Z. Phys. D 40, 403–406 (1997) ZEITSCHRIFTFUR PHYSIK Dc© Springer-Verlag 1997

A kinetic model of carbon clusters growthA.L. Alexandrov, V.A. Schweigert

Institute of Theoretical and Applied Mechanics, 630090 Novosibirsk, Russian Federation(Fax: 7-3832-35-22-68, Phone: 7-3832-35-36-41, e-mail: schweig@site.itam.nsc.ru)

Received: 5 July 1996 / Final version: 28 August 1996

Abstract. A kinetic model of carbon cluster growth is sug-gested, including the processes of polycyclic clusters forma-tion and their isomerization to fullerenes, discovered recentlyin experiments. The model is based on Smoluchowski equa-tions, describing the rates of clusters formation in variouspossible reactions. The simulation results are in agreementwith the experimental observations.

PACS: 36.40

1 Introduction

Since the discovery of fullerenes in 1985, the description oftheir formation mechanism in condensing carbon, explaininghigh mass yield of C60, C70 and some others, still remains anunresolved task. The most important information about theprocess of carbon clusters growth and fullerenes formationhad been obtained in experiments where structure of clusterswas investigated by methods of gas chromatography [1]. Thenew class of clusters, called polycyclic rings and containingof few clipped carbon rings, was discovered. It was alsoshown [2–5], that these structures are fullerenes precursorsand easily transform to fullerenes during annealing.

The following path of carbon condensation can be im-aged: at first, small linear chains are formed, and when theyovergrow 10 atoms, they close into monocyclic rings, whichis the most stable configuration for this size [6, 7]. The ringscontinue growth, but also coalescence between them occurs,forming the different kinds of polycycles. The large enoughpolycycles are efficiently transforming to fullerenes, whichare also the most stable form for clusters larger than 30atoms. Since no advantages were noticed for C60 or C70fullerenes formation from polycyclic precursors [5], theirlarge mass yield may be explained by relatively low reactiv-ity of perfect fullerene structure, that helps them to survivein condensing vapor, while the ‘unperfect’ fullerenes (con-taining adjacent pentagons) continue growth and finally formthe soot.

We have developed a kinetic model of carbon condensa-tion by the mechanism suggested above. In previous works

we had considered the reactions between linear and cyclicclusters using MINDO/3 method [8], which allowed to de-termine the main types of reactions and to estimate theirenergetic barriers. We also performed a theoretical consider-ation of the kinetics of polycycles isomerization to fullerenes[9].

A few kinetic models of carbon cluster formation aredescribed in literature [10–12], but the kinetics of polycyclesformation and their further conversion to fullerenes werenever considered.

2 Model description

The kinetic model, considering formation of different typesof clusters, was developed as a system of Smoluchowskirate equations, including reactions of coalescence betweenvarious clusters and polycycles isomerization to fullerenes.

We supposed, that the temperature of oversaturated car-bon vapor is low enough (we considered temperatures lessthan 2000 K) and the reactions of dissociation, having bar-riers larger than 5 eV, together with other high-barrier re-actions, proceed very slow and can be neglected. Anotherassumption was that the buffer gas concentration is highenough to use the high-pressure limit approximation, whenits concentration doesn’t affect the reaction rates, and theclusters temperatures are equal to the buffer gas one.

The system includes the next types of clusters, like ob-served in [1]:

Cn - linear chains, n<10, including monomer C1;Rn - monocyclic rings, n≥10;Bn - bicyclic rings, n≥20;Tn - tricyclic and more complex clusters, n≥30;Fn - fullerenes, n≥30.After consideration of various reactions between clusters

in [8] we have made next basic conclusions:a) Coalescence of two linear chains leads to forma-

tion of chain or monocyclic ring (if resulting cluster sizeis larger than 10). The various intermediate structures, ap-pearing through different kinds of chains attachment, are notstable, and in usual condensation conditions (pressure lower

