Intermetallic Grains through Nucleation and ... - ACS Publications

3266

J. Phys. Chem. C 2007, 111, 3266-3274

Nanodusting of RENi5 Intermetallic Grains through Nucleation and Growth of Carbon Nanotubes (RE t Rare-Earth) M. Tomellini,† D. Gozzi,*,‡ and A. Latini‡ Dipartimento di Scienze e Tecnologie Chimiche, UniVersita` di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy and Dipartimento di Chimica, UniVersita` di Roma La Sapienza, Piazzale Aldo Moro 5, 00185 Roma, Italy ReceiVed: October 6, 2006; In Final Form: December 18, 2006

The kinetics of the methane decomposition at NdNi5 catalyst powder has been studied by means of the thermogravimetry in the temperature range from 475 to 525 °C. The catalytic reaction yields the dusting of the catalyst particles, the size of which is in the domain of microns, in nanocrystals, which are encapsulated at one of the ends of the multiwalled carbon nanotubes with these last being the only carbon-containing byproduct of the reaction. Microscopy and spectroscopic techniques have been employed to characterize both the catalyst nanocrystals and multiwalled carbon nanotubes. The experimental kinetics exhibit an induction period, which is temperature dependent, before the steady state is reached. The behavior of these curves has been described by a kinetic model, the physical basis of which lays on the interplay between the nucleation process of the carbon nanotubes and the detachment of the intermetallic nanosingle crystal from the catalyst grain with these being the locus of the catalytic process.

1. Introduction Since the discovery of carbon nanotubes (CNTs) by Ijima in 1991,1 many papers have been dedicated to this subject and several efforts have been done to understand the main aspects related to their structure,2,3 chemical and physical properties,4-6 as well as their immediate and potential applications. The growth mechanism of the single-walled (SW)CNTs and multiwalled (MW)CNTs has been studied in several synthesis conditions7-9 by making use of different techniques.10,11 The decomposition of light hydrocarbons has been one of the main routes followed for the synthesis of both SWCNTs and MWCNTs, even if this process has been performed by using a large variety of catalysts operating in very different experimental conditions.12-14 Because of this, the formation of the new carbonaceous phase could occur through different pathways,9,15 and at present time there is not an unified scenario describing the nucleation and growth mechanism of the CNTs. To produce large quantities of molecular hydrogen for the envisaged needs16 of the hydrogen economy, it is recognized worldwide that hydrogen cannot be produced through the steam reforming of methane as it occurs today for satisfying the annual world request of 50 Mtons, which implies 275 Mtons of CO2 greenhouse gas. By the wait for H2 produced completely from renewable sources, a way to obtain H2 without releasing CO2 is the methane and light hydrocarbon decomposition that implies also a mass production of CNTs, as recently obtained in using Ni-Mo-MgO catalyst17. Several research groups are involved in this subject by basic18-22 and engineered23-25 studies. Three main problems should be solved to perform the process at industrial scale: (i) the catalyst poisoning, (ii) the decrease of the reaction temperature, and (iii) the conversion yield. The latter aspect appears to be mostly dependent on the reactor * Corresponding author. E-mail: [email protected]. † Universita ` di Roma Tor Vergata. ‡ Universita ` di Roma La Sapienza.

design. Our group gave a contribution to the field by performing some basic studies on a class of nonsupported catalysts able to overcome the first two problems.26 Here, we present a novel approach to study the methane decomposition using NdNi5 micrometric powder instead of Ni-containing film or supported catalyst, as widely reported in the literature. It is found that in the temperature window of 475-525 °C, which is very close to the thermodynamic threshold of the methane decomposition, no poisoning of the catalyst occurs provided that carbon grows only as MWCNTs. It will be shown that the feasibility of this process depends on some co-operative effects at the catalyst surface that allow the detachment of a nanosingle crystal from a catalyst micrometric grain induced by each growing MWCNT. This phenomenon, here named nanodusting, has been found during the corrosion of steels in a strong carburizing environment at high temperatures.27 Different from supported catalysts in which the nanoparticles pre-exist in heterogeneous systems, the nanoparticles are detached in the dusting process from the homogeneous system during the reaction. The aim of this work is to present a combination of experimental data and their theoretical modeling with particular concern on nucleation and growth processes of MWCNTs arising from nanodusting of NdNi5. 2. Experimental The kinetics of carbon atom production via the CH4 decomposition has been measured by means of the thermogravimetric technique (TG). The TG measurements have been carried out by a Netzsch STA PC 409 Luxx instrument (Germany) (resolution is 2 µg, temperature measurement range is from room temperature (RT) to 1500 °C, temperature sensor is S-type thermocouple). The catalyst26 was a ball-milled NdNi5 powder. The catalyst was synthesized28 by electron beam melting (electron beam gun model EV1-8, Ferrotec, Germany) of a pellet made of Nd (99+% Aldrich) and Ni (99.99%, Aldrich) powders in stoichiometric amounts, which were carefully mixed

10.1021/jp0665731 CCC: $37.00 © 2007 American Chemical Society Published on Web 02/07/2007

