Carbon Nanotube Growth by Catalytic Chemical Vapor Deposition

J. Phys. Chem. C 2010, 114, 4773–4782

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Carbon Nanotube Growth by Catalytic Chemical Vapor Deposition: A Phenomenological Kinetic Model N. Latorre, E. Romeo, F. Cazanã, T. Ubieto, C. Royo, J. I. Villacampa, and A. Monzoń* Institute of Nanoscience of Aragon, Department of Chemical and EnVironmental Engineering, UniVersity of Zaragoza, 50009 Zaragoza, Spain ReceiVed: July 21, 2009; ReVised Manuscript ReceiVed: February 24, 2010

A Phenomenological Kinetic Model has been developed that includes all the relevant steps involved in CNT growth by CCVD,that is, carbon source decomposition, nanoparticle surface carburization, carbon diffusion, nucleation, CNT growth, and growth termination by catalyst deactivation or by the effect of steric hindrance. Here we emphasize the importance of using a proper kinetic description of all the stages, in particular the initial carburization-nucleation and the growth cessation. We have discussed the different mechanisms proposed to explain the critical step of carburization-nucleation and have used an autocatalytic kinetic model to describe it. The two parameters involved in this autocatalytic equation allow a very good fit of the initial induction period usually observed during the growth of CNTs. In addition, rigorous formulations of the main causes of CNT growth cessation (catalyst deactivation by several causes and steric hindrance) have been proposed. The developed model is a versatile tool of potentially general application. In this paper, we have applied it to fit data obtained in our lab, and also to super growth of VA-SWNT experimental data published in the literature. In all cases the values obtained for the kinetic parameters have realistic physical meaning in good agreement with the mechanism of CNT formation. 1. Introduction 1,2

The attractive possibilities of using the outstanding properties of carbon nanotubes and other new carbonaceous nanomaterials in a broad range of new applications3-5 is motivating substantial research effort in practically all the fields of nanoscience and nanotechnology. However, extensive use of these materials requires the development of scalable and selective production processes. Catalytic chemical vapor deposition (CCVD) has become probably the main technique for the synthesis of carbon nanotubes, including selective production of single-walled nanotubes (SWNTs) using many carbon sources and catalysts.6,7 In addition to the usual CCVD method, that uses large surface area porous supported metallic catalysts,8-10 the production of layers of vertically aligned VA-SWNT is assuming increasing importance.11,12 VA-SWNTs can also be produced by CCVD if the catalyst composition and the operating parameters are optimized for the synthesis conditions.13,14 Furthermore, this technique can be improved, for example combining a dip-coat catalyst loading process15 with the alcohol catalytic chemical vapor deposition (ACCVD) method.12,16-18 Other methods used are water-assisted CVD,13,19 oxygen-assisted CVD,20 point-arc microwave plasma CVD,21 molecular-beam synthesis,22 and hotfilament CVD.23 These new advances have succeeded in significantly increasing the overall yield, but improvements in SWNT quality and control over chirality are still necessary, particularly when considering electrical and optical applications.18 Even though the formation and growth mechanisms of carbon nanotubes by CCVD have been extensively studied in the past,24-35 there is no general agreement about what the critical steps are. Most authors propose that this mechanism includes the stages of hydrocarbon (or another carbon source such as * To whom correspondence should be addressed. E-mail: amonzon@ unizar.es. Tel: +34 976 761157. Fax: +34 976 762142.

CO) decomposition over the metal surface, carbon diffusion through the particles27,31-35 and/or atomic carbon surface transport,36,37 and finally carbon precipitation forming CNTs. Although this form of carbon accumulation allows the catalyst to maintain its activity for an extended period of time, catalyst deactivation can occur through the formation of encapsulating carbon over the surface of the metal particles.34,36 The deactivation phenomenon can be reversible as a consequence of gasification of this type of carbon by oxygen,19 water,38-41 or hydrogen42-45 that can be added, or be present, in the feed. Recently, Amama et al.,46 have shown that another deactivation cause may be the metallic nanoparticles sintering by an Ostwald ripening mechanism. These authors also demonstrate that the presence of water vapor inhibits the sintering, extending the catalyst lifetime. Additionally to catalyst deactivation, it has been considered other causes of carbon growth cessation as steric hindrance,47 defect diffusion to the growth front48 or when the forming CNTs are not able to make new bonds because are close up, and no more sites are available for incorporation of more acetylene molecules.49 Other phenomena such as initial catalyst activation can also occur if the catalyst is not previously reduced before the reaction.47,50 The physical-chemical description of the steps involved in the CNT growth process has usually been tackled by kinetic models that take into account only some of the stages (e.g., nucleation and initial growth51). Other proposed models describe the process in steady state,31,33,52 without considering that it is strongly time-dependent. These models are usually used to fit experimental data of (i) stationary carbon formation rate as a function of the operating conditions,42,43,33,52 (ii) evolution over time of the carbon formation rate,53-55 and (iii) evolution over time of mass (or length of CNTs) of carbon accumulated.15-18,44-47,56 Obviously, a rigorous description of all these stages necessarily implies obtaining very complex mathematical models with too many parameters, which eventually hinders their application

