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Page 1: Monitoring the chemical vapor deposition growth of multiwalled carbon nanotubes by tapered element oscillating microbalance

Monitoring the chemical vapor deposition growth of multiwalled carbon nanotubes bytapered element oscillating microbalanceV. Švrček, I. Kleps, F. Cracioniou, J. L. Paillaud, T. Dintzer, B. Louis, D. Begin, C. Pham-Huu, M.-J. Ledoux, andF. Le Normand Citation: The Journal of Chemical Physics 124, 184705 (2006); doi: 10.1063/1.2192515 View online: http://dx.doi.org/10.1063/1.2192515 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/124/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simultaneous catalyst deposition and growth of aligned carbon nanotubes on Si O 2 ∕ Si substrates by radiofrequency magnetron sputtering J. Appl. Phys. 102, 114905 (2007); 10.1063/1.2818368 Low temperature growth of carbon nanotubes by alcohol catalytic chemical vapor deposition for field emitterapplications J. Vac. Sci. Technol. B 25, 579 (2007); 10.1116/1.2433964 Effect of supporting layer on growth of carbon nanotubes by thermal chemical vapor deposition Appl. Phys. Lett. 89, 183113 (2006); 10.1063/1.2382735 Vertically aligned carbon nanotube field emission devices fabricated by furnace thermal chemical vapordeposition at atmospheric pressure J. Vac. Sci. Technol. B 24, 1190 (2006); 10.1116/1.2190671 Carbon nanotubes synthesized by Ni-assisted atmospheric pressure thermal chemical vapor deposition J. Appl. Phys. 91, 3847 (2002); 10.1063/1.1448877

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Page 2: Monitoring the chemical vapor deposition growth of multiwalled carbon nanotubes by tapered element oscillating microbalance

Monitoring the chemical vapor deposition growth of multiwalled carbonnanotubes by tapered element oscillating microbalance

V. Švrčeka�

Nanoarchitectronics Research Center, AIST, Central 5, 1-1-1 Higashi, Tsukuba, Ibaraki 305 8565, Japanand Laboratoire des Matériaux, Surfaces et Procédés pour la Catalyse (LMSPC), Ecole Supérieurede Chimie-Physique-Matériaux (ECPM), 25 rue Becquerel, UMR 7514 CNRS, F-67087 Strasbourg, France

I. Kleps and F. CracioniouNational Institute for Research and Development in Microtechnologies (IMT), P.O. Box 38-160,Bucharest, Romania

J. L. PaillaudLaboratoire des Matériaux à Porosité Contrôlée, UMR CNRS 7016, Ecole Nationale Supérieure de Chimiede Mulhouse, 3 rue Alfred Werner, 68093 Mulhouse, France

T. Dintzer, B. Louis, D. Begin, C. Pham-Huu, and M.-J. LedouxLaboratoire des Matériaux, Surfaces et Procédés pour la Catalyse (LMSPC), Ecole Supérieure de Chimie-Physique-Matériaux (ECPM), 25 rue Becquerel, UMR 7514 CNRS, F-67087 Strasbourg, France

F. Le NormandInstitut de Physique et Chimie des Matériaux (IPCMS), UMR 7504 CNRS, BP 43, 23 rue du Loess,F-67034 Strasbourg, France

�Received 9 September 2005; accepted 9 March 2006; published online 10 May 2006�

The growth of multiwalled carbon nanotubes �MWCNTs� produced by a catalytic chemical vapordeposition �CCVD� process has been monitored using a tapered element oscillating microbalance�TEOM� probe. This technique displays a high sensitivity ��1 �g�. Growths in the TEOMmicroreactor are investigated with catalytic particles �Fe, Ni� dispersed on different supports. First,high surface area Fe/Al2O3 or Fe �Ni� exchanged on zeolite powders is used. Second, growths areperformed on array of nickel dots or Fe/Si-nc particles dispersed on large holes patterned on Si�100�substrates. An accurate monitoring of the early stages of growth permits a precise evaluation of thegrowth rates and shows substantial differences between these samples which greatly differ by thesurface area. On catalysts dispersed on Si�100� the mass uptake is linear throughout the process. Onhigh surface area catalysts, however, a saturation of the mass uptake is indifferently observed. Thissaturation is explained either by diffusion limitation by the growing MWCNTs or by internaldiffusion through the pores or external diffusion through the grains of the catalyst. The kineticdependence with partial pressure of the incoming C2H6:H2 gas mixture is then explored on theFe/Al2O3 catalyst. A linear dependence of the MWCNT growth an �PC2H6

/PH2�1/2 is found. A simple

model is then developed that accounts for this dependence only if an associative and competitiveadsorption of ethane is the rate determining step of the overall process. These results thus bringinsight to improve and control the CCVD growth kinetics of MWCNTs. © 2006 American Instituteof Physics. �DOI: 10.1063/1.2192515�

I. INTRODUCTION

The single walled carbon nanotubes �SWCNTs� andmultiwalled carbon nanotubes �MWCNTs� were subjected toan increased attention as a unique one-dimensional �1D� na-nomaterial during this last decade.1–3 The reasons are simplythat they possess geometrical characteristics, and physicaland chemical properties which make them a very promisingmaterial in the future for several application fields. One ofthe most promising approaches to the MWCNT production atan industrial scale are the catalytic chemical vapor deposi-

tion �CCVD� methods.4–7 A gas mixture containing a carbonprecursor such as methane, carbon monoxide, acetylene, etc.,and a diluent gas such as hydrogen flows over metallic cata-lytic particles such as Fe, Ni, and Co dispersed on a supportat temperatures ranging between 600 and 900 °C.MWCNTs, and eventually SWCNTs with a severe control ofthe catalyst size, can then be grown.