Nanodusting of RENi5 Intermetallic Grains

J. Phys. Chem. C, Vol. 111, No. 8, 2007 3267

and pressed at room temperature. The catalyst powder has been characterized through the surface area measurement (Micromeritics ASAP), X-ray diffraction (XRD) (Panalytical X’Pert Pro, Cu KR1 radiation, angular resolution of 0.001°) and scanning electron microscopy (SEM) analysis (Zeiss LEO 1450 VP with 3.5 nm resolution). The Brunauer-Emmett-Teller (BET) surface area was found 0.6 m2 g-1. The catalyst samples, 2025 mg in weight, are inserted into a pure BN crucible (GE Advanced Ceramics, UK) that is placed on the head of the thermobalance. The measurement chamber was evacuated up to 10-2 mbar by rough vacuum pump and refilled with CH4 (99.95% pure, Rivoira, Italy). Once the chamber was at atmospheric pressure, the CH4 flow rate was set at 21 cm3 min-1 (referred to 1013 mbar and 21 °C) using the Netzsch gas control box of the thermobalance. The kinetic measurements were recorded at constant temperature. Specifically, four sets of experiments were performed at 475, 500, and 525 °C. In all the experiments, the catalyst powder was heated up to the working temperature at a heating rate of 30°C min-1. The data were acquired and elaborated through a PC by making use of the bundled Netzsch software package. The carbon containing species have been characterized by XRD, SEM, and transmission electron microscopetem (TEM, JEOL JEM 2010, 200kV, point resolution 0.23 nm) techniques. 3. Results and Discussion 3.1. Characterization of the Reaction Products. The carbon containing species produced during the catalytic decomposition of CH4 has been characterized by SEM and TEM. Figure 1a,b indicates unambiguously the formation of MWCNT structures as the main solid byproduct of the catalytic reaction. The TEM and SEM images also show that catalyst nanoparticles are embedded at one of the ends of the MWCNT structure (Figure 2a,b). The thermograms recorded during the CH4 decomposition are displayed in Figure 3 as mass increase, M, of the sample vs the running time t. It appears evident that the process requires an induction period that is shorter as the temperature is higher. After this time, the reaction rate reaches a value that remains nearly constant in time up to the end of the experiment (i.e., the decomposition process occurs under quasisteady-state condition). The induction time is found to be much longer than the characteristic time recorded when CNT production occurs by different methods. For instance, in the CNT synthesis by chemical vapor deposition (CVD) technique on metal catalysts29 the characteristic time for the CNT growth is found to be of the order of minutes. However, at odds with our experiments in ref 29 the metal nanoparticles are dispersed on a substrate, and a stream of a H2-C2H6 gas mixture is forced to pass through the catalytic bed. In other words, the nanoparticles and H2 are available at the beginning of the reaction. From a purely qualitative point of view, the induction time in the kinetic curves of Figure 3 can therefore be identified with the time required to detach the nanoparticles from the catalyst grains (i.e., a dusting process at nanoscale level occur). In this context, the key importance of these nanoparticles is witnessed also by the SEM image of the catalyst as well as by the TEM images of the nanotubes shown in Figure 4 and in Figure 2a,b and Figure 5, respectively. The granulometry of the catalyst powder is in the range 1-20 µm, whereas the average size of the catalyst particles encapsulated in the MWCNTs is about 5 × 10-2 µm (i.e., from 20 to 400 times smaller than the size of the initial catalyst powder. Furthermore, the spacings and angles measured on the electron diffraction pattern (Figure 6) of the nanosingle crystals embedded at the end of the MWCNTs have been found

Figure 1. Panel (a) shows the SEM image of the carbon-containing product of the reaction. As it appears, the only product consists in CNTs. The inset shows the profile of a single tube. Panel (b) shows the TEM image of the same product. The cross section of a MWCNT (magnified in the inset) is shown in the middle of the explored area.

perfectly matching with NdNi5. The detachment of the particle by the growing nanotube can be explained according to the arguments that follow. Because of the CH4 decomposition, C and H atoms form at the catalyst surface at which the H atoms can either recombine as H2 or diffuse into the catalyst. The latter process is expected to cause the lattice expansion of the NdNi5 compound that is known to be a hydrogen absorber.30,31 These phenomena, together with the C diffusion at the grain boundaries, are thought to facilitate the detachment of the single crystal nanoparticle from the powder specimens through the nucleation of MWCNT.32 After the detachment has occurred, new crystallites are exposed to the gas phase providing a new active center for the MWCNT nucleation. This point will be further discussed in the next section. Unlike other systems investigated in the literature, the formation of carbide species during the reaction does not occur in the present experiment because of the lower temperature values. The formation of Ni3C also should be excluded as confirmed recently in the literature.27 In fact, this is proven by the electron diffraction pattern of the catalyst particle that rules out the formation of carbide species at the interface. Nevertheless, this does not rule out the formation of the solid solution and C segregation at grain boundaries. As it appears from the TEM micrographs, the MWCNTs grow on a tip-shaped part of the nanoparticles only. This suggests a “template like” effect

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Figure 4. SEM picture of the catalyst powder before use. The inserts show the typical sizes of the particles that are in the domain of microns.