10.1021/jp906893m  2010 American Chemical Society Published on Web 03/02/2010

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for the analysis of kinetic data. On the other hand, the simplest models are frequently used owing to the fact that they are easy to apply and understand, although obviously these models are unable to describe the complete process accurately. A compromise solution is to find models that consider the main critical stages of the process without excessively increasing its mathematical complexity and therefore the number of parameters. In any case, it is necessary to have a sufficient quantity of precise experimental data, for which in situ techniques are especially suitable. Among the in situ techniques, the most commonly used to follow the growth of carbon nanotubes are (i) environmental HR-TEM,35-37,56-59 (ii) optical absorbance,15-18,48 (iii) Raman scattering60-62 and Global Raman Imaging (GRI),63,64 (iv) time-resolved reflectivity (TRR) of laser beams,56,65-67 (v) gas analysis by mass spectrometry of time-evolved gases,47,68 (vi) thermogravimetric techniques,44,45,55,69,70 (vii) videography using optical microscopy,71,72 and (viii) differential electrical mobility (DMA), applicable to the floating catalyst CVD method.73,74 In addition to these techniques, SEM images taken after different reaction times have also been used to follow the kinetics of vertically aligned single wall carbon nanotube (VASWCNT) growth.75,76 Recently, our group has developed kinetic models to investigate the growth of carbon nanotubes. These models were successfully applied to studying the data obtained with in situ thermogravimetric systems in reactions with different carbon sources and catalysts.44,45,69,70,77 In the different examples studied, we have considered the steps of deactivation-regeneration,45,70 catalyst activation,47 or growth termination by steric hindrance.47 Despite these efforts, an adequate description of the critical initial induction period of time has not been attained. In fact, the evolution of the rest of the steps is determined by this initial period, when some surface carburization of the metallic nanoparticles and carbon nanotube nucleation takes place. It has been observed that the length of this period is strongly dependent on the operating conditions.44,60 Thus, it has been found that the induction period becomes longer as the partial pressure of the carbon source diminishes,44,60,68,70 indicating that the initial carburization-nucleation step is controlled by the feeding of carbon atoms. With the aim of obtaining a more complete description of the growth process, in this work we present a kinetic model including all the above considerations, considering especially the initial carburization-nucleation step and the growth termination, by catalyst deactivation and/or by the effect of steric hindrance. However, we have tried to reduce the number of empirical equations using parameters with a physical-chemical meaning. To show that the developed model could be a versatile tool for general application, we have used it not only to fit data from our lab, but also other data as for example super growth of VA-SWNT experimental data. 2. Kinetic Model of CNT Growth In Figure 1 is presented a simplified scheme of the interfaces developed in the system “gas-phase/catalyst/CNT” during the CNT growth. After hydrocarbon (or other carbon source) decomposition, adatoms of carbon and gaseous hydrogen (or CO2) and other hydrocarbons are left on the surface of the metallic nanoparticles. These adatoms diffuse from interface 1 to interface 2 where, after nucleation, they there are incorporated to the CNT. In addition, other phenomenon like catalyst deactivation or steric hindrance can occurs simultaneously decreasing, or even finishing, the CNT growth. In the next paragraphs is presented the kinetic description of all these stages.

Latorre et al.

Figure 1. Scheme of the interfaces developed in the system gas-phase/ catalyst/CNT during the CNT growth.

Formation of Surface Carbide and Carbon Nanotube Nucleation. According to the mechanism proposed by Alstrup,29 after hydrocarbon decomposition the remaining carbon atoms react with the metallic nanoparticles at the surface forming a metastable carbide, which in the reaction conditions decomposes leaving carbon atoms at the metallic subsurface. After this decomposition-segregation step, the carbon atoms are introduced inside the metal particles,29 determining in this way the value of the carbon concentration at the carbide-metallic nanoparticle interphase. However, for this stage Puretzki et al.56 consider that the carbon atoms are dissolved on the metallic nanoparticles forming a highly disordered “molten” layer on their surface. Because of much higher carbon diffusivity in the disordered layer compared to the ordered solid phase, the carbon atoms diffuse along this layer and precipitate into a nanotube. Helveg et al,36 based on density functional theory calculations and on in situ TEM observations, propose that the growth of CNFs on a Ni-Mg-Al catalyst involves surface diffusion of both carbon and nickel atoms. Thus, the graphene layer nucleation and growth is explained as a dynamic formation and restructuring of monatomic step edges at the nickel surface. In the case of SWNT growth, in a recent paper47 we have considered that the carbon atoms enter into the metallic nanoparticles through the clean surface (interface 1) and that they leave the metallic phase through interface 2 when forming the SWNT. The driving force for the surface or bulk diffusion from interface 1 to interface 2 is the difference in chemical potential between the two interfaces. Comparing these two diffusion pathways, Lin et al.37 consider that the surface diffusion is dominant due to its lower activation barrier, as a consequence of a lower coordination number. After the carbon concentration has reached a certain threshold, nucleation of ordered forms of carbon (e.g., hexagons) occur forming nuclei of graphene caps.38,79,80 The development of the graphene caps is driven by the need to minimize the energy associated with the nucleation of a graphene layer when constrained to grow and form a nanotube.37 Supposing that the process of nucleation follows an autocatalytic kinetics47,81 the rate of carburization-nucleation can be expressed as

CNT Growth by CCVD

rS )

J. Phys. Chem. C, Vol. 114, No. 11, 2010 4775

dCS ) ψS(1 + KSCS)(CSm - CS) dt

(1)

where CS is the concentration of surface carbide and has units of (g · C/g · cat). In this equation, the autocatalytic contribution can be related to an autoassembled process81 that, for example, occurs during the formation of the caps of the SWNT.38,79,80 The term ΨS represents the intrinsic kinetic function of carburization, and for a given catalyst its value depends on the reaction conditions. ΨS has units of time-1. CSm represents the maximum surface carbide concentration attainable on the surface of metallic particles at the side gas phase (interface 1), and this value determines the thickness of the carbide layer. This thickness has been associated56 to the number of layers of the MWNT precipitated from the metallic particles. On the other hand, the value of the parameter KS determines the weight of the autocatalytic effect on the carburization kinetics, and has units of (g · cat/g · C). Integration of eq 1 gives the evolution of CS over time in terms of degree of carburization, θS, as follows