However, in spite of the importance to control the pro-cess for further optimization of the MWCNT properties �sizedistribution, growth rate, density, helicity control, defect andimpurity incorporation, etc.�, there are surprisingly fewworks that investigate the kinetics of MWCNTs in a system-atic �determination of the kinetic partial orders, the activationenergies, etc.� and comprehensive �implication on the nucle-ation and growth mechanisms� way.8,9 Some reasons to this

a�Author to whom correspondence should be addressed. On leave from In-stitute of Physics, Academy of Sciences of the Czech Republic, Cukrovar-nická 10, Praha 6, 16253 Czech Republic. Fax: �81�29-861-6355. Elec-tronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 124, 184705 �2006�

0021-9606/2006/124�18�/184705/12/$23.00 © 2006 American Institute of Physics124, 184705-1

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can be found in the inherent difficulties of accurate nanos-caled instrumentation with nanomaterials produced fromnanoparticules. By contrast the corresponding literature onthe growth of larger carbon nanofibers �CNFs� is betterdocumented.10,11 Two methods are mainly used to in situinvestigate the growth of CNFs: controlled atmosphere elec-tron microscopy10–12 �CAEM� and thermogravimetricmeasurements.13–19 In this last case, however, the experimen-tal setup has to simulate a packed-bed reactor in differentialconditions where the reactive hydrocarbon gas flows throughthe catalyst bed under favorable hydrodynamic conditions,i.e., avoiding at the same time inhomogeneity in flow distri-bution, large pressure drop, and mass and heat transfers. In aconventional balance, however, most of the gas feed by-passes the catalytic bed and the differential conditions cannotbe firmly established. This shortcoming can be avoided byusing a tapered element oscillating microbalance �TEOM�setup as the gas flow is forced to pass through the catalyticpacked bed. This probe allows achieving homogeneity incontact time with the catalyst and determining of absolutereaction rate.20,21 In addition TEOM has a unique capacity torecord relatively small weight changes �around 1 �g at1023 K� and to monitor the first steps of the growth pro-cesses of MWCNTs.

In the present paper, we report on a kinetic study of CNTgrowth using a TEOM setup. The samples involve catalyticparticles deposited on high surface area powders such as alu-mina or zeolites as well as flat or patterned Si�100� sub-strates. These quite different samples are used in order toaccurately investigate limitations due to gas diffusion inhighly porous media such as the catalytic powders with highsurface area. As a main result of this study, it is found thatindeed limitations due to gas diffusion may occur in thecourse of the CNT growth. Thus whatever the sample, theinitial growth rate is the significant data for absolute com-parisons. Subsequently, the kinetic orders relative to the con-stituents of the gas have been determined on Fe/Al2O3. Aroot square dependence versus PC2H6

/ PH2is obtained, which

is consistent only with a slow competitive and dissociativeadsorption of hydrocarbon on the catalytic surface as the ratedetermining step.

II. EXPERIMENTAL DETAILS

A. Tapered element oscillating microbalance

To achieve the synthesis of MWCNTs, a commerciallyavailable TEOM �Rupprecht & Patashnick Co., Inc.� pulsemass analyzer apparatus commonly used for thermogravi-metric studies21,22 has been implemented. To ensure repro-ducible results the TEOM is provided with an automatic flowand heating controls. A purge gas flow passes through thetapered element. Two heating control zones maintain thetemperatures at their set points. The preheating zone controlsthe temperature of the gas stream in the upper part of thesensor. The lower heat zone controls both the temperature ofthe tapered element and the temperature of the sample bedwhere the reaction takes place. The gas stream reaches itsmaximum temperature as it passes through the test material,and then the gas stream temperature drops as it flows toward

the gas outlet. The vibration frequency of the tapered ele-ment is monitored by an optical way, by means of both atransmitter and a receiver setup located on the opposite sidesof the tapered element. The system determines the masschange �m of the sample bed between times t0 and t1 usingthe following equation:

�m = K� 1

f12 −

1

f02� , �1�

where f0 and f1 are the natural oscillating frequencies attimes t0 and t1, respectively, and K is a constant that dependson the geometry of the experiment �in T2M units�.

When a new experiment is planned, the frequency f0 issettled after the catalyst maintained between wool quartz hasbeen introduced into the microreactor. The change of thefrequency f1 is measured by comparison with f0; thereforethe TEOM permits to evaluate the absolute values of themass changes. If the mass of the sample bed increases, theoscillating frequency f1 of the tapered element decreases.Sensitivity of this oscillating system can be lower than 1 �g.Since the volume of the sample bed markedly increases inthe course of the CNT growth, a relatively large void volumeis initially let in the catalytic bed. Blank experiments wereperformed with the same weight of quartz wool and no in-crease of mass was observed at all. Limitations in the dura-tion of the experiment are more serious, due to the filling ofthe bed in the course of the growth. We have checked, how-ever, that the reactor initially was operating in differentialconditions. In addition the decomposition of ethane is almostathermic in the temperature range considered.

B. Synthesis of MWCNTs on transition metalsdispersed on Al2O3 or zeolite powders

One of the supports used in this study was a high surfacearea �-Al2O3 �CK 300B Ketjen with a surface area of220 m2 g−1� which was mainly made up of a mesoporousnetwork of typical size of 8 nm. Iron deposited on alumina�Fe/Al2O3� was used as catalyst to achieve the growth ofMWCNTs. The alumina support was crushed and sieved anda fraction of 40–80 �m was retained for catalyst prepara-tion. The catalyst was prepared using an aqueous solution ofFe�NO3�3 ·9H2O, with an Fe concentration fixed at 40 wt %�Fe/Al2O3 catalyst�. The wet solid was dried at 373 K andfurther calcinated in air at 623 K for 2 h in order to obtain anoxide form of the catalyst precursor.

Metal coated on zeolite supports through ion exchangewas used as another type of porous catalyst. Two sampleswere prepared by ion exchange with low concentration ofiron �Fe/Z at 0.14 wt %� and nickel �Ni/Z at 0.38 wt %�.23

The zeolites, of faujasite structure, exhibit small pores withdiameter of 0.74 nm with faceted grains.

15 mg of the catalyst were placed in the weighting panof TEOM between quartz cotton. Synthesis of MWCNTs wasachieved with a gas mixture of ethane and hydrogen�C2H6:H2� at a constant total flow rate �60 ml/min� and atatmospheric pressure with a ratio R= �C2H6/H2�. Heliumwas used as a vector gas with the same flow rate. The cata-lysts were previously reduced in situ before the CVD process

184705-2 Švrček et al. J. Chem. Phys. 124, 184705 �2006�

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with pure hydrogen at 673 K. Figure 1�a� shows the varia-tion of the temperature in the weighting pan as a function oftime. The temperature was increased about 40 min under ahydrogen flow from room temperature to 673 K; then thistemperature was set constant for 1 h to reduce the oxideform of the metal catalyst. This resulted in a loss of mass inthe weighting pan �Fig. 1�b��. Once this loss mass was sta-bilized, the temperature was then continuously increased upto 1023 K. It must be noted that the loss mass decrease wasstabilized at the reaction temperature before ethane introduc-tion, indicating that the metal particles were reduced. At thisreaction temperature, the hydrogen flow was switched with amixture of ethane and hydrogen, and the CVD growth ofcarbon could start. The synthesis was carried out for differenttimes, catalyst or gas mixtures �see below�, but an increase ofthe total mass was observed as soon as carbon was intro-duced in the gas mixture �Fig. 1�b��. The average variationsof the total mass in the weighting pan of TEOM are shown inFig. 1�b� as a function of temperature for 15 mg of Fe/Al2O3

catalyst. A mass increase during the 5 min duration of ethaneintroduction corresponding to MWCNT growth clearly ap-pears. It is expected henceforth that the net mass uptake wasdirectly related to carbon deposition on the catalyst. At thevery beginning of the process, an incubation time is observedthat may be explained by the transient state needed for thefront of ethane to reach the catalyst.

