THE JOURNAL OF CHEMICAL PHYSICS 134, 134704 (2011)

Toward CH4 dissociation and C diffusion during Ni/Fe-catalyzed carbonnanofiber growth: A density functional theory study

Chen Fan,1 Xing-Gui Zhou,1 De Chen,2 Hong-Ye Cheng,1 and Yi-An Zhu1,a)

1State Key Laboratory of Chemical Engineering, East China University of Science and Technology (ECUST),Shanghai 200237, China2Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), N-7491Trondheim, Norway

(Received 11 October 2010; accepted 15 March 2011; published online 7 April 2011)

First-principles calculations have been performed to investigate CH4 dissociation and C diffusionduring the Ni/Fe-catalyzed growth of carbon nanofibers (CNFs). Two bulk models with different Nito Fe molar ratios (1:1 and 2:1) are constructed, and x-ray diffraction (XRD) simulations are con-ducted to evaluate their reliability. With the comparison between the calculated and experimentalXRD patterns, these models are found to be well suited to reproduce the crystalline structures ofNi/Fe bulk alloys. The calculations indicate the binding of the C1 derivatives to the Ni/Fe closest-packed surfaces is strengthened compared to that on Ni(111), arising from the upshift of the weightedd-band centers of catalyst surfaces. Then, the transition states for the four successive dehydrogena-tion steps in CH4 dissociation are located using the dimer method. It is found that the energy barriersfor the first three steps are rather close on the alloyed Ni/Fe and Ni surfaces, while the activationenergy for CH dissociation is substantially lowered with the introduction of Fe. The dissolution ofthe generated C from the surface into the bulk of the Ni/Fe alloys is thermodynamically favorable,and the diffusion of C through catalyst particles is hindered by the Fe component. With the combi-nation of density functional theory calculations and kinetic analysis, the C concentration in catalystparticles is predicted to increase with the Fe content. Meanwhile, other experimental conditions, suchas the composition of carbon-containing gases, feedstock partial pressure, and reaction temperature,are also found to play a key role in determining the C concentration in bulk metal, and hence themicrostructures of generated CNFs. © 2011 American Institute of Physics. [doi:10.1063/1.3575193]

I. INTRODUCTION

Carbon nanofibers (CNFs) have recently attracted in-creasing attention owing to the unique physical and chemicalproperties, such as high resistance to strong acids and bases,high electronic conductivity, large surface area, and highmechanical strength,1–3 which give rise to many potentialapplications in polymer fillers,4 adsorbents,5 electrodesfor fuel cell,6 catalysts,7, 8 and catalyst supports.9, 10 Theindustrial large-scale synthesis of CNFs is currently achievedthrough catalytic chemical vapor deposition (CCVD) for itshigh selectivity, high yield, and low cost.11–13 In the process,the catalytic CNF growth can be described as follows:14–16

(i) carbon-containing compounds adsorb dissociatively ontransition metal surfaces to form atomic C and release H2;(ii) the generated C atoms dissolve into catalyst particles anddiffuse through the bulk metal or along the metal surfaces;and (iii) the C atoms precipitate eventually on the rearsurfaces of the catalyst particles in the form of graphenelayers. It is now generally accepted that the second stepis the rate-limiting step for the whole process because theexperimentally measured activation energy for CNF growthis rather close to that for C diffusion.14, 17

a)Author to whom correspondence should be addressed. Electronic mail:yanzhu@ecust.edu.cn.

Because of the abundance and cheapness of natural gas,CH4 is among the most attractive feedstock in the chemical in-dustry. As the step involved in both steam methane reformingand CNF production, CH4 dissociation on Ni-based alloyedcatalysts has been extensively investigated by both experi-mental and theoretical means, for purposes of elucidating thereaction mechanisms and designing new catalysts with highcoke resistance.18–29

Through molecular beam experiments, the sticking co-efficient of CH4 on Ni surfaces was found to increase expo-nentially with its translational energy along the surface nor-mal when the ratio of the number of adsorbed CH4 to thatof impinging CH4 is less than 10−3, which suggested thatCH4 dissociation is a direct, highly activated process.18 Withthe same technique, Beebe et al. reported the energy barriersof 0.58, 0.28, and 0.55 eV for CH4 dissociation on Ni(110),Ni(100), and Ni(111), respectively.19 More recently, Egeberget al. reported an energy barrier of 0.77 eV on Ni(111) bymeans of the selective blockage of surface defects with inertAu atoms.20