Figure 2. Panels (a) and (b) show the TEM images of the nanocrystals of catalyst attached to the MWCNTs. Figure 5. TEM image of a nanocrystal of a catalyst attached to a MWCNT. The templating effect is evident in connection with the cavity and diameter of the nanotube.

Figure 3. The experimental thermograms recorded at the temperatures 475, 500, and 525 °C are reported, respectively, in curves a, b, and c. The marks indicate the calculated induction times. These points were computed from the intercepts with the time axis of the curve tangents after the slope change.

of the tip on the C atom generated by the CH4 decomposition in a similar way as observed at Ni nanoclusters.32 As a matter of fact, the MWCNT formation could imply the self-organization of C atom in polyine rings around the tip in which the stability of the rings is gained thanks to the coordination with the metal atoms at the catalyst surface.33 These rings are very reactive and condense forming a wall of the MWCNT. Only the rings having similar diameter can condense evolving in a MWCNT. Once a ring is taken away by the growing CNT, the surface is again active for the CH4 decomposition and ring production so that the CNT growth proceeds. Detailed drawings of the proposed mechanism are reported in Figure 7a-c.

Figure 6. Electron diffraction pattern of the catalyst nanoparticle reported in Figure 2b. The pattern shows clearly that the particle is a single crystal of the initial catalyst (NdNi5 has a hexagonal structure, and the spacing is perfectly in agreement). This finding rules out any carbide formation.

To demonstrate the peculiar activity of the catalyst, some experiments were carried out separately using Nd and Ni powders having approximately the same granulometry of NdNi5



Figure 8. XRD pattern of catalyst purified MWCNTs obtained from the methane decomposition. The interplanar distance between the graphene walls is slightly larger than that one of graphite, indicating a certain degree of turbostraticity.

Figure 7. Sticks and balls models of the polyine ring formation and carbon nanotubes growth are shown in panels (a-c). The rings are assumed to nucleate at grain boundaries leading to the detachment of the particles.

powder. Though the experiments were performed in the same experimental conditions, no detectable reactivity was observed. As far as the structure of the nanotube is concerned, it is typical of MWCNT with a diameter value that is related to the size of the detached single crystal that pre-exists in the catalyst grains. The XRD pattern of Figure 8 shows a certain degree of turbostraticity of the walls, being that their spacing is 46.1 pm higher than that of graphene sheets in graphite. For comparison purposes, the XRD pattern of graphite also is reported in the same figure. 3.2. Kinetic Model. 3.2.1. Phenomenological Rate Equation for CNT Formation. It has been established in the literature34-36 that the formation of carbonaceous phases, such as the diamond deposited by the CVD, the single and the multiwalled carbon nanotubes, and the glassy carbon, occur by nucleation and growth processes. The former refers usually to the formation of the smallest (kinetically) stable clusters of the new phase that once generated can only grow at the expenses of the mother phase. In fact, in the nucleation theory subcritical clusters have

a larger chance to decrease their size by losing a monomer than to increase their size by the capture of one monomer.37 Conversely, supercritical clusters have a larger probability to grow rather than to lose monomers. The definition of the smallest stable cluster (nucleus) is based on a cut off criterion, which is useful for analytical modeling, according to the following: a critical cluster (made up of e.g., j-1 atoms) is the cluster whose probability of growth after acquiring a monomer becomes larger than 1/2.38 Therefore, the size of the smallest stable cluster is equal to j. In the following we refer also to the critical cluster as “germ”. With the purpose of modeling analytically the processes aforementioned, two quantities are defined, namely the nucleation density, N(t), and the growth law of the stable clusters. The former is equal to the total number of stable clusters at time t; the latter gives the dependence of the cluster size on actual time, t, and birth time of the cluster t′ < t.34,37 Usually the growth law is a function of the growth time (i.e., the difference t - t′). As far as the carbon nanotube is concerned, the phenomenological growth law can be written in the form m(t - t′) ) Fl(t - t′), where m(t - t′) and l(t - t′) are, respectively, the mass and the length of the CNT at running time t, which started growing at time t′, and F is the linear density of the CNT. The time dependence of the total mass of CNT is expressed by the convolution

M(t) ) F

∫0t I(t′)l(t - t′)dt′

(1)

where I(x) ≡ dN(x)/dx is the nucleation rate. The changing rate of the total mass is given by the derivative of eq 1

dM(t) )F dt

∫0t I(t - ξ)

dl(ξ) dξ ) dξ FI(0)l(t) + F

∫0t l(t - ξ)

dI(ξ) dξ (2) dξ

where ξ ) t - t′ and I(0) is the nucleation rate at the beginning of the reaction. Furthermore, as far as the kinetics of dM/dt is concerned, the experimental data indicate that this function after a transient period of duration, such as t0, becomes nearly constant with time (i.e., the kinetics reaches the steady-state condition). On the basis of the phenomenological eqs 1 and 2 and experimental kinetics, some considerations can be done about the general behavior of the nucleation rate and growth law of the CNT. To begin with let us consider the two cases where the nucleation process is assumed to be either constant or simul-