C. Synthesis of MWCNTs on Si„100… flat substrates

Catalytic nanoparticles were introduced by two differentways on Si�100� substrates �n-type Si�100� heavily doped�Sb� with an electrical resistivity of 3 m� cm�. First ironnanoparticles were coated on silicon nanocrystals �Si-ncs�.The Si-ncs were prepared by electrochemical etching as de-scribed elsewhere �Si �111� Czochralski �Cz� grade, p-typeboron doped at 1 � cm�24 with a size distribution rangingbetween 2 and 40 nm. The Si-ncs were further diluted into anaqueous iron nitrate solution �Fe�NO3�3 ·9H2O� at a concen-

tration fixed to 20% in weight.25 The Si-ncs coated with iron�Fe/Si-ncs� were introduced by spin coating into the largeholes of a Si�100� substrate dipped with large holes by alithographic process that is elsewhere described26 �mean holedistance of 3 �m, diameter of 7 �m, and thickness of300 nm�. The clusters of Fe/Si-nc introduced into these Sigrooves were further dried at 373 K and calcinated into air at623 K for 2 h.

In a second way, an array of nickel dots on top of a TiNdiffusion barrier �7 nm thick� was designed by UV lithogra-phy on �100� silicon substrate.27 The nickel dots were 7 nmthick, 1 �m in mean diameter, and were spaced with eachother by 4 �m.

A small piece of such flat Si�100�-based samples wasvertically fixed between quartz wool in the TEOM microre-actor. The iron catalytic nanoparticles and the nickel dotswere reduced in situ in flowing hydrogen at 673 K for 1 h.After this reduction step the temperature was continually in-creased up to 1023 K, i.e., the reaction temperature at a rateof 10 °C/min. At 1023 K the hydrogen flow was switchedby a gas mixture of ethane and hydrogen �C2H6:H2

=20:40� and the MWCNT growth was started as describedabove for 5 min.

D. Scanning electron microscopy observations

After the CVD synthesis the zeolites were removed with20 wt % of HF solution for scanning electron microscopy�SEM� observations. The samples were examined by SEMon a Phillips field emission gun �FEG� operating within3–10 kV.

III. RESULTS

A. MWCNT growth by TEOM on Fe/Al2O3 powder

Figure 2 shows the time dependence of the weight in-crease in the weighting pan of TEOM at 1023 K, when theC2H6:H2 gas mixture �1:3 at 60 ml/min� is established. Ineach case, the increase of mass is related exclusively to car-

FIG. 2. Time dependence of the average mass increase in the weighting panof a TEOM at 750 °C when the ethane:hydrogen �ratio of 1:3; 60 ml/min�gas mixture was introduced and mass increase appears. The plot shows threedifferent durations of the MWCNT growth: t=2 min �squares�; t=5 min,�open circles�, and t=10 min �triangles�. The curves of mass increase werearbitrarily translated for a better visualization.

FIG. 1. �a� Variation of the temperature in the weighting pan of the taperedelement oscillating microbalance �TEOM� for the growth of MWCNTs byCVD process. �b� Parallel overall variation of the total mass in the weightingpan of TEOM as a function of temperature. The initial short peak is due tothe transient state that follows the settlement of the frequency f0.

184705-3 Growth of multiwalled carbon nanotubes J. Chem. Phys. 124, 184705 �2006�

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Page 5: Monitoring the chemical vapor deposition growth of multiwalled carbon nanotubes by tapered element oscillating microbalance

bon growth on a new catalyst at three different times ofdeposition. The derivation of the initial linear increase ofmass �at t=0� permits an evaluation of the carbon growthrates quoted to 0.059±0.002 mg/mg�cat� min for t=2 min,0.062±0.003 mg/mg�cat� min for t=5 min, and0.066±0.015 mg/mg�cat� min for t=10 min, which arequite reproducible, with an average value G=0.062±0.004 mg/mg�cat� min. This initial linear increasealso allows estimating the yield of carbon deposition. In anideal case assuming that �i� the hydrocarbon flow fully con-tacts the catalyst, �ii� the initial mass increase is pure carbon,�iii� the carbon selectivity is fully towards carbon formation,and �iv� perfect gas for the gas mixture, the carbon yield RM,defined as the ratio of the carbon deposited on the catalyst nd

over the initial hydrocarbon input ni, can be expressed as

RM = nd/ni = RTG/�2PC2H6DMc� , �2�

with PC2H6the partial pressure of ethane �105 Pa�, T

=1023 K, R=8.31 USI, D=60�10−6 m3/min, Mc

=12 g/mole. A carbon yield RM of 22.2% is then derivedfrom Eq. �2�. This value will be further useful when compar-ing the carbon yield at different temperatures or different gascompositions.

It is clearly seen that after about 4 min of deposition �seeopen circles or triangles in Fig. 2�, saturation starts and be-comes more pronounced after 10 min deposition. One expla-nation would be the coking of the catalyst surface that de-creases its reactivity. This point is supported by SEMobservations of the carbon deposit after these three differentdeposition times �Fig. 3�. Not only are MWCNTs depositedbut also some amorphous carbon can be observed. After2 min of deposition, fine MWCNTs with an average outerdiameter of 30 nm and lengths estimated as longer than900 nm are exclusively seen. After 5 min of deposition, anincrease of both the MWCNT outer diameter at around45 nm and lengths that can be now longer than 2 �m areobserved. When the deposition is prolonged to 10 min, how-ever, the formation of thin MWCNTs �25 nm� in addition tothe thicker MWCNTs is reported accompanied with someaggregates of amorphous carbon. It is expected that this ap-parition of amorphous carbon can be related to the saturationin the mass uptake plot seen in Fig. 2.