CS S0 - exp(-ψCt) θS ) ) CSm S0 + KS′ exp(-ψCt)

(2a)

where the terms ΨC, KS′, θS0 and S0 are given by

CS θS ) ) CSm

(

)

1 exp(-(t - tS)/τS) KS′ ; (1 + exp(-(t - tS)/τS))

1-

τS )

1 ψC

(5) The term τS can be considered as the lifetime of the carburization process. Considering the eq 1 as the kinetic model of carburization, several particular cases can be considered. Thus, if the value of K′S is zero, or its value is very low compared to ΨS, the autocatalytic contribution is negligible, and then the kinetics of carburization will be described by a first order law

dCS CS ) ) ψS(CSm - CS) S θS ) dt CSm 1 - (1 - θS0)exp(-ψSt) (6) In this case, rS does not pass through a maximum, decreasing monotonically over time. In the opposite case, if the autocatalytic contribution is the dominant effect, that is, KS′ is very high compared to ΨS, then the global kinetic function of carburization is simplified to ψC ) ψSKS′ , and the kinetics of carburization will be

ψC ) ψS(1 + KS′ ); KS′ ) KSCSm CS0 1 + KS′ · θS0 θS0 ) ; S0 ) ; 1 < S0 < ∞ CSm (1 - θS0)

(2b) K′S is an adimensional parameter, and ΨC represents the global kinetic function of carburization that has units of time-1. The term S0 is a lumped adimensional parameter related to the initial degree of catalyst carburization. CS0 is the initial concentration of the surface carbide, and therefore the term θS0 is the initial degree of carburization. Usually, θS0 is zero or negligible, then S0 ) 1, and the eq 2a is simplified to

θS )

CS 1 - exp(-ψCt) ) CSm 1 + KS′ exp(-ψCt)

(2c)

In terms of time evolution, the rate of surface carburization can be now expressed as

rS )

CSmS0 · ψC(1 + KS′ )exp(-ψCt)) dCS ) dt (S0 + KS′ exp(-ψCt))2

(3)

From the above equation it is deduced that rS passes through a maximum, (rS)max, at a time, tS, equal to

ln(KS′ /S0) tS ) ; ψS(1 + KS′ )

(rS)max )

CSmS0ψS · (1 + KS′ )2 2KS′

(4)

Taking into account the above equation, eq 2a can be rewritten as

Figure 2. Effect of the parameters ψS (Figure 2a) and KS′ (Figure 2b) on the degree and rate of carburization, θS. The values of the parameters used in the simulations are: K′S ) 100 (Figure 2a), ΨS ) 1 min-1 (Figure 2b), Ψd ) 0.08 min-1, Ψr ) 0.005 min-1 and jC0 ) 1 gC/g cat · min. To present in the same scale, the values of rS in Figure 2b are divided by the following factors: curve 1, factor ) 1; curve 2, factor ) 5; curve 3, factor ) 25; curve 4, factor ) 125; and curve 5, factor ) 625.

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dCS ) ψSKSCS(CSm - CS) S dt CS 1 ) θS ) CSm 1 + β exp(-ψCt) β ) (1 - θS0)/θS0

Latorre et al.

(7)

In this particular case, θS0 must be have a nonzero value in order to the carburization process can go forward. Finally, if the carburization of the metallic nanoparticles occurs very fast, that is, ΨC takes a very high value, then CS will attain the value of CSm almost instantaneously. Then CS ) CSm and θS ) 1. The influence of carburization parameters, ψS and KS′ , on the evolution of both the rate and the degree of carburization is presented in Figure 2a,b. The simulation results in Figure 2a indicate that as the value of the kinetic function of carburization increases, for example, due to an augment of the hydrocarbon concentration and/or of the reaction temperature, the surface carburization step occurs before and more rapidly. Furthermore, as it is deduced from eq 4, the maximum rate of carburization, (rS)max, appears at shorter times attaining higher values. Obviously, if the operating conditions (feed composition, reaction temperature, etc.) are changed, besides ψS, the rest of the kinetic parameters also will vary and the global effect on the CNT growth rate may be masked. However, in the case of comparison of two different catalysts working at the same operating conditions probably their intrinsic carburization functions will be different and finally the rate of CNT growth also will be different. In the last term, the affinity of each catalyst to be carburized will depends on their physical and chemical properties. These considerations are valid for all the parameters that appear in the kinetic model. The ultimate goal in the developing of more fundamental models is to establish the relationship between the physical-chemical properties of the catalyst, and the intrinsic kinetic parameters which appears on each of the stages of the CNT growth. In regard to the effect of KS′ , as the value of this parameter increases, the carburization occurs at higher times, and the value of (rS)max is boosted up notably; see Figure 2b. Therefore, KS′ can be considered as a measure of the resistance to which the autocatalytic event occurs during the carburization step. The higher it is this parameter, the more difficult seems to carry out this stage, and therefore, more retarded is the growth of the CNTs; see Figures 3 and 4. Rate of Carbon Nanotube Growth. This stage begins with the formation-precipitation of the carbon nanotubes at the interface 2. When carbon concentration has reached a certain threshold, nucleation occurs forming graphene caps.38,79,80 This fact generates the interface CNT-metal, interface 2, and the carbon flux is maintained because the nanotube structure provides a thermodynamic sink for the carbon, and as a consequence, the carbon concentration at the interface 2 is kept low. The rate of the diffusion-precipitation process determines the rate of formation of carbon nanotubes. In addition, bearing in mind the possible effect of catalyst deactivation, the rate of CNT formation can be expressed in terms of catalyst activity, a, as follows78,82