Tomellini et al.

taneous in time. Specifically, for a constant nucleation rate, eq 2 gives dM/dt ) FIl(t) whereas in the case of simultaneous nucleation, being I(x) ) N0δ(x) where N0 is the number of nuclei and δ the Dirac delta function, the eq 2 leads to dM/dt ) FN0 dl(t)/dt. Consequently, at steady state dl/dt ) 0 and dl/dt ) constant in the case of constant and simultaneous nucleation, respectively. In the real systems, the nucleation rate is neither a constant nor a Dirac delta function. In particular, at the beginning of the phase transformation I(0) ) 0, dI/dt|t)0 > 0 and, after an induction period (of duration for example ts), the nucleation rate reaches its steady-state value (dI/dt ) 0). According to eq 2, the possibility to have a constant value of dM/dt is linked to the steady-state regime of the nucleation process. In fact, by evaluating eq 2 for t > ts one gets (I(0) ) 0); dM(t)/dt ) F ∫0ts l(t - ξ) dI(ξ)/dξ dξ, which is constant as long as the constraint l(t - ξ) ) constant, at t > ts, is fulfilled. We note that the minimum value of the argument of the length function, within the integration domain, is just equal to ˜t ) t - ts; this means that the growth rate has to vanish for time longer that of ˜t. Moreover, the minimum value of the actual time for which this condition is fulfilled can be easily identified with the time t0 defined above. This argument leads to the important conclusion according to which the growth law of the CNT satisfies the condition dl(t)/ dt|t>t0-ts ) 0. In other words, the growth of a CNT reaches completion within the time t0 - ts. It is worth to point out that this result is in agreement with the experimental findings, which indicate a termination of the CNT growth process, usually attributed to the poisoning of the catalyst (ref 29 and references therein). 3.2.2. Modeling of the CH4 Decomposition Kinetics. By considering the decomposition of methane to occur at the catalyst nanoparticles, the decomposition rate (i.e., the rate of C atom production) is given by the convolution

RC(t) ) mCΩKR

∫0t

dNp(ξ) θCH4(t - ξ)dξ dξ

(3)

where dNp(ξ)/dξ is the rate of nanoparticle (i.e., catalyst) detachment, θCΗ4 is the CH4 coverage at the catalyst particle, KR is the rate constant for the CH4 decomposition, Ω is the number of adsorption sites at the bare surface of the particle (i.e., active sites), and mC is the mass of the carbon atom. In eq 3, the adsorption of CH4 has been taken into account, because CH4 is assumed to be the main species in the gas phase. However the model can be equally applied in the case of adsorption of gas species other than CH4. As reported above, the analysis of the morphology of the CNT by means of TEM microscopy indicates clearly the presence of a catalyst nanoparticle in each CNT structure. It also has been shown in ref 35 that the mechanism of the MWCNT nucleation on a catalyst surface (at low temperature) results in the formation of encapsulated nanoparticles. The nucleation of the CNT initiates through the formation of twodimensional carbon germs at the particle surface owing to the segregation of C atoms at the interface.35,39 On the basis of this argument, we infer that the dNp(ξ)/dξ quantity in eq 3 can be identified with the nucleation rate of the CNTs, I ) dN(ξ)/dξ. Implicit in this statement is the simplified hypothesis according to which a single nucleation event takes place on the catalyst nanoparticle. (However, the present kinetic model can be modified to include multiple nucleation events by considering a multiplicative factor in eq 3). In addition, from the discussion above, it follows that since the CNT growth is bounded in time,

the decomposition process on the single nanoparticle is expected to reach completion after some time (e.g., ∆t) when the catalytic activity of the particle vanishes. In other words, ∆t should be considered as the “life time” of the catalytic activity. This can be ascribed to a poisoning like process, leading to a “blockingeffect” of the active sites for the CH4 decomposition. In particular, we note that such an effect can be due to either the blocking of active sites by the growing new phase or reduction in the accessibility of the gas molecule at the metal site owing to the “shielding” effect connected with the CNT growth.29,39 For the sake of simplicity, in eq 3 we set the function θCΗ4 (t - ξ) equal to θCΗ4(t - ξ) ) Θ(t - ξ) - Θ(t - ξ - ∆t) where Θ(x) is the step function [Θ(x) ) 1 for x > 0, Θ(x) ) 0 for x < 0]. Inserting this last equation in eq 3 and considering ∆t to be much shorter than the characteristic time for nucleation, we end up with

RC(t) = (mCΩKR∆t)

dN dt

(4)

Equation 4 shows that the decomposition rate is proportional to the nucleation rate. This equation could also be derived from eq 2 for dl/dξ ∝ Θ(ξ) - Θ(ξ - ∆t) and considering the characteristic time for nucleation longer than ∆t. In addition, ∆t can be identified with time interval t0 - ts defined in the previous section. To model the nucleation kinetics, we resort to the classical theory of nucleation40 for cylindrical critical nuclei of radius r* and height h. Kuznetsov et al.41 have computed the free energy change for the formation of the CNT nucleus of radius r as

