriers~2.0 and 0.4eV! along the dynamic pathways as shown in Fig. 2. Forthese models, we estimate the formation rates at temperaturerangeT51000– 2000 K as listed in Table I. Estimated for-mation rates provide an ample opportunity of the moleculeformation within the short period;1023 s. In the presentstudy, we observe that two conditions~a low internal energyand a low barrier! on the precursor model lie at the root ofthe archetypal topology containing hexagons and/or chainsas a portion of the precursor model.

We find that the values of the activation energy barrierare less than the typical value of a single C–C bond energy(;3.6 eV), which is in good agreement with experimentalestimation by Hunteret al.20 This fact directly indicates thatthe molecule formation is derived from the concerted atomicmotions. We also find that the existence of chains in thepresent models of tangled polycyclics and open cages arebeneficial for the formation C60 molecule. The reason for thisis that chains attached to the cluster body are energeticallyfavorable and show lithe movements along the dynamic

FIG. 2. Potential energy fluctuations of twelve precur-sor models along the C60 formation pathway in theADMD method are shown. The related dynamical tran-sition pathways are obtained though the ADMD methodwith a chosen simulation time intervalt. The simula-tion time intervalt50.897 ps is discretized into 151steps. We usem5108 a.u. andn51010 a.u. as in theRef. 17. The energy of the system is calculated usingthe tight-binding method~Ref. 22!.

4675J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 The formation of C60

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pathway. These characteristic atomic movements are wellrepresented in the corresponding potential energy evolutionsalong their dynamical pathways as shown in Fig. 2. Theseresults are in marked contrast to the cases of the planar formof precursor models.

Finally, the mechanism of the C60 formation based onour calculations is as follows. The C60 formation starts withfusions of a few clusters which are mainly derived from theentropy of the system at the condition of graphitevaporization.31 As the carbon clusters are cooled down byinert gas flow, cluster restructuring reactions occur. In thisprocess the clusters evolve into the shape of tangled polycy-clics or open cages with significant amount of energy loss,forming a kinetically favorable configuration for furtherstructural transformation. This step is important to avoid in-stantaneous dissociation of the carbon molecules from theexcess kinetic energy due to the exothermic nature of the C60

formation. Subsequently, clusters in the shape of tangledpolycyclics or open cages can transform spontaneously intothe C60. The estimated activation energy barrier is in therange of 0.4–3.3 eV. These precursor models are of rathersmall excess energies (DE,46 eV) and large formationrates so that they can be trapped in the forward reaction ofC60 formation.

We have presented dynamical pathway models for theC60 molecule using ADMD simulations. By combining twoconditions of a small energy difference (DE) and a low ac-tivation energy barrier (Ea) for precursor models we select aset of probable configuration. Various planar polycyclic mod-els are in a disadvantageous state where they cannot betrapped in the forward reaction due to their high excess in-ternal energies. Our simulation results show that precursorsin the shape of a tangled polycyclic or in the shape of anopen cage are kinetically favored over precursors in theshape of planar hexagonal graphite fragments. A typical pla-nar graphite fragment has higher activation energy barrier~15.1 eV! than non-planar tangled polycyclic precursors~3.3–0.4 eV!. The calculated activation energies for the pro-posed precursor models are in good agreement with experi-ments. Existence of chains in the models of tangled polycy-clics and open cages is beneficially for the formation of C60

molecule. Chains attached to the precursor model are ener-getically favorable and provide lithe movements along thedynamic pathway. The present theoretical approach is appli-cable to various important reaction pathway studies.

This work was supported by ‘The Swiss-Korean Out-standing Research Efforts Award Program’ of the Ministry ofScience and Technology and by ‘‘The 4th SupercomputingApplication Support Program’’ of the Korea Institute of Sci-ence and Technology Information. I.H.L. and H.K. acknowl-edge support by the Ministry of Science and Technology of

Korea through the National Science and Technology Progi-ram Grant No. M1-0213-04-0002. J.L. acknowledges sup-port by Grant No. R01-2003-000-11595-0 from the BasicResearch Program of the Korea Science & EngineeringFoundation.

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4676 J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 Lee, Kim, and Lee

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