First, MWCNTs are believed to form outside the poresof alumina support as the catalytic particles are very rapidlyreached by the gas hydrocarbon precursor. In a second step,the hydrocarbon reaches the catalytic particles which are in-side the Al2O3 microporosities with the formation of smallerMWCNTs.

Finally encapsulation of the metal catalyst occurs andaggregates of amorphous carbon are formed, which both se-lectively change the nature of the carbon deposited and limitthe rate of carbon deposition. This leads consecutively to aprogressive saturation of the mass increase. In the case ofTEOM, it has to be pointed out that at synthesis times longerthan 10 min the microreactor is generally filled, furtherdeposition leaving MWCNTs to fall down out of the reactor.Thus the experiment is not extended above 10 min. In con-clusion, it can be stated that the mass increase must be con-

sidered in the initial times, as rapid poisoning of the catalystmay occur with formation of amorphous carbon agglomer-ates.

B. Partial pressure dependence on Fe/Al2O3 catalyst

To investigate the partial pressure dependence at1023 K, the total flow D=DC2H6

+DH2=60 SCCM �SCCM

denotes cubic centimeter per minute at STP� is kept constantwhile varying the ratio r=DC2H6

/DH2= PC2H6

/ PH2. Figure 4

shows the time variations for two different ratios r. It isobserved that an increasing content of ethane also increasesthe initial growth rate from 0.050 mg/mg�cat� min for a 1:5C2H6:H2 to 0.115 mg/mg�cat� min for a 1:1 C2H6:H2 gasmixture, respectively. On the other hand, with an increasedconcentration of ethane in the gas flow, saturation starts at a

FIG. 3. Corresponding scanning electron microscopy images of MWCNTsobtained by CVD in TEOM with a gas mixture of ethane and hydrogen �1:3�at 60 SCCM/min and 1023 K. The reaction was carried out at three differ-ent times: �a� t=2 min, �b� t=5 min, and �c� t=10 min.

184705-4 Švrček et al. J. Chem. Phys. 124, 184705 �2006�

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Page 6: Monitoring the chemical vapor deposition growth of multiwalled carbon nanotubes by tapered element oscillating microbalance

shorter time �Table I�. With a 1:5 C2H6:H2 gas mixture,saturation is not observed whereas with a 1:1 mixture satu-ration starts after around 3 min of MWCNT synthesis. Thisis in agreement with a promoted coking process at higherhydrocarbon pressure. In addition, as the concentration ofethane increases, more but thinner MWCNTs are obtained.This can clearly be seen from the SEM images displayed inFig. 5. When the ethane:hydrogen ratio is increased from 1:3to 1:1, the average outer diameter of the MWCNTs decreasesfrom about 45 to 21 nm. The corresponding data are gath-ered in Table I as a function of DC2H6

.In Fig. 6 the initial growth rate G is plotted as a function

of gas ratio r in a semilogarithmic scale. The linear behaviorallows determining the power law of rate growth G vs r thatcan be expressed as

G = krx exp�− Ea

RT� , �3�

where k is a kinetic constant and Ea is the activation energyof the mass increase process. From the linear extrapolation ofthe plot, an exponent x=0.50±0.03 is derived. Thus the cor-rected expression is

G = k�PC2H6/PH2

�1/2 exp�− Ea

RT� . �4�

The signification of this power law is clear. The kineticsof carbon mass increase is first order with regard to carbonpartial pressure and is prevented to the first order by hydro-gen partial pressure. The fact that hydrogen is competingwith the hydrocarbon source strongly supports a rate limitingstep due to the adsorption on iron active catalytic sites. Amodelization of this behavior is further discussed. The car-bon yield RM �%� calculated according to expressions �2� and�4� is now

RM = �RTk/2DMc�PC2H6PH2

�1/2�exp�− EA/RT� . �5�

Carbon yields fall between 20% and 25%; that meansthat 1

4 to 15 of the input carbon is decomposed and used for

the CNT growth. A transmission electron microscopy �TEM�image clearly displays graphitic multiwalls �Fig. 7�a��around the tubule when the mixture of ethane and hydrogenwas C2H6:H2=1:1 with D=60 SCCM at 923 K for 5 min.Similar observations are obtained on each sample. Ramanspectrum �Fig. 7�b�� of the carbon deposit exhibits the twonarrow bands D and G at 1340 and 1570 cm−1 with a cleartailing around 1610 cm−1. These contributions are quite char-acteristic of the presence of MWCNT with some extent ofdefects as also evidenced by the TEM image. Similar spectraare recorded on each sample.

In addition the outer diameter of the MWCNTs is clearlydecreasing with an increasing ratio r �Fig. 8�. It should beremembered that this behavior can also qualitatively be cor-

FIG. 4. Time dependence with the average mass increase in the weightingpan of TEOM at 1023 K for 5 min for two different ratios of C2H6:H2.Circles for 1:1 and squares for 1:5 ratio.

TABLE I. Kinetic data on the rate of carbon mass increase G as a function of DC2H6at D=60 SCCM/min,

T=1023 K, and P=1 atm.

DC2H610 15 20 30

G �mg/mg�cat� min� 0.050±0.0035 0.0635±0.0025 0.082±0.0018 0.117±0.0016RM �%� 26.2 22.2 21.4 20.4

tsaturation �min� �5 4.3 4 3Outer diameter �nm� 46.7±2.4 44.7±1.8 29.4±2.1 21.1±1.6

FIG. 5. Scanning electron microscope images of MWCNTs obtained byCVD in TEOM when the mixture of ethane and hydrogen was varied. �a�C2H6:H2=1:3 and �b� C2H6:H2=1:1. The reaction was carried out at1023 K for 5 min.

184705-5 Growth of multiwalled carbon nanotubes J. Chem. Phys. 124, 184705 �2006�

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related with an increasing density of MWCNTs. New Fecatalytic sites for MWCNT nucleation and growth are activewith higher carbon content. As the density of tubes can besignificantly increased with larger carbon content, it could beinferred that less carbon is available for each catalytic par-ticle for CNT growth.