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than 1 atm,T ≥1000 K) due to low isomerization barri-ers would relax to the ground state before next coalescencereaction occurs.

b) Similarly, a chain attachment to a ring fragment leadsto its rapid insertion enlarging the ring size, so it doesn’tchange the topology of cluster.

c) The attachment of rings forms relatively stable poly-cyclic structures. Their further low-barrier isomerizationmay proceed only by atoms exchange between ring frag-ments. Change of their topology requires a bond breakingand the energetic barrier for this is 4÷5 eV and higher, sosuch event may be considered as rare. But their isomeriza-tion to fullerenes proceeds avoiding bond breaking [9] andits energetic barrier, evaluated in [5], is about 2.4 eV only.

From all this we supposed that the next bimolecular re-actions must be included in the system (for simplicity allpolycycles with three or more ring fragments are consideredtogether):

Cn + Ck ⇒ Cn+k (or Rn+k when n+k≥ 10);Rn + Ck ⇒ Rn+k;Rn + Rk ⇒ Bn+k;Bn + Ck ⇒ Bn+k;Bn + Rk ⇒ Tn+k;Bn + Bk ⇒ Tn+k;Tn + (each cluster of size k, except F)⇒ Tn+k.Besides, the monomolecular reactions of polycycles con-

version to fullerenes are included, ifn ≥ 30, with odd poly-cycles transforming to even fullerenes [2–4]:

Bn ⇒ Fn, evenn;Tn ⇒ Fn, evenn;Bn ⇒ Fn−1 + C1, oddn;Tn ⇒ Fn−1 + C1, oddn.Finally, we supposed, that during fullerenes coalescence

with every type of cluster the fullerene structure always sur-vives:

Fn + (each cluster of size k, including F)⇒ Fn+k.So for each cluster type and size a balance equation can

be written in the form of Smoluchowski rate equation, whichincludes the formation of a given cluster in all contributingreactions and its consumption in all bimolecular reactions.The decay due to isomerization to fullerene is also addedfor polycycles:

dNXk

dt=

1V

(∑Y,Z

k−1∑i

KY Zi,k−iN

Yi N

Zk−i

−NXk

∑Y

Imax∑i

KXYk,i N

Yi

)−KXF

izo NXk +

∑Z,i

KZXizo N

Zi

HereNXk is the total number of clusters of type X (it

may be C, R, B, T, F) and size k,V is the total volume ofcondensing vapor,Imax is the maximum k considered (thelarger clusters are related to soot). The sum over Y,Z meanstaking all the reactions where Y+Z⇒X. The sum over Z,imeans all the clusters of type Z and size i whose isomeriza-tion leads to appearance of cluster X,k.

The bimolecular reactions rate constantsKXYij were

taken as:

KXYij = ασXY

ij

√T (i+j)ij exp(−Eb/kT )

where i,j are cluster sizes,σXYij is a cross section of co-

alescence between cluster of type X and size i with clusterof type Y and size j;Eb is the energetic barrier of the coa-lescence reaction;T is temperature;α = (8k/πMc)1/2 (Mc

is carbon atomic mass);k is the Boltzmann constant.The rate of isomerization was taken as:KBF,TFizo = ν exp(−Eizo/kT ),

whereν is an unknown pre-exponential factor,Eizo-theactivation energy for isomerization.

The coalescence cross sectionsσXYij were estimated us-

ing Monte-Carlo numerical approach. For a cluster of typeX, size i and a cluster Y,j the relative motion with thermalrotation was simulated with the initial positions and veloc-ities chosen with Monte-Carlo technique. The atoms weretaken as hard spheres and the cross section was found fromthe probability of successful collision (for example, in [8] wefound that attachment of chain by its inner atom is highlyimprobable and such collisions were neglected). After av-eraging over 20–50 thousands of initial data sets the crosssections converged within 1% accuracy. As it is too difficultto calculate cross sections for all i,j pairs, for each type ofreactionσij were calculated for 10–15 values of each size,taken in wide enough range, with interpolation to other sizes.