(rC)t ) (rC)0a;

a ) (rC)t /(rC)0

(8)

The term (rC)0 corresponds to the CNT growth rate in the absence of any deactivation phenomenon, which can be

Figure 3. Effect of the parameter KS′ on the reaction rate and CNT concentration. Case with catalyst deactivation. The values of the parameters used in the simulations are ΨS ) 1 min-1, Ψd ) 0.08 min-1, Ψr ) 0.005 min-1 and jC0 ) 1 gC/g cat · min.

Figure 4. Effect of the parameter KS′ on the reaction rate and CNT concentration. Case without catalyst deactivation or any steric hindrance. The values of the parameters used in the simulations are ΨS ) 1 min-1 and jC0 ) 1 gC/g cat · min.

expressed in terms of difference of chemical potential force between the interfaces 1 and 2 as83

(rC) )

∂mC ) DCe∇CC ∂t

(9)

In this equation, mC represents the mass of the CNTs accumulated over the catalysts and has units of (g · C/g · cat). The term DCe is the effective diffusivity of the carbon and it depends on the type of the diffusion phenomenon (bulk, surface of subsurface) that takes place at the metallic nanoparticles from the interface 1 to interface 2. Assuming unidirectional diffusion, the above expression can be simplified obtaining

(rC)0 )

dmC | ) kC(CS - CF) dt t)0

(10)

The term kC is the effective coefficient of carbon transport, has units of time-1, and depends on the average size of the metallic crystallites, the metallic exposed area, and the carbon atom diffusivity on the metallic nanoparticles.25,26,32,33,47 CF is the carbon concentration at the interface CNT-metal.

CNT Growth by CCVD


Catalyst Deactivation. The deactivation rate, rd ) -da/dt, must be described according to an appropriate deactivation kinetic model related to the deactivation cause.82,84-86 For example, if the activity decay is caused by the reversible formation of encapsulating carbon, the catalyst does not suffer a complete deactivation, maintaining a residual level of activity. In these conditions, the net rate of activity variation can be expressed as87

rd ) -

da ) ψd · ad - ψr(adm - a) dt

(11)

The terms ψd and ψr are respectively the “deactivation and regeneration kinetic functions”, and both depend on the operating conditions during the CCVD. If the activity decay is irreversible, then Ψr ) 0 and the deactivation rate will be78,82

rd ) -

da ) ψdad dt

(12)

It should be noted that in eq 11 or 12 the values of the kinetic orders d and dm depend on the reaction mechanism, that is, the number of sites involved in the controlling steps of the main (m) and deactivation (h) reactions:78,87 d ) (m + h - 1)/m and dm ) (m + 1)/m. Thus, only values with a real physical meaning for m and h, that is 1 or 2, were assessed. Therefore, cases involving 3 or more active sites in an elemental step were not considered since they are quite improbable.88 This means that m and h are not fitting parameters, and therefore their values must be selected before the data fitting. For the case of deactivation by sintering of the metallic, Bartholomew84,85 had showed that most of the cases can be described by the following empirical equation

rd ) -

da ) kdS(a - aS)n ; dt

n ) 1 or 2

(13)

The term aS is the residual activity of the catalyst. kdS is the deactivation kinetic constant of sintering, and n is the deactivation kinetic order. Monzoń et al.89 have shown that when n ) 1, eq 13 is a particular case of eq 11 for a reversible sinteringredispersion process. Steric Hindrance. For the SWNT it has also been considered that the growth on porous catalysts can be slowed down as a consequence of the steric hindrance.46,47 Thus, for the typical high-surface-area porous catalysts, the steric hindrance can appear when the length of the CNT is comparable to the diameter of the pores of the catalyst, which can be on the order of dozens of nanometers or even smaller. In this situation, the growth process is ultimately obstructed due to the lack of space for displacement inside the pores and to the tube-tube interactions.47 The degree of interaction of a growing SWNT with the catalyst support, and/or with other nanotubes, depends on the catalyst pore sizes. The larger the available space on the catalyst, the less hindered will be the insertion of new carbon atoms at the interface. Similar considerations have been made to explain the deactivation of catalytic carbonaceous materials used to produce hydrogen by methane decomposition.90,91 In other cases, such as in the growth of VA-SWNT, Amama el al.46 argue that growth termination occurs as a result of a chemical-mechanical coupling of the top surface of the carpet, which causes an energetic barrier to the relative displacement between neighboring nanotubes. Similar considerations are made by Meshot and

Hart67 that found that growth termination is accompanied by loss of alignment among the CNTs. Recently, Vinten et al.71 have found that for tall forests termination is a combined effect between the build-up of tension on individual nanotubes and the forces present in the forest that keep the nanotubes attached to the substrate and to one another. All these observations can also be considered as phenomenon of steric hindrance that delays the insertion of new carbon atoms at interface metal-SWNT. We assume here that this hindrance effect can be expressed as a potential function of the accumulated mass of carbon

CF ) ξHmCp

(14)