C. Growth on patterned Si„100… substrates

1. MWCNT growth in Si holes on iron catalyst

It is expected that TEOM is sensitive not only on alu-mina powders but also on patterned Si�100� surfaces wherecatalytic nanoparticles are either deposited or patterned andsubsequent MWCNT growth is achieved. Moreover studieson such surface will give more insight on the real nature ofthe saturation observed on powders in the mass uptake. Thus

a piece of silicon-based substrate has been put into the mi-croreactor. Two different types of catalysts were used.

It has been shown recently that carbon nanotubes can begrown on single silicon nanocrystal loaded with iron nano-particles �Fe/Si-ncs�.28 In Figs. 9�b� and 9�c� bunches ofMWCNTs are exclusively growing from the bottom of somelarge holes dipped into Si�100� where Fe/Si-nc clusters ascatalysts have been deposited by spin coating. The growthprocess of such MWCNTs can be monitored by TEOM �Fig.9�a��. The growth rate is much slower �23.9 �g/min� bycomparison with growth rate on Fe/Al2O3 catalyst undersimilar conditions �1.21 mg/min�.

It could be seen in addition that Fe/Si-nc clusters areexclusively filling holes and are not spread on the flat sur-face. This low rate is due to the much smaller concentrationof catalyst, because only a few holes were filled. More im-portantly the carbon mass increases now linearly along thesynthesis time �Fig. 9�a��. This means that limitations due todiffusion do not occur, unlike Fe/alumina powder. This canbe understood if we consider the mean free path of carbonspecies in the gas phase, which in the limits of an ideal gascan be expressed as

�c = kBT/�2P� , �6�

where kB is the Boltzmann constant, the collisional crosssection quoted to 10−19 m2, P=1 atm, and T=1023 K. Then�c yields around 1 �m, much lower than the interdistancebetween each holes. The MWCNTs are more than twicelarger in average outer diameter �48±4 nm� compared withFe/Al2O3 powders under similar experimental conditions�21 nm in Table I�. It is also observed in detail that a Si-nc�size around 30 nm� is stuck on the tip of a MWCNT �Fig.9�d��. The morphological and structural studies of these hy-brid nanostructures have been reported elsewhere.28 One canspeculate that such structure can be useful either to absorbphotons by single Si-nc and to transfer the electric charges tosilicon substrate, or as effective photodetectors and photovol-taic structures with an optimized surface area.

2. MWCNT growth on Si„100… substrates patternedwith Ni dots

A similar procedure of MWCNT growth �1023 K for5 min with a mixture of C2H6:H2=1:2 at P=1 atm and D=60 SCCM� was applied on a regular array of Ni dots de-signed by UV lithography on Si�100�. In Fig. 11�a� a linear

FIG. 8. Mean outer diameter of the MWCNTs grown as a function of theratio of partial pressure r. The line is a guide for the eye.

FIG. 6. Initial rate growth G as a function of gas flow ratio RM

= PC2H6/ PH2

.

FIG. 7. �a� Transmission electron microscope image and �b� Raman spec-trum of a MWCNT obtained by CVD in TEOM when the mixture of ethaneand hydrogen was C2H6:H2=1:1 with D=60 SCCM at 923 K for 5 min.

184705-6 Švrček et al. J. Chem. Phys. 124, 184705 �2006�

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increase during synthesis of MWCNTs is again obtained. Thetotal mass uptake increases linearly without saturation with aweak growth rate �4.9 �g/min� �Fig. 10�a��. Again we canconclude that the growth rate after 5 min is not limited bydiffusion as the distance between each dot exceeds 1 �m.This very low growth rate can be explained by a strong de-activation of the Ni particles as only a few of them can grow

MWCNT �Fig. 10�b��. Thus a maximum of one longMWCNT is obtained on each dot �Fig. 10�c��.

A fast carbon covering of the Ni particles, as in this case,cannot explain this deactivation of the Ni particles. A rapidinitial mass increase followed by saturation would be ob-served. More probably etching of the Si substrate poisonsmost of the Ni particles by hydrogen giving nickel silicide.From the SEM micrographs, it appears that MWCNTs weregrown with a random growth direction and lengths that donot exceed 1 �m �see Fig. 10�c��.

D. Growths on zeolite-based catalysts

In order to ascertain that the saturation observed on theFe/Al2O3 catalyst �Fig. 4� can be explained by the high sur-face area of the support, the MWCNT growth is investigatedon zeolite-based catalysts. They display homogeneous nan-opores with an average porous diameter of 0.74 nm. Corre-sponding SEM images of MWCNTs are shown on Ni/Z andFe/Z samples in Figs. 11�a� and 11�b�, respectively. Figures12�c� and 12�d� represent the samples after further etching ofthe zeolite support with a 20 wt % HF solution. In the latter

FIG. 10. �a� Time dependence of average mass increase in the weightingpan of TEOM at 1023 K for 5 min for C2H6:H2 gas mixture �1:2�. Ni dotsare used as catalyst on Si�100� substrate. �b� Corresponding scanning elec-tron microscopy overview of MWCNT growth on an array of nickel dots.The thin veil seen on some dots might be due to remaining resin. �c� Indetail, we can see one single MWCNT grown from one dot with an outerdiameter of about 50 nm and a length of several hundred nanometers.

FIG. 11. SEM images of �a� nickel/zeolite �Ni/Z� and �b� iron/zeolite�Fe/Z� after MWCNT growth on TEOM. Subsequent zeolite etching by HFon �c� Ni/Z and �d� Fe/Z.

FIG. 9. �a� Time dependence of average mass increasein the weighting pan of TEOM at 1023 K for 5 min forC2H6:H2 �1:2� gas mixture with iron catalyst intro-duced into silicon holes. �b� Corresponding scanningelectron microscopy overview of the sample with someholes filled with a bunch of MWCNTs. �c� In detail ofone bunch of MWCNTs with �d� an Fe/Si-nc particlestuck on top of a MWCNT with a 30 nm size.

184705-7 Growth of multiwalled carbon nanotubes J. Chem. Phys. 124, 184705 �2006�

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figures the MWCNTs can clearly be evidenced with an aver-age outer diameter of about 40 nm, similar to porous alumina�Fig. 3, 1023 K�.