In order to improve coke resistance in steam methanereforming or to relate the catalyst composition with the mi-crostructures of CNFs, Ni catalysts have been modified withother metals, such as Cu,21 Sn,22 Ag,23 and Fe.24–26 With thecombination of experimental studies and density functionaltheory (DFT) calculations, Nikolla et al. demonstrated that

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134704-2 Fan et al. J. Chem. Phys. 134, 134704 (2011)

the decrease in C deactivation on Sn/Ni catalysts is attributedto the Sn-induced lowering in the C binding energy on low-coordinated sites, which serve as the C nucleation centers.22

Parizotto et al. reported that alloyed Ag/Ni catalysts with theAg loading more than 0.3 wt. % show high coke resistancein steam methane reforming.23 On the other hand, our grouphas synthesized a series of CNFs on Ni/Fe alloyed catalystswith different Fe to Ni molar ratios by the decomposition ofCH4 and found that the CNF microstructures are rather sensi-tive to the catalyst compositions. As the Fe content increases,the graphite layers in CNFs are more parallel to the growthaxes.24–26 However, the underlying reason is unknown.

With the use of periodic slab DFT calculations, CH4 dis-sociation was examined from a molecular perspective. Takinginto account the spin-polarization effect, Henkelman et al. re-ported an energy barrier of 0.82 eV, using PW91-GGA func-tional and ultrasoft pseudopotentials.27 Our group reportedan energy barrier of 0.91 eV [0.79 eV with zero-point en-ergy corrections considered] on a p(3×3) slab model, usingthe generalized gradient approximation functional proposedby Perdew, Burke, and Ernzerhof (GGA-PBE) and projectoraugmented wave (PAW) pseudopotentials.28 More recently,Xu and Saeys reported an activation energy of 0.94 eV ona p(2 × 2) slab model, using PW91 functional and PAWpseudopotentials.29 With the introduction of Cu on Ni(111),An et al. found that the activation energy for the activationof CH4 is 1.3 times higher than that on Ni(111) and the de-hydrogenation of CH is 1.8 times higher. They claimed thatthe relative changes in the energy barriers for the four suc-cessive dehydrogenation steps play a key role in impeding Cformation in steam methane reforming.21

As for the dissolution and diffusion of C in Ni catalystparticles, most of the related research was carried out the-oretically. Through DFT calculations, Abild-Pedersen et al.claimed that the diffusion of C on the Ni surfaces and in thesubsurfaces is dominant.30 The bulk diffusion is unlikely tooccur owing to the considerably high energy barrier. Xu andSayes recently reported that the C dissolution in the subsur-faces and bulk of Ni particles is energetically more favorablethan the C adsorption on the surface,31–33 and at high C sur-face coverages, the C diffusion through the bulk of catalystparticles becomes feasible. Our group also performed spin-polarized DFT calculations on C diffusion through bulk Ni. Itwas found that the activation energy for C hopping dependsstrongly on the C concentration in Ni particles, decreasingfrom 1.64 eV in Ni32C to 1.26 eV in Ni4C.34

Despite the numerous studies focusing on CH4 dissoci-ation as well as C diffusion, the mechanism for C precipi-tation and CNF formation remains elusive. In particular, thekey factors that determine the CNF microstructures are stillunclear. Through the previous experimental investigations,it was found that the CNF microstructures can be alteredby varying individual experimental conditions including thecompositions of carbon-containing gases, reaction tempera-ture, feedstock partial pressure, active metal components, andcatalyst supports.3, 14, 17 Merkulov and co-workers recentlyproposed a self-consistent mathematical model to describe thegrowth process of CNFs by CCVD.35 In their model, the car-bon concentration in catalyst particles was highlighted and

found to be the most important parameter that characterizesthe growth process. It controls the shape, size, and topol-ogy of the interface between catalyst particles and CNFs. Asthe slope of graphene layers in CNFs duplicates the interfaceslope, the microstructures of CNFs depends strongly on the Cconcentrations in bulk metal. It is, therefore, of vital impor-tance to examine how