The term ξH is called the “hindrance factor”. In addition, the value of the hindrance kinetic order, p, determines if this effect is progressive over time, p > 1, or decays over time, p < 1. As commented above, the values of both parameters depend on the catalyst texture. Finally, combining eqs 2a, 8, 10, and 14, the rate of CNTs formation can be calculated from the following expression

jC0(S0 - exp(-ψCt)) dmC a + ξHCmCp a ) dt (S0 + KS′ exp(-ψCt))

(15)

where the terms ξHC and jC0 are defined as

ξHC ) ξHkC ;

jC0 ) CSmkC

(16)

jC0 has units of (g C/g cat · min) and can be considered as the intrinsic CNT growth rate for the fresh catalyst. Similarly, the term ξHC represents the intrinsic constant rate of steric hindrance, and has units of (g cat(1-p)/g C(1-p) · min). In summary, the set of differential eqs 11 (or 13) and 15 define the kinetic model for studying the CNT growth, as a function of the operating conditions and of the potential effects of catalyst deactivation and/or steric hindrance. The parameters involved are ξHC and p, accounting for the steric hindrance, Ψd and Ψr (or kdS and aS) relating to catalyst deactivation and jC0, that is the intrinsic growth rate. This model can be simplified into many particular cases; some of them are described in the following paragraphs. One of the most relevant aspects of this model is the description of the carburization stage. In regard to this point, Figures 3-5 show some examples of the effect of ΨS and KS′ on the CNT growth. In Figures 3 and 4, it can be seen that as the value of K′S increases, the induction period of CNT formation becomes longer, because the more demanding the autocatalytic effect, the more time is needed to achieve the initial stage of nucleation. In contrast, if KS′ is zero or takes a low value, the induction period is not appreciable because the carburization steps occur very quickly. In regard to the effects of catalyst deactivation, comparison between Figures 3 and 4 shows up that the deactivation causes a diminution in the reaction rate and consequently on the mass of CNTs accumulated over the catalyst. Furthermore, the maximum rate of reaction appears later, and with a lower value (Figure 3). In relation to the evolution of mC, an increase in KS′ produces a decrease in the amount of CNTs formed because the catalyst deactivation occurs before the carburization step has been completed and the nanoparticles can not transport the

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Latorre et al. used also to describe other deactivation causes such as sinteringredispersion of supported catalysts,89 which could include the case of sintering by Ostwald ripening.46 At this point, mC could be calculated integrating numerically the following equation

jC0(S0 - exp(-ψCt)) dmC ) (a + (1 - aS) × dt (S0 + KS′ exp(-ψCt)) S exp(-(ψd + ψr)t)) (19)

Figure 5. Effect of the parameter ΨS on the reaction rate and CNT concentration. The values of the parameters used in the simulations are KS′ ) 10, Ψd ) 0.08 min-1, Ψr ) 0.005 min-1, and jC0 ) 1 gC/g cat · min.

carbon atoms. On the other hand, if there is no catalyst deactivation, after the initial induction period, the reaction rate increases until it reaches a constant value (Figure 4), and consequently mC increases linearly along time (Figure 4). In Figure 5, the results relating to the impact of the parameter ΨS are presented. Thus, an increase in ΨS causes a shortening of the induction period and simultaneously an increase in the maximum reaction rate. Both phenomena are explained considering that the termination of the carburization stage occurs before than the catalyst will be deactivated. The results in Figures 2 to 4 correspond to the case where there is no effect of steric hindrance (ξHC ) 0). However, qualitatively similar results will be obtained if the growth termination is caused by steric hindrance, or by both causes acting simultaneously. In the following sections we present the equations corresponding to some particular cases of the kinetic model developed here. Case 1. CNT Growth Termination by Catalyst Deactivation Alone. If the CNT growth is not hindered by any steric impediment, ξHC ) 0, or the value of CF is very low compared with CS, eq 15 can be simplified to

rC )

jC0(S0 - exp(-ψCt)) dmC ) a dt (S0 + KS′ exp(-ψCt))

(17)

The value of the catalyst activity, a, is calculated from eqs 11 or 14 depending on the deactivation cause, and then the fitting parameters in this case are Ψd, Ψr (or kdS and aS), jC0, KS′ and ΨS. In the activity, decay is caused by the reversible formation of encapsulating carbon, the explicit relationship for the activity versus time relationship depends on the values of m and h. For example, in the simplest case, m ) h ) 1, the deactivation rate and the catalyst activity are given by the following expressions

-

da ) ψda - ψr · (1 - a) S a ) aS + (1 - aS) × dt aS ) ψr /(ψd + ψr)

exp(-(ψd + ψr)t)

(18) As it is has been said before, the term aS represents the residual activity of the catalyst. This equation has been also

If the deactivation is irreversible, Ψr ) aS ) 0, and the carburization step is very rapid, that is, KS′ f 0 and ΨS is very high, then CS ) CSm, In these conditions eq 19 results in a simpler case

dmC ) jC0 exp(-ψdt) dt

(20a)

After integration, it is obtained that

mC(t) )

jC0 ψd

(1 - exp(-ψdt))

(20b)

Equation 20b is formally equal to that used to describe many studies of the production of layers of vertically aligned singlewalled nanotubes, VA-SWNT, including the so-called supergrowth proces,17-19,48,61,75 that is receiving much attention in the last years

(

( ))

H(t) ) γ0τ0 1 - exp -

t τ0

(21)

The equivalence between the kinetic parameters of both models is given by

H(t) ∝ mC(t); γ0 ∝ jC0 ; Hmax ) γ0τ0 ∝ jC0 /ψd

τ0 ) 1/ψd

(22)

If the VA-SWNT are not totally straight, the real relation mass ∼ height could be measured and correlated experimentally, and then use this correlation as a calibration. In summary, eq 21 can be considered as a particular case of the model described in this work. It is obvious that for the cases described by eqs 15, 20a, and 20b, there are many others that can be easily deduced and applied for each specific particular study. For example, if θS0 ) 0, and the carburization step is described by eq 6, instead of considering that CS ) CSm, the following intermediate case is obtained

(

mC ) jC0

)

(1 - exp(-ψSt)) (1 - exp(-(ψd + ψS)t)) ψd + ψS ψS (23a) rC ) jC0(1 - exp(-ψSt))exp(-ψdt)

(23b)

These equations will be applied later in the analysis of VASWNT supergrowth data.