∆G(r) ) -

∆µ hπr2 + ∆σπr2 + 2πr υm

(5)

where υm is the atomic volume of graphite, ∆µ ) kBT ln a/a* is the supersaturation (a* being the carbon activity in the CNT and a the C activity in the “mother” phase), ∆σ ) σng + σns σsg where σng, σns, and σsg are the excess free energies of the nucleus-gas, nucleus-substrate, and substrate-gas interfaces, respectively. In eq 5, is the specific edge free energy. The phase that is supersaturated by C (i.e., the phase referred to above as mother phase) can be identified, depending on the deposition method with either the solid solution made up with C species dissolved in the catalyst or the C atoms in the gas phase.35,42 In the former case, that should apply to our experiment C atoms segregate at the grain boundary interface and combine leading to the formation of a first graphitic layer. In the absence of sufficient mobility of the system, this layer does not bend to form a small cup but rather it continues to grow resulting in the MWCNT nucleus once the bending of the graphitic planes stabilizes the sp2 orbitals at the border of the layer through the coordination at metal sites.35 Further growth of the tube gives rise to the process of detachment of the nanoparticle from the micrometer size catalyst grain. As discussed in the following, the supersaturation of the mother phase, the nucleation, and the nanoparticle detachment processes are considered in the kinetic model. The modeling does not allow to establish, quantitatively, how far the MWCNT growth proceeds prior to the particle detachment. Nevertheless, on the basis of the present experimental results concerning the morphology of the fibers, together with those of ref 27, we consider the nucleation and the particle detachment processes to be nearly simultaneous. In other words, the growth of



MWCNTs prior to detachment has been neglected compared to the whole kinetics of growth. The maximum of the ∆G(r) function gives the critical size of the cluster, r*, from which the free energy change for critical cluster formation is promptly attained

∆G(r*) )

-π2 (∆σ - υm-1hkBT ln a/a*)

(6)

where ∆σ can be expressed in terms of the adhesion work, W, according to ∆σ ) 2σgraphite - W in which σng has been taken as the surface free energy of the graphite. Under steady-state conditions, the nucleation rate is given by37,40

dN ) N0K*Ze-β∆G(r*) dt

(7)

where β ) (kBT)-1, N0 is the number of nucleation centers on the surface of the catalytic bed, K* is the rate at which C atoms join the critical nucleus, and Z (that is in the range from 0.01 to 1) is the Zeldovich factor that gives account of the difference between the nucleation rates under steady state and equilibrium conditions.37 The nucleation model discussed so far (eqs 5-7) does not take into account the phenomenon of particle detachment. As already stated, the nucleation process precedes the growth of the new phase. On the other hand, our results show that the growth of the MWCNT starts with the detachment of the particle single crystal. This nanoparticle is anchored on the top of the freestanding side of the tube.32 Interestingly, this behavior is similar to that observed in ref 43 in the case of liquid metal nanoparticles. On the basis of this argument, the physical process we are dealing with is the detachment of nanoparticle driven by the nucleation process (i.e., the onset of MWCNT growth). Consequently, the energy of the particle fragmentation is expected to affect the free energy change for the critical nucleus formation. If the nucleation occurs at the interface between the nanocrystals of the powder (grain boundary), the ∆σ term should be replaced by ∆σ′ ) 2σns - σgb ) 2(σng + σsg) - 2W - σgb, where σgb is the surface energy of the grain boundary. In addition, this quantity is related to the work (per unit area) required to detach the nanoparticle according to W′ ) 2σsg σgb. Once inserted in the ∆σ′ expression, this last equality leads to ∆σ′ ) 2σng + W′ - 2W = 2σgraphite + W′ - 2W. This equation shows that the free energy for nucleation depends on the “strength” of the interface, which in turn should depend on temperature, microstructure of the catalyst powder, and H content in the catalyst. For instance, if W′ < W, then ∆σ′ < ∆σ and the nucleation is easier at the interface as in this case ∆G*(∆σ′) < ∆G*(∆σ). The larger the (W′ - W) difference, the larger the supersaturation required to attain a sizable rate of nucleation. The free energy change can therefore be rewritten as

∆G(r*) )

[

-π2υm ∆συm a -β∆E e - T ln hkB hkB a*

(

)]

-1

(6b)

where ∆σ ) 2σng - W and ∆E ) (W′ - W)υm/h is the contribution linked to the particle detachment. It is worth emphasising that eq 6b is equivalent to eq 6 provided the C activity function is properly rescaled. To this end, it is convenient to define the “effective” activity, f ) a/a* e-β∆E, which takes into account the effect of the interface on the nucleation process. The lower the ∆E value the larger the

Figure 9. Right scale: typical behavior of the nucleation rate according to eq 7 and normalized to N0K*Z. Left scale: carbon atom production as a function of the dimensionless time, t/τ ≡ t/λ2, calculated according to eq 9. M(t) is normalized to M0 ) mCΩKR∆t. The parameter values are: k1 ) 80 K, k2 ) 81 K, and T ) 700 K. Case a: λ1 ) 100 K. Case b: λ1 ) 500 K.