The energetic barriers for coalescence between linear orcontaining ring fragments clusters were estimated in [8]. Weassumed thatEb=0.5 eV for all reactions involving chains(excepting chain-fullerene reaction) andEb=0.4 eV for coa-lescence between cyclic or polycyclic clusters.

The activation energy of polycycles isomerizationEizo

(which is 2.4 eV for clusters of 60 and more atoms andslightly increasing for smaller ones) was taken from [5],with interpolation to other sizes.

The unknown three-body dimerization rate constant wasestimated by the order of magnitude from the well-knownconstants for nitrogen and oxygen, which order is1015 cm6/(mol2 s). In calculations we approximated thedimerization rate as:

dNC2

dt=

1VβKCC

11 (NC1 )2

whereEb in KCC11 is zero. The value ofβ= 10−5÷ 10−4

corresponds to three-body dimerization rate for N or O in in-ert buffer gases in temperature range ofT = 1000÷2000 K,buffer and reactant gases concentration of 1017÷ 1018 cm−3.

The unknown energetic barriers of coalescence reactionsinvolving fullerenes were treated as adjustable parameters.

3 Results of simulation

The model was tested with the use of the experimental results[1], where the structure of clusters obtained by laser ablationof graphite into helium stream was analyzed.

The main calculational problem was that we don’t knowthe behavior of the volume and the temperature of the ex-panding carbon vapor, produced by laser pulse. Note that ina kinetic equations system, consisting of bimolecular reac-tions only, the effect of volume increase may be reduced to

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a varying time-scaling factor and so wouldn’t affect the be-havior of the solution. As can be seen from our rate equation,in our system, including also the volume-independent uni-molecular reactions rates, the effect of volumeV increaseis a deceleration of coalescence reactions relatively to theisomerization reactions.

The influence of temperature is similar, especially at theearly stage of condensation, when the non-fullerenes clustersare predominant. The values of their coalescence barriers areclose and the bimolecular reactions rates should vary withthe temperature similarly, which also reduce its effect toa time-scaling factor. As the isomerization reactions barri-ers are higher, the temperature also affects the relation be-tween isomerization and coalescence rates (the temperatureincrease, like the volume expansion, is equivalent to the rel-ative acceleration of isomerization, and vice versa). We cancharacterize this relation as the main factor determining thebehavior of the solution.

We considered the solution within the assumption of con-stant volume and temperature, but varied the relation of ratesfor the two main classes of reactions. The volume was as-sumed to unity and the initial condition wasNC

1 =C0 (thismeans starting monomer concentration), which we varied.After this the described system of kinetic equations was in-tegrated. The maximum cluster sizeImax in our calculationswas limited to 150–200, the larger clusters were transferredto unreactive soot. The influence ofImax value is weak (thesolution variation is less than 1% forImax=150 and 200).The perfect fullerenes C60, C70 and some others were char-acterized by decreased reactivity.

We assumed the pre-exponential factorν in the iso-merization rate as 1014 s−1, as the atomic oscillations fre-quency. In the assumption of constant volume and temper-ature the value of C0 determines the coalescence reactionsrates. Hence the relation between isomerization and coales-cence rates is determined by the parameterν/C0, which wevaried, assuming different C0. It is convenient to choose thecalculation time units asτizo =1/Kizo =ν−1exp(2.4 eV/kT ),which is the characteristic time of isomerization. Then thesolution depends onν/C0 relation only, not on their absolutevalues.

We have chosen the next values of the adjustable parame-ters: the dimerization rate factorβ=10−4; the Eb=0.8 eV forfullerenes reactions with other clusters and Eb=1.2 eV forfullerene-fullerene reactions. For the reactions with perfectfullerenes (C60, C70 and some others) the rate constants weremultiplied by factor of 0.05 (this was the estimation of C+

60cation reactivity in relation to C+56,58 in [13]).