CNT Growth by CCVD


Case 2. CNT Growth Cessation by Steric Hindrance Only. If the catalyst does not suffer deactivation, that is, a ) 1, and the growth cessation is only caused by steric hindrance, rC is calculated as

jC0(S0 - exp(-ψCt)) dmC + ξHCmCp ) dt (S0 + KS′ exp(-ψCt))

(24)

In the above equation the fitting parameters are ξHC, p, jC0, KS′ , and ΨS. Similarly to the above case, if p ) 1, KS′ f 0, ΨS is very high, then CS ) CSm and therefore eq 24 is simplified to

dmC + ξHCmC ) jC0 dt

(25a)

The analytical solution of the above equation is

mC(t) )

jC0 ξFC

(1 - exp(-ξFCt))

(25b)

The above equations are identical to eqs 20a and 20b, replacing Ψd by ξHC. This result also indicates that by means of kinetic studies alone it is not easy to distinguish what is the true underlying reason, or reasons, for CNT growth decay. In fact, both causes considered here can act simultaneously, clearly complicating the elucidation of the parameters and requiring a previous design of a careful experimental study specifically designed to obtain them. Case 3. No Effect of Steric Hindrance or Catalyst Deactivation. This case corresponds to the simplest situation where the growth rate is maintained along the reaction.38,51 In this situation, eq 15 is simplified to

(rC)t )

(

dmC S0 - exp(-ψCt) ) jC0 dt S0 + KS′ exp(-ψCt)

)

(26a)

The analytical integration of the above expression gives

[ (

mC(t) ) jC0 t +

)(

KS′ + 1 S0 + KS′ exp(-ψCt) ln ψCKS′ S0 + KS′

)]

(26b)

This solution is represented in the curves corresponding to the evolution of mC in Figure 4. 3. Application of the Kinetic Model to Experimental Data In Figures 6-8, we present different results of the application of the kinetic model developed in this contribution and some comparisons with other simpler models used in the literature. Thus, Figure 6a,b shows two examples taken from a kinetic study of VA-SWCNT growth, synthesized by catalytic chemical vapor deposition of ethanol.18 In that paper, the authors study the effect of the partial pressure of ethanol and of the synthesis temperature over the evolution over time of the VA-SWCNT thickness. The kinetic analysis of the results obtained was made using a kinetic model that corresponds to eq 21, here called the Exponential Model. As has been discussed before, the Expo-

Figure 6. Fittings comparison of VA-SWCNT film thickness and growth rate data taken from Figure 2a,b of ref 18. (Panel a) Data corresponding to γ0 ) 2.7 µm/min and τ ) 3.6 min; (Panel b) data corresponding to γ0 ) 5.2 µm/min and τ ) 6 min.

nential Model is a particular case of the Phenomenological Model described by eq 15. The implicit simplifications contained in eq 21 do not allow, for example, an explanation of the presence of a maximum in the growth rate, as is shown in Figure 6a,b. However, this maximum is usually observed in this type of experiment.18,44-47,56,68 Furthermore, as has also been discussed before, another important phenomenon usually observed is the presence of an initial induction period that the Exponential Model is unable to predict. Table 1 shows the values of the kinetic parameters obtained with both models. To achieve more statistical consistency, the parameter estimations have been made fitting simultaneously both (i) the VA-SWCNT thickness data and G-band (i.e., the measurement proportional to the amount of CNT accumulated on the catalyst in each case); and (ii) the growth rate data. Thus, the objective functions minimized are defined as follows

SSRTotal ) SSR1 + SSR2

(27a)

∑ (yexp - ycalc)2

(27b)

SSR )

yexp

In the case of SSR1, the values of the variable “yexp” are the VA-SWCNT thickness or the G-band intensity, while for SSR2 “yexp” corresponds to the growth rate data.

4780


Latorre et al. TABLE 1: Kinetic Parameters of Experimental Data of VA-SWCNT Growth: Comparison of the Models Phenomenological Model, described by eq 15 parameter jC0a ψS (min-1) KS′ (-) ψd (min-1) ψr (min-1) SSRTotal

data in Figure 6a data in Figure 6b data in Figure 7 (ref 18) (ref 18) (ref 60) 3.118 ( 0.041 7.333 ( 0.767

5.131 ( 0.046 890.01 ( 12.45

0.311 ( 0.005

0.167 ( 0.003

0.38

0.505 ( 0.085 0.00021 ( 0.00012 456.4 ( 338.75 0.015 ( 0.001 0.0015 ( 0.0002 283.8b

1.18

Exponential Model, described by eq 21 2.453 ( 0.111 5.252 ( 0.133 0.094 ( 0.002 γ0 ∝ jC0Z(a) τ0 ) 1/Ψd (min) 4.111 ( 0.304 5.766 ( 0.300 204.5 ( 5.8 SSRTotal 7.86 13.19 999.8b

Figure 7. Comparison of fittings of the G-band intensity data obtained with the Phenomenological and Exponential Models. The experimental data corresponds to that in the Figure 4b of ref 60.