“effective” supersaturation and in turn the larger the probability of nanotube nucleation (i.e., the probability of the nanocrystal detachment). As from eqs 5 to 7, the application of the kinetic theory requires the knowledge of the C supersaturation. On the other hand, in the experiment under consideration this quantity is not constant during the reaction. As a matter of fact, its value is associated to the process of methane decomposition on the catalyst powder, and as a consequence it is expected to depend on time. Besides, as anticipated in the previous section the hydrogen absorption by the catalyst powder is thought to facilitate the fragmentation process due to the embrittlement of the catalyst material. In the framework of the modeling developed here, this should also entail a time dependent ∆E because of its dependence on the H content of the catalyst. The proposed mechanism therefore implies a strong interplay between CH4 decomposition and the catalyst nanoparticle detachment. By assuming quasisteady-state conditions37 for the CNT nucleation, eq 7 can be employed provided a time dependent effective activity is used. To this end, we make use of the following function:

f(t) ≡

a -β∆E ) f(∞)(1 - e-t/τ) + f(0)e-t/τ e a*

(8)

where τ is the time constant. To be specific, τ can be interpreted as the characteristic time of the kinetics that is rate determining between the kinetics of the C supersaturation (a/a*) and catalyst embrittlement by H absorption (e-β∆E). The functional form adopted in eq 8 will be discussed shortly. Figure 9 shows the typical behavior of the nucleation rate and of the M(t) curve as obtained by the present approach. As it appears, the induction time of the M(t) function is ascribed to the characteristic time of the nucleation process that is dictated by the kinetics of the C supersaturation. According to ref 41, typical values of the parameters are: = 7 × 10-10 J/m, h ) 0.34 nm, υm ) 5.3 cm3/mol, σgraphite ) 0.08 J/m2, and W ) 0.1 J/m2. The modeling also has been used for describing the experimental M(t) kinetics by means of a fitting procedure. To be specific, the changing rate of the carbon weight is computed by means of eq 4 above where dN/dt is given through eq 7. By substituting eq 8 in eq 6b, the time dependent free energy change


Tomellini et al.

Figure 10. Kinetics of the mass uptake as modeled by the present approach. The full line is the experimental TG data (see Figure 3) and the dashed line the curve obtained by integrating the best fit of eq 9 to the data.

for nucleation is eventually computed. The dM/dt rate can therefore be rewritten in the form

{

}

λ1 dM ) λ0 exp dt k1 - T ln[2(1 - exp(-t/λ2)) + exp(k2/T)]

(9)

where the difference f(∞) - f(0) has been set equal to 241 and f(0) ) exp(k2/T). In eq 9, λis are the fitting parameters whereas k1 and k2 are constants. In particular, a k1 value equal to 100 K was used as estimated from the relationship k1 ) υm∆σ/hkB. As far as the k2 term is concerned, we note that it is related to the supersaturation value at the beginning of the process. However, it is worth to point out that the classical nucleation theory can be applied provided the critical cluster is sufficiently large,37,40 a condition that can be fulfilled for positive values of the supersaturation only, being in this case r* ) (∆µ/υm h - ∆σ′)-1 > 0. Consequently, in eq 9, the inequality k2 > k1 has to be verified. A value of k2 ) 110 K has been adopted in all the fits. One notices that the product kBk2 is just equal to the initial value of the effective supersaturation. The experimental M(t) kinetic curves, measured in the temperature interval from 475 to 525 °C, are displayed in Figure 10a-c together with the functions obtained by integrating the