In Fig. 11 the time dependence of the average mass up-take in the weighting pan is shown. Similarly to porous alu-mina, a saturation of the signal is observed due to the inter-grain diffusion, more pronounced on the Ni/Z sample thanon the Fe/Z sample. In agreement with the SEM images, thegrowing rate in the case of Ni/Z catalyst is much higher. Thehigher loading of Ni and thus the higher density of Ni par-ticles can explain this. The initial growth rates on Fe/Zand Ni/Z are evaluated to be 0.01±0.0031 and0.035±0.0013 mg/mg�cat� min and saturation starts at t�2 min and t3 min, respectively. However, it is interest-ing to relate the carbon mass uptake with the concentrationof catalyst. This will give an expression of the growth ratethat is related to the turnover number of the catalyst ex-pressed as

v = G�Mcat/2MC��1/mcat� , �7�

where Mcat and mcat are the mass molar and the mass of ironcatalyst, respectively. The turnover frequencies are quoted tobe 0.42 �at.�C� / s at.�Fe�� and 0.57 �at.�C� / s at.�Ni��, respec-tively. Thus it is estimated that the Ni catalyst is most effi-cient to convert the hydrocarbon into carbon nanotube thanFe. This is in agreement with the literature data.29

IV. DISCUSSIONS

It is shown here that the initial growth rate obeys a firstorder kinetic relative to the square root of the ratio r= PC2H6

/ PH2. Surprisingly, data are yet very scarce about the

dependence on partial pressures of the growth of carbonnanotubes, and none has been collected to the best of ourknowledge. By contrast, some are available on the growth ofcarbon nanofibers, either on foils or on supported catalysts ofFe, Co, Ni, or any alloys including one of these elements.The results point out a complex behavior according to thenature of the hydrocarbon and to the presence of hydrogen.Zero order kinetics relative to olefin partial pressures andeven positive orders relative to hydrogen were often re-ported. By contrast the kinetics is first order relative to par-affinic carbon precursors such as and C6H14.

30 However, thisbehavior is drastically changed with temperature with a clearmaximum in the growth rate on iron monocrystals or foils.

Thus the rate limiting step is inferred to be the carbon diffu-sion through Fe3C into Fe nanoparticle at low temperature,whereas the step of decomposition of the hydrocarbon pre-cursor is limiting the rate at higher temperatures.31 Somerecent reports on the growth of nanofibers performed with athermobalance point out a complex dependence on partialpressure of hydrogen or carbon. In addition growth of thesenanofibers is observed generally at lower temperatures, typi-cally 723–973 K.15–17 Finally the kinetic order may be dif-ferent according to the nature of the substrate, i.e., foils, flatsurfaces, or powders such as supported catalyst. We will dis-cuss first this effect of nature of the catalyst as it governs thereliability of the data presented. We will further address thekinetic implications of the results obtained on the depen-dence on the partial pressures.

Samples exhibiting quite variable gas phase accessibili-ties to the surface have been used. Patterned Ni dots or widepatterned holes filled with Fe/Si-nc catalysts on Si�100� areassumed to be quite accessible to the gas phase, whereas onhigh surface area catalytic nanoparticles supported on alu-mina or zeolites uneasy accessibility is supposed. Rapid satu-ration of the carbon uptake occurs with high specific areasamples �Table II� whereas a linear rate is invariably ob-served on flat surfaces. This infers that nonsteady state kinet-ics occurs when encompassing a large time period of massuptake over high surface area catalysts. This nonstationarystate has been explained by a selective coking of the catalyticparticles either located outside or inside the pores of the sup-port. Over a long period of mass increase, the measured ki-netics tends rapidly to saturation and the determination of thekinetic order may be strongly altered. Indeed it is tempting toconsider that the initial rate at t=0 would be the true rate ofcarbon incorporation. However, it was shown independentlyon the same setup that the initial carbon uptake also involvesthe irreversible carbon bulk saturation of the catalystparticle.32 This step corresponding to carbon adsorption fol-lowed by diffusion either into or over the surface of thecatalytic particle is, however, quite rapidly established. It isafter this initial carbon absorption and nucleation at the in-terface between the support and the particle that carbonnanotube starts to grow, as shown by the growth at shortsequential times. Thus care has to be taken that the trueinformation on the carbon growth rate can be extracted onlywithin the time period after carbon saturation of the catalystbut before the possible saturation due to coking of the cata-lyst or to deactivation of the catalyst by encapsulation. Otherexplanations to the saturation of the mass uptake at longtimes on high surface area powders may be an external �in-

FIG. 12. Time dependence of the average mass increase in the weightingpan of TEOM at 750 °C �ethane: hydrogen �ratio of 1:2�� on zeolite support.Full squares correspond to the Ni/Z catalyst and open circles to the Fe/Zcatalyst. The full lines correspond to the initial mass rate.

TABLE II. Summary of the results.

Catalyst Support Kinetic behaviorType ofsupport

Ni �patterned� SiO2/Si�100� Linear FlatFe/Si-nc Holy Si�100� Linear FlatFe Al2O3 Saturation or deactivation PowderFe Zeolite Saturation or deactivation PowderNi Zeolite Saturation or deactivation Powder

184705-8 Švrček et al. J. Chem. Phys. 124, 184705 �2006�

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tergrain diffusion� or internal �through the pores� gas phasediffusion. External gas phase diffusion is strongly mini-mized, owing to the small amount of sample. Internal gasphase diffusion is governed both by the pore size distributionand by the location of the catalytic particles on the surface ofthe grain. In addition, it must be noted that a rapid termina-tion is sometimes reported and CNT growth abruptly ceases.Some collective poisoning of the surface of the catalyst canbe invoked, as we note that this poisoning involves denseand interacting CNTs.9 It is believed, however, that thisabrupt termination step cannot explain the progressive satu-ration of the carbon rate increase observed here.

We will derive now a simple kinetic model that couldexplain the first order kinetics relative to the square root ofthe C2H6/H2 ratio of partial pressures. Provided that the ratelimiting of the process is not gas phase diffusion, the gasphase carbon precursor concentration PC2H6

S over the surfaceof the catalyst is not limited by gas phase diffusion and isrelated to the incoming ethane partial pressure PC2H6

B throughFick’s law of gas phase diffusion in the limiting case of alaminar flow mode:

PC2H6

B − PC2H6

S = �PC2H6= KI , �8�

where l is a characteristic diffusion length and K the diffu-sion coefficient of ethane in the C2H6/H2 mixture.