Figure 8. Simultaneous fitting of CNT concentration (panel a) and growth rate (panel b). Data taken from ref 44.

As can be seen in Figure 6a,b, the fittings obtained with the Phenomenological Model are more realistic than those obtained with the Exponential Model because the former is able to explain

a Units corresponding to the data in the figures where the data are taken. b In this case the SSR2 is zero because there is no growth rate data.

the presence of a maximum value of the reaction rate. According to our model, initially the growth rate is zero, and as the carburization step progresses, the reaction rate increases quickly reaching a maximum. Then the rate decreases as a consequence of the catalyst deactivation, or the steric hindrance. The different behavior observed in Figure 6a,b is a consequence of the different operating conditions used in both cases.18 Thus, in the case of Figure 6b the carburization step is very fast, ΨS ) 890 min-1, and the maximum rate appears very soon, in accordance with the tendency presented in Figure 5. In this case, eq 18 can be simplified to eq 21 and as a consequence the values of the kinetic parameters obtained with both models are very similar (see Table 1). The number of parameters used with our model, in this case 3, is higher that with the Exponential Model, 2. However, the inclusion of this additional parameter, ΨS, is totally justified from a physical point of view. The data in Figure 7 corresponds to a study of SWNT synthesis by CCVD followed by in situ RAMAN spectroscopy60 also carried out by Maruyama’s group. As can be seen, the evolution of the G-band signal intensity, proportional to the amount of SWNT formed, shows a very long induction period due to the very low value of the partial pressure of ethanol (0.1 Torr) used during the experiment. In these operating conditions, the rate of carburization of the metallic particles is low, and therefore the time needed to complete the nucleation is greater causing an enlargement of the induction period. Obviously, from Figure 7 and the values presented in Table 1 it is clear that the fitting obtained with the Exponential Model is clearly worse than that obtained with the Phenomenological Model. With respect to the values of the parameters, the fitting of the long induction period is solved by the present model giving a low value for the ΨS parameter and a high value for the KS′ parameter, in agreement with the trends presented in Figures 3 and 4. Figure 7 also shows the evolution of the two rates, SWNT growth and carburization, calculated with the parameter values estimated using the Phenomenological Model. The initially low reaction rate of CNT growth is a consequence of the necessity of previous carburization of the catalyst nanoparticles. This

TABLE 2: Application of Phenomenological Model to Data in Figure 8a,b44 % CH4 jC0 (g C/g cat.min) ψS (min-1) KS (g cat./g C) ψd (min-1) ψr (min-1)

2.5% -3

5.0% -4

5.91 × 10 ( 1.14 × 10 1.16 × 10-3 ( 3.58 × 10-5 502.372 ( 12.790 0.018 ( 4.453 × 10-4 0

-2

7.5% -5

1.11 × 10 ( 7.27 × 10 2.34 × 10-3 ( 6.85 × 10-5 502.372 ( 14.653 0.032 ( 8.467 × 10-5 0

-2

10.0% -4

1.53 × 10 ( 4.19 × 10 3.91 × 10-3 ( 7.56 × 10-6 502.371 ( 13.231 0.046 ( 2.366 × 10-3 0

-2

1.81 × 10 ( 4.47 × 10-4 8.40 × 10-3 ( 1.64 × 10-4 502.368 ( 11.123 0.062 ( 1.792 × 10-3 5.448 × 10-3 ( 1.88 × 10-4

CNT Growth by CCVD figure effectively shows that the maximum CNT growth rate (rCmax) only is attained after that the catalyst is almost completely carburized. This result is only explained by our model, while the Exponential Model predicts that initially the reaction rate is the higher. Finally, the results presented in Figure 8a,b correspond to a study of CCVD of CH4 on a Ni-Al catalyst, carried out by our group.44 These figures correspond to the results obtained analyzing the influence of the methane concentration on the CNT production and on the reaction rate. Figure 8a shows that the length of the induction period is strongly dependent on the methane concentration, that is, on the methane flow rate,40,49,68 obtaining that the higher methane concentration, the lower the induction period. This result is in accordance with the conclusions obtained from Figure 7. Table 2 shows the values of the parameters of the Phenomenological Model, after simultaneous fitting of the experimental data in Figure 8a,b. In this case, the values of the variable “yexp” on eq 27a are carbon concentration, while for SSR2 “yexp” corresponds to the growth rate data (eq 27b). The evolution of the parameters with respect to the methane concentration indicates that as the CH4 partial pressure increases, the values of jC0 and Ψd also increase. These results explain that at the end of each experiment, the total amount of CNT formed is similar because, in this particular case, the higher reaction rate is compensated by a higher deactivation.68 However, usually it is obtained45,69,70 that the concentration of CNTs increases when partial pressure of methane augment. In regard to the parameters related to the carburization step, the value of ΨS also increases with the CH4 concentration and the parameter K′S remains practically constant. Therefore, in this case the Phenomenological Model can explain the variation of the length of the induction period only modifying the parameter ΨS. This fact indicates that, probably, the parameter K′S is related to the nature of the catalyst, while the parameter ΨS also depends on the operating conditions. The values in Table 2 can be used to estimate the kinetic orders of these parameters with respect to CH4. Thus, the orders for jC0, Ψd, and ΨS are respectively 0.8, 0.9, and 1.3. The kinetic orders for jC0 and Ψd indicate that both stages, the CNT growth and the deactivation, involve an adsorption step of the methane molecules on the metallic particles surface, according to the mechanism proposed for this reaction.31-33 In addition, the value of 1.3 for ΨS can be explained assuming a second order process according to an autocatalytic phenomenon considered for the carburization step. In summary, this work shows that the “Phenomenological Kinetic Model” can be applied to analyze a wide variety of experimental kinetic studies of CNT growth by CCVD. Thus, as a proposal for future work, the specific dependence of the kinetic parameters on the different carbon sources, operating conditions, type and catalyst composition, can be now established using this model. Once this relationship was established, it was possible to select the optimum operating conditions that maximize the CNT production avoiding, or minimizing, catalyst deactivation. 4. Conclusions We have developed a kinetic model that includes all the relevant steps involved in CNT growth. As an improvement of previous models, in the present paper we have stressed the importance of an adequate description of carburization, nucleation, growth, and termination steps. In addition, it has been considered that the termination of the CNTs growth could be