best fit of eq 9 to the data. The agreement between the data and analytical model is more than satisfactory. Equations 6b and 9 indicate that λ1 ) π2υm/hTk2B and this allows an estimate of the 2/h ratio. The average value of 2/h has been found to be < 2/h > ) 1.2 × 10-12 J/m3, which by assuming the thickness of the critical nucleus just equal to the interlayer distance of graphite, leads to the specific edge free energy = 1.1 × 10-11 J/m. This value is in reasonable agreement with those computed for Ni and Fe by using the formation enthalpies of the C-C and C-metal bonds.41 The parameter λ2 (eq 9), namely the time constant τ in eq 8, is found to be of the order of magnitude of 104 s in all the experiments and it can be identified with the induction time mentioned above. A simple model for describing the kinetics of C supersaturation in the catalyst (i.e., a modeling of the time constant τ) can be developed by means of rate equations. In the Appendix, kinetic equations are solved for the surface coverage of the catalyst particle by both CH4 molecules and C atoms, as well as for the C dissolved into the particle. To this end, the rate constants for C atom insertion (Kdis) and segregation at the grain boundary (Kseg) and for CH4 decomposition (KR) have been defined. Furthermore, as discussed in refs 41 and 13, the supersaturation is linked to the activity of the C atom in the catalyst-C solid solution. According to eqs A5 and A7, if 1/Kseg < 1/Kdis < 1/KR + 1/J with J being the flux of CH4 incoming on the surface, the time constant of eq 8 can be identified with (Kdis)-1, provided the catalyst embrittlment kinetics is fast. This means low probabilities for molecule decomposition and/ or adsorption, when compared to those for C atom segregation and solubilization. Under these circumstances the kinetics of supersaturation is given by eq A7 whose asymptotic value is found to be directly and inversely proportional to Kdis and to the particle size, respectively. This outcome is in qualitative accord with the results of ref 14 on the effect of the particle size on the kinetics of CNT formation. In fact, the rate constant Kdis is expected to be proportional to the diffusion coefficient of C in the particle. Notably, in the kinetic approach presented here, the rate determining step in the formation of stable MWCNT cluster can be either the process leading to the supersaturation of the mother phase or the nucleation process. The characteristic time for nucleation is just proportional to τN ≈ (K*)-1eβ∆G* where ∆G* is given by eq 6b at t ≈ τ (eq 8), namely at about the highest value of the effective supersaturation. The relative magnitude between τN and τ determines which of these two processes is rate determining. Before concluding, a comment is in order on the interplay between the catalytic activity and the dusting phenomenon. The efficiency of the detachment process is described by the extra energy term ∆E in eq 6b that is linked to the grain boundary cohesion. On the other hand, to be active toward the CH4 decomposition the surface coverage of the particle ought to be lower than unity during the whole reaction. This requires an efficient consumption of carbon from the active site; in this instance, the MWCNT growth could represent a facile route for C atoms removal that is subjected however to the detachment of the catalyst particle. Conclusions The main results reported in this communication can be summarized as follows: 1. The decomposition of CH4 on NdNi5 powder yields the formation of carbon nanotubes through the fragmentation of the initial catalyst particles (with typical size in the domain of µm’s) in nanoparticles. These nanoparticles are attached to the MWCNT structures.



2. The structural characterization of the catalyst nanoparticles indicates that they are single crystals of the initial compound. The MWCNT nucleation initiates at the tip-shaped region of the catalyst and brings about the detachment of the nanocrystal. 3. A kinetic model is presented for describing the experimental kinetics. The physical basis of the model lays on the nucleation of the MWCNT byproduct, which governs the detachment of the nanoparticle. The catalyst activity that is connected to the number of active sites at the nanoparticle surface increases with the number of nanoparticles. The model has reproduced the induction period by considering a time dependent effective supersaturation of carbon atoms. Phenomenological rate equations do show that the growth law of the MWCNTs is bounded in time. 4. Nanosingle crystals of materials of high-technological added value, which are very difficult to produce by other means, can be obtained through the growth of MWCNTs. It is worth noticing that most of the binary intermetallics constituted by rare-earth and transition metals are very strong magnets. Furthermore, it is expected that their catalytic activity increases when nanostructured. Acknowledgment. The authors are very grateful to Professor G. Capannelli and Mr. C. Uliana of the Department of Chemistry and Industrial Chemistry of the University of Genoa, Italy for the TEM analyses they performed. Appendix The rate equations for the surface coverage of the catalyst powder (size of the order of µm) by methane carbon units read

dθCH4 dt

) J(1 - θC - θCH4) - KRθCH4

(A1)

dθC ) KRθCH4 - KdisθC dt

(A2)

where J is the flux of CH4 incoming on the surface. θC and θCH4 denote, respectively, the coverage of C and CH4. KR and Kdis are, respectively, the rate constants for the CH4 decomposition and for C atom dissolution in the catalyst. By assuming CH4 decomposition to occur under steady-state conditions (i.e., θ˙ CH4 = 0) eq A1 yields

θCH4 )

J(1 - θC) J + KR

(A3)

and dθC 1 ) J˜ - θC dt τ′

(A4)

with J˜ ) KRJ/(KR + J) and (τ′)-1 ) Kdis + JKR/(KR + J). The kinetics of carbon coverage therefore reads as

θC ) J˜τ′(1 - e-t/τ′)

(A5)

The number of C atoms, n, for interstitial site into the catalyst particle can be estimated using the rate equation

dn ) K′disθC - Ksegn dt

(A6)

where Kseg is the rate constant for the segregation of C atom and K′dis ∝ Kdis/(R/a) where R is of the order of magnitude of the particle size and a is the lattice space (for a spherical particle

K′dis ) 3Kdis/(R/a)). By inserting eq A5 into eq A6, the n(t) function is eventually obtained according to

n(t) )

K′disJ˜

[

]