The next step of the overall nanotube growth process canbe defined as follows:

�i� Hydrocarbon and hydrogen competitive adsorptionson the catalyst surface. At this point the competitionof the ethane adsorption with the H2 adsorption mustbe emphasized since it preserves the surface from arapid poisoning of the catalyst surface by carboncoverage.30 Many reports consider a dissociative hy-drocarbon adsorption in their model of carbon nano-tube growth �a Langmuir-type adsorption�.14–20 Thisassumption is pertinent in cases where the carbon par-tial pressure is not negligible with regard to the hy-drogen partial pressure and at high temperature. How-ever, in the cases where the carbon partial pressure islow with regard to the hydrogen partial pressure, thehydrogen coverage �H is high and an associative ad-sorption scheme �Rideal-type adsorption� would beenvisaged.33–36 Owing to the high temperatures usedhere and the fast desorption of hydrogen, it is as-sumed that the adsorption is dissociative:

C2H6 + 2�S → C2H5ads + Hads, �9�

H2 + 2�S → 2Hads, �10�

C2H5ads → 2Cads + 5Hads, �11�

where the index “ads” means an adsorbed state, and�S is a free surface catalytic site proper for ethaneadsorption. The notion of a surface catalytic site is acomplex one with a rather large molecule such asethane as it may involve an ensemble of active metal-lic atoms on a terrace or at a step edge of the catalyticsurface.33–35 The meaning of chemical reaction �9�

relative to �S is that at most two free surface sites canbe blocked by adsorption of molecular hydrogen forfurther ethane adsorption despite that the ensemble ofsites required for dihydrogen adsorption is smallerthan for ethane adsorption. Reaction �10� simplystates that hydrogen is in competition with ethane forthe multisite adsorption on a proper ethane adsorptionsite �S. Reaction �11� is a generic equation involvingthe full decomposition of adsorbed hydrogenated car-bon to carbon through dehydrogenation and carbon-carbon bond rupture. Strictly speaking, each dehydro-genation step as well as the carbon-carbon bondrupture would be taken into account; however, it isbelieved that these steps are fast for the followingreasons. The formation of carbon nanofibers is welldocumented with the gas mixture CH4+H2 on transi-tion metal catalysts, but at lower temperatures �T�873 K�.14–20,37 It is then found that the rate deter-mining step is invariably the chemisorption step �9�rather than one of dehydrogenation step �11�.15,18 Athigher temperatures the dehydrogenation rate is ex-pected to be even faster than the chemisorption. Acatalytic facet of the particle, i.e., part of the particlesurface with surface area AC, is devoted to decomposecarbon according to Eqs. �9�–�11�. It must be recalled,however, that the hydrocarbon chemisorption and de-hydrogenation and hydrogen chemisorption are highlystructure sensitive. Thus these steps are generallymuch faster on opened surfaces such as �110� and�100� facets than on dense facets such as �111�. Fi-nally we will not consider in this simple model thepoisoning of the catalyst through encapsulation of thecatalytic particles for the following reasons: First, theinitial constant rates are considered, as carbon satura-tion of the catalytic particle and nucleation rapidlyoccur at high temperature, and second, it has beenshown that the encapsulation process is depleted athigh temperatures.16–20,38,39

�ii� Subsequent to the hydrocarbon adsorption and de-composition on the catalytic facets AC, carbon dif-fuses either into the bulk or over the surface of thecatalyst, towards the growing facets with surface areaAG where CNT growth can start. Recent results in theliterature strongly supports, however, the occurrenceof surface diffusion rather than bulk diffusion, whichis more activated.12,32,40,41 This will lead to a carbongradient concentration that in first approximation canbe expressed as

Cads�AC� → Cads�AG� + �S. �12�

�iii� Carbon adsorbed on the growth facets AG is incorpo-rated into the graphene sheets and thus leads to thegrowth of carbon nanotubes with a rate G. At thispoint we must point out that this growth concernsmultiwall carbon nanotubes and thus carbon can con-tribute to incorporation into an existing rolled upgraphene shell or to the creation of a new graphenesheet. We will not further discuss this point.

184705-9 Growth of multiwalled carbon nanotubes J. Chem. Phys. 124, 184705 �2006�

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Cads�AG� → CNT. �13�

However, the surface area of the particle must be par-titioned between areas where catalytic carbon adsorp-tion and decompositions �AC� occur, growth area AG

where CNT growth occurs, and finally area of theparticle where no carbon growth nor catalytic decom-position occurs, given the hollow part of the nanotube�AH� with

AC + AG + AH = 1. �14�

We believe that the rate of CNT growth G is governedby the rate of one of the reactions in �9�–�13� as therate determining step �rds�. Thus we have calculatedthe expression of the rate growth G for each casewhere one of the reactions in �9� or �11�–�13� is therate determining step. ki and Ki are the kinetic rateand the equilibrium constants for the reaction i, re-spectively. The equations are reported in the Appen-dix. If �C, �C2H5

, and �H �%� are the surface convergesby carbon, ethane, and hydrogen, respectively, on thecatalytic surface AC of the particle,

�C + �C2H5+ �H + �S = AC. �15�

It is found that the rate of CNT growth G, when theethane adsorption step determines the overall kinetics,can be expressed as �see Appendix for some details ofthe calculation�

G = k5k4�ACk3K1/k2�1/2�H1/2�PC2H6

/PH2�1/2. �16�

The other expressions with different rdss can be brieflycommented on. If the growth is limited by the ethane decom-position�11�, it is believed that the growth rate behaves as�k5k4 /AC

2 ��K3k1 /k2�1/2�H−3/2�PC2H6

/ PH2�1/2. In the case of

diffusion-limited process, G is proportional to �PC2H6�1/2. In

the case of the growth-limited process, G is proportional to�H �PC2H6

/ PH2�1/2. Whatever the rds of the process, the

growth rate is proportional to �PC2H6�1/2. The difference be-

tween each rds arises from the dependence on PH2and the

hydrogen coverage �H. If it is assumed that the hydrogencoverage at high partial pressure of hydrogen is almost con-stant ��H��H

0 �, where �H0 is the hydrogen coverage in the

absence of hydrocarbon, then there are very few differencesbetween expressions �A10�, �A11�, and �A13� reported in theAppendix. At low hydrogen partial pressure the dependenceof �H on PH2

can be effective, but most obviously encapsu-lation of the catalytic particles occurs. In cases where eitherCNT growth interfacial or dehydrogenation of the adsorbedhydrocarbon reactions is slow, the dependence with partialgas pressure is in both cases �PC2H6

/ PH2�1/2. However, the

dependence on �H is more pronounced in these two last casesthan in the case where the hydrocarbon adsorption is slow.