J. Phys. Chem. C, Vol. 114, No. 11, 2010 4781 due to different reasons like catalyst deactivation, or the effect of the steric hindrance. The results presented in Figures 6-8 and in Tables 1 and 2 indicate that the Phenomenological Model presented here is a useful instrument. The modular structure of the proposed model, allows take into account directly the influence all these phenomena, allowing at the same time to discriminate the influence of each stage. Thus, as a relevant result of the model application, it has been found that the extent of the initial induction period observed during the growth of CNTs can be modulated modifying the operational conditions, especially the concentration (or molar flux) of the carbon source. The length of this period can be quantified through the values of parameters ΨS, KS′ and jC0. In all cases the values obtained for the kinetic parameters have realistic physicochemical meaning derived from the mechanism of CNT formation. Finally, the developed model can be simplified, and successfully applied, to many different situations like, for example, the supergrowth of VA-SWNT forests by CCVD. The rigorous application of the Phenomenological Model to these kinetic studies will allows attaining a more realistic description of this process, which is the first necessary step to optimize the production of these materials. Acknowledgment. The authors acknowledge financial support from MICINN (Spain)-FEDER, Project CTQ 2007-62545/ PPQ, and the Regional Government of Aragoń, Departamento de Ciencia, Tecnologıá y Universidad, Project CTP P02/08. Also, the authors thank Professor S. Maruyama for sending the data shown in Figures 6 and 7. References and Notes (1) Saito, R.; Dresselhaus, G.; Dresselhaus, M. S. Physical Properties of Carbon Nanotubes; Imperial College Press: London, 1998. (2) Carbon Nanotubes: Synthesis, Structure, Properties and Applications; Dresselhaus, M. S., Dresselhaus, G., Avouris, Ph., Eds.; Springer: Berlin, 2001; Vol. 80. (3) Delgado, J. L.; Herranz, M. A.; Martıń, N. J. Mater. Chem. 2008, 18, 1417. (4) Jiao, L.; Zhang, L.; Wang, X.; Diankov, G.; Dai, H. Nature 2009, 458, 877. (5) Terrones, M. Nature 2009, 458, 845. (6) Joselevich, E.; Dai, H.; Liu, J.; Hata, K.; Windle, A. H. Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications. In Topics in Applied Physics; Jorio, A., Dresselhaus, G., Dresselhaus, M. S., Eds.; Springer-Verlag: Berlin, 2008; Vol. 111, p 101. (7) Fu, Q.; Liu, J. In Carbon Nanotechnology; Dai, L., Ed.; Elsevier B.V.: New York, 2006; p 81. (8) Kitiyanan, B.; Alvarez, W. E.; Harwell, J. H.; Resasco, D. E. Chem. Phys. Lett. 2000, 317, 497. (9) Herrera, J. E.; Balzano, L.; Borgna, A.; Alvarez, W. E.; Resasco, D. E. J. Catal. 2001, 204, 129. (10) Bachilo, S. M.; Balzano, L.; Herrera, J. E.; Pompeo, F.; Resasco, D. E.; Weisman, R. B. J. Am. Chem. Soc. 2003, 125, 11186. (11) Hata, K.; Futaba, D. N.; Mizuno, K.; Namai, T.; Yumura, M.; Iijima, S. Science 2004, 306, 1362. (12) Murakami, Y.; Chiashi, S.; Miyauchi, Y.; Hu, M.; Ogura, M.; Okubo, T.; Maruyama, S. Chem. Phys. Lett. 2004, 385, 298. (13) Noda, S.; Sugime, H.; Osawa, T.; Yoshiko, T.; Chiashi, S.; Murakami, Y.; Maruyama, S. Carbon 2006, 44, 1414. (14) Zhang, L.; Tan, Y.; Resasco, D. E. Chem. Phys. Lett. 2006, 422, 198. (15) Murakami, Y.; Chiashi, S.; Miyauchi, Y.; Maruyama, S. Chem. Phys. Lett. 2003, 377, 49. (16) Maruyama, S.; Einarsson, E.; Murakami, Y.; Edamura, T. Chem. Phys. Lett. 2005, 403, 320. (17) Einarsson, E.; Kadowaki, M.; Ogura, K.; Okawa, J.; Xiang, R.; Zhang, Z.; Yamamoto, T.; Ikuhara, Y.; Maruyama, S. J. Nanosci. Nanotech. 2008, 8, 1. (18) Einarsson, E.; Murakami, Y.; Kadowaki, M.; Maruyama, S. Carbon 2008, 46, 923.

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Carbon Nanotube Growth by Catalytic Chemical Vapor Deposition

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