1 (1 - e-Ksegt ) - τ′(1 - e-t/τ′) (1/τ′ - Kseg) Kseg (A7)

with the asymptotic value n(∞) ) K′disJ˜τ′/Kseg, which scales as R-1. Equations A5 and A7 show that the time constant of the θ(t) and n(t) kinetics are equal to τ′ and max (τ′, 1/Kseg), respectively. For instance, if the mother phase is the C-NdNi5 solid solution in the limit Kseg , 1/τ′ and 1/τ′ , Kseg the time constant of eq 8 is 1/Kseg and τ′, respectively, provided the embrittlment kinetics is fast. Note Added After ASAP Publication. This article was published ASAP on February 7, 2007. Due to a production error, eq A7 and the caption for Figure 9 were incorrect. The correct version was posted on February 12, 2007. References and Notes (1) Iijima, S. Nature 1991, 354, 56. (2) Dresselhaus, M. S.; Dresselhaus, G.; Saito, R. Carbon 1995, 33, 883. (3) Iijima, S. Mater. Sci. Eng., B 1993, B19, 172. (4) Tasis, D.; Tagmatarchis, N.; Bianco, A.; Prato, M. Chem. ReV. 2006, 106, 1105. (5) Krupke, R.; Hennrich, F.; Lo¨hneysen, H. V.; Kappes, M. M. Science 2003, 301, 344. (6) Yao, Z.; Kane, C. L.; Dekker, C. Phys. ReV. Lett. 2000, 84, 2941. (7) Lee, C. J.; Park, J. J. Phys. Chem. B 2001, 105, 2365. (8) Wagg, L. M.; Hornyak, G. L.; Grigorian, L.; Dillon, A. C.; Jones, K. M.; Blackburn, J.; Parilla, P. A.; Heben, M. J. J. Phys. Chem. B 2005, 109, 10435. (9) Little, R. B. J. Cluster Sci. 2003, 14, 135. (10) Sharma, R.; Iqbal, Z. Appl. Phys. Lett. 2004, 84, 990. (11) Perez-Cabero, M.; Royo, C.; Monzon, A.; Guerrero-Ruiz, A.; Rodriguez-Ramos, I. J. Catal. 2004, 224, 197. (12) Valentini, L.; Armentano, I.; Kenny, J. M.; Lozzi, L.; Santucci, S. Diamond Relat. Mater. 2003, 12, 821. (13) Jeong, H. J.; Jeong, S. Y.; Shin, Y. M.; Han, J. H.; Lim, S. C.; Eum, S. J.; Yang, C. W.; Kim, N.; Park, C.; Lee, Y. Chem. Phys. Lett. 2002, 361, 189. (14) Tsai, T.; Chuang, C.; Chao, C.; Liu, W. Diamond Relat. Mater. 2003, 12, 1453. (15) Gorbunov, A.; Jost, O.; Pompe, W.; Graff, A. Carbon 2002, 40, 113. (16) Dresselhaus, M. Basic Research Needs for the Hydrogen Economy; U.S. Department of Energy, Office of Science, May 13-15, 2003; Argonne National Laboratory: Argonne, IL, 2004. (17) Li, Y.; Zhang, X. B.; Tao, X. Y.; Xu, J. M.; Huang, W. Z.; Luo, J. H.; Luo, Z. Q.; Li, T.; Liu, F.; Bao, Y.; Geise, H. J. Carbon 2005, 43, 295. (18) Qian, W.; Liu, T.; Wang, Z.; Wei, F.; Li, Z.; Luo, G.; Li, Y. Appl. Catal., A 2004, 260, 223. (19) Wang, H.; Baker, R. T. K. J. Phys. Chem. B 2004, 108, 20273. (20) Takenaka, S.; Shigeta, Y.; Tanabe, E.; Otsuka, K. J. Catal. 2003, 220, 468. (21) Li, J.; Lu, G.; Li, K.; Wang, W. J. Mol. Catal. A: Chem. 2004, 221, 105. (22) Reshetenko, T. V.; Avdeeva, L. B.; Ushakov, V. A.; Moroz, E. M.; Shmakov, A. N.; Kriventsov, V. V.; Kochubey, D. I.; Pavlyukhin, Y. T.; Chuvilin, A. L.; Ismagilov, Z. R. Appl. Catal., A 2004, 270, 87. (23) Muradov, N.; Chen, Z.; Smith, F. Int. J. Hydrogen Energy 2005, 30, 1149. (24) Lee, K. K.; Han, G. Y.; Yoon, K. J.; Lee, B. K. Catal. Today 2004, 93-95, 81. (25) Weizhong, Q.; Tang, L.; Zhanwen, W.; Fei, W.; Zhifei, L.; Guohua, L.; Yongdan, L. Appl. Catal., A. 2004, 260, 223. (26) Gozzi, D.; Latini, A. PCT Int. Appl. 2006, 21 pp., CODEN: PIXXD2 WO 2006040788 A1 20060420. (27) Zheng, Z.; Natesan, K. Chem. Mater. 2005, 17, 3794. (28) Latini, A.; Di Pascasio, F.; Gozzi, D. J. Alloys Compd. 2002, 346, 311. (29) Svrcek, V.; Kleps, I.; Cracioniou, F.; Paillaud, J. L.; Dintzer, T.; Louis, B.; Begin, D.; Pham-Huu, C.; Ledoux, M.-J.; Le Normand, L. J. Chem. Phys. 2006, 124, 184705. (30) Takakuchi, Y.; Tanaka, K. J. Alloys Compd. 2000, 297, 73.

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