Of course the nature of the reactant must be pointed out.It is known that the absorption of saturated hydrocarbon suchas methane or ethane is more activated than the easier sur-face decomposition of alkenes or acetylene or CO decompo-sition and quite different results can be observed accordingto the nature of the reactant.30 CH4 decomposition has been

studied on iron foils with formation of carbon nanofibers, butat lower temperatures. The results cannot distinguish be-tween the chemisorption step and the first dehydrogenationstep as rds.42 However, more recent results on Ni catalystspoint out that the chemisorption of methane is slower thanthe first dehydrogenation step.37 Cooper and Trimm cannotdetermine the order relative to propylene on monocrystallineor polycrystalline iron either fresh or used.31 Moreoverethane adsorption on iron leads to a complete hydrogen-deuterium exchange which indicates that the dehydrogena-tion to carbon fragments is rapid once the ethane ischemisorbed.42 Recent studies with the parallel developmentof a modelization point out a complex dependence behavioron the hydrogen and hydrocarbon partial pressures.43 How-ever, reconsidering the results at low carbon partial pressurePC with regard to the hydrogen partial pressure PH, a lineardependence of the growth rate with the ratio PC/ PC can beinferred.15–17 These results, similar to this work, support sucha simple model, provided we are in the domain where themass uptake is governed by the nanotube growth and that thepartial pressure of hydrogen is high with regard to the partialcarbon pressure. In this general discussion we have not dis-cussed the nature of the iron catalyst as it is well known thatthe formation of bulk or subsurface carbide could beenvisaged.2,3,10,11 However, further studies point out that thereal nature of the catalyst is not decisive for the growth ofCNT.32

V. CONCLUSION

In this paper, it has been shown that the use of the ta-pered element oscillating microbalance �TEOM� setup canmonitor with a high sensitivity ��1 �g� the dynamics ofmass increase due to the carbon deposition. The selectiveformation of carbon nanotubes is observed at high tempera-ture �973 K�. So far this technique allows to determine thenet rate of carbon incorporation into carbon nanotube andthus to evaluate the growth rate of MWCNTs.

Further it is shown that the initial rate of mass increasemust be considered to determine the initial growth rate ofMWCNT as illustrated by comparison of the MWCNTgrowth on Fe/Al2O3 powders and on patterned Si�100� sub-strates covered with Fe or Ni catalysts. On flat surfaces, thisrate is linear over the whole time range as it is neither per-turbated by long-term poisoning of the catalyst, nor limiteddue to diffusion, nor by the rapid initial transient steps due tocarbon saturation of the metal particle and nanotube nucle-ation.

We have shown that the growth rate increases linearlywith the square root of the ethane-to-hydrogen ratio of partialpressures. Thus a first order kinetics with regard to carbonconcentration can be deduced from these results, whereashydrogen competes with hydrocarbon for the adsorption onthe catalytic sites of the catalyst. These results are coherentwith a simple model where the dissociative adsorption of thehydrocarbon is the rate determining step of the overall pro-cess within the assumed limitation that the carbon partialpressure is low compared with the hydrogen partial pressure.

Thinner MWCNT formation at high carbon content can

184705-10 Švrček et al. J. Chem. Phys. 124, 184705 �2006�

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also be explained by an increasing density of sites availablefor MWCNTs on iron catalyst with increasing carbon partialpressure. Further results concerning the thermal effect ongrowth rate and the determination of the activation energiesare in progress.

ACKNOWLEDGMENTS

This work was supported by CNRS �Centre National dela Recherche Scientifique� and by the European NAN-OTEMP project �Contract No. HPRN-CT-2002-00192�. Tha-les RT �Orsay� is acknowledged for purchasing the Si�100�samples. The authors thank Dr. F. Garin for helpful discus-sion.

APPENDIX: CALCULATION OF THE RATE OF CNTGROWTH G FROM THE REACTION SYSTEMS„9…–„13…

We give some details in the case where the adsorption�Eq. �9�� is the rds. From chemical reactions �12� and �13�,the net CNT growth rate G can be expressed as

G = AGk5�C�GF = k5ACk4�C. �A1�

�C is given from reaction �11� as

�C = k31/2AC

−1/2��C2H5�1/2. �A2�

The ethane coverage is given by the equilibrium of adsorp-tion �13� given by

K1 = �C2H5�H/�S

2PC2H6, �A3�

whereas the concentration of free sites ��S� and constituent��C2H5

, �C, and �H� surface coverages are given by

�C2H5+ �C + �H + �S = 1. �A4�

Generally the carbon and ethane coverages are low com-pared with the hydrogen coverage when a hydrocarbon/hydrogen mixture flows over a metallic catalyst. Carbon cov-erage of 0.2 on iron at lower temperatures has beenreported,42 which is of the same order of magnitude with thecarbon yield reported in this study. According to

�H = 1 − �C2H5− �C − �S � 1 − �S �A5�

and

�H = �H0 �1 − �C2H5

− �C� , �A6�

where �H0 is the hydrogen coverage without hydrocarbon, it is

believed that the overall carbon coverage is rather low. Oneindication of this can be found from the carbon yield RM

determined around 0.2 that can be quoted to the carbon cov-erage if it is expected that each carbon adsorbed contribute tothe CNT growth. From reaction �10�

�H2 = �S

2�k2PH2� . �A7�

Then the ratio �H/�S can be expressed as

�S = �H/�k2PH2�1/2. �A8�

Thus from expressions �A1�–�A3� and �A8�,

G = k5k4�ACk3K1/k2�1/2�H1/2�PC2H6

/PH2�1/2. �A9�

In Table III, we have summarized the rate of CNT growth asa function of the rds in function of the partial pressures PC2H6and PH2

and the hydrogen coverage �H.

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TABLE III. Expressions of the CNT rate growth according to the rds de-fined by reactions �9� and �11�–�13�.

rds G Equation

1 k5k4�ACk3K1 /k2�1/2�H1/2�PC2H6

/ PH2�1/2 �A10�

3 �k5k4 /AC2 ��K3k1 /k2�1/2�H

−3/2�PC2H6/ PH2

�1/2 �A11�4 k5K4�k3k1 /AC�1/2�PC2H6

�1/2 �A12�5 K5k4�k3k1AC/k2�1/2�H�PC2H6

/ PH2�1/2 �A13�

184705-11 Growth of multiwalled carbon nanotubes J. Chem. Phys. 124, 184705 �2006�

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