Figure 1 shows the simulated relative abundances of thenon-fullerene isomers with comparison to results in [1]. Thecalculations were performed forT=1000 K and the casesof fast coalescence (C0=1019 cm−3, Fig. 1a) and fast isomer-ization (C0=1016 cm−3, Fig. 1b). The results are shown forcalculation time of 0.1τizo (dashed lines) and 10τizo (solidlines). It is seen, that the calculated isomer composition issimilar to the measured one in wide range of time and val-ues ofν/C0. The best quantitative agreement is observed forcalculation time 2÷10 τizo (in this interval the abundancesvary very slightly and are close to solid curves on Fig. 1).For the longer simulation times the abundance of policyclesat sizes larger than 30 atoms (minimal fullerene size) begins

Fig. 1. Relative abundances of the non-fullerene isomers, calculated forC0=1019 cm−3 (a) and 1016 cm−3 (b). Dashed linesshows the result fortime t=0.1τizo, solid lines– for t=10τizo, the points are taken from [1].The isomers shown are: linear chains – curves (1) and (�); monocyclicrings – curves (2) and (◦); bicyclic rings – curves (3) and (4); tricyclicrings – curves (4) and (N)

to decrease because of growing amount of fullerenes. The‘teeth’ on curves appear due to strong odd/even alternationof fullerenes abundance, also seen in experiment.

The fullerenes relative abundance is growing with timeand for some moment fits the results of [1] well. On Fig. 2we compare the simulated and experimental fullerenes abun-dances, obtained for the same two values of C0 and the cal-culation time of 4τizo). The data for odd and even fullerenesare shown separately. The drop of the curves atn=61 is ex-plained by low reactivity of C60, embarrassing the forma-tion of C61. For the longer times the fullerenes abundancebecomes more abrupt and finally turns into a stepwize one.

During the next stage of condensation, the highly reac-tive isomers (linear, cyclic and polycyclic) disappear andonly fullerenes remain. At the final stage, the ‘unperfect’fullerenes (except C60, C70 and any others, characterized byrelatively low reactivity) coagulate to the soot. Only perfectfullerenes are the remainders.

The weak dependence of the isomer abundances simu-lation results on theν/C0 parameter points that the resultsobtained within the assumption of constant volume and tem-perature are reliable. It can be expected that providing of realV andT behaviour (mathematically it means that the time-scaling factor and the relation between isomerization andcoalescence rates would no more be constant but changed atevery time step) the model would also give agreement withexperiment [1].

The model is able to describe the bimodal shape of car-bon cluster mass spectra. Unlike the isomer abundances, themass spectrum shape is sensitive to the relation betweenisomerization and coalescence rates. For relatively fast iso-merization, a sharp second maximum appears in size region

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Fig. 2. Relative abundances of fullerenes for evena and oddb size ofcluster att=4τizo, C0=1019 cm−3 (upper curves) and 1016 cm−3 (lowercurves). The points are taken from [1]

n =50÷60, like on the spectrum reported in [14]. For moreslow isomerization, the maximum becomes low and wide,shifting to larger sizes region, as on the example shown in[11]. The broad variety of observed experimental spectramay possibly be explained by their sensitivity to the coolingand expansion regime of carbon vapor.

The final fullerenes mass yield obtained in our modeldue to their lessened reactivity reaches 20% for someν/C0values and temperatures. The yield shows abrupt temperaturedependence, similar to observed in experiment [15].

It must be noted that, unlike the isomer abundances, themass spectra and final yield in our model are strongly depen-dent on the relation between isomerization and coalescencerates. That means, that for their more precise simulation theassumption of constantV andT is bad and the behaviourof vapor temperature and volume must be known.

We can conclude that the model is able to describe theprocesses occurring in the condensing carbon.

This work was supported by the Russian Foundation for Basic Researches(Grant 94-03-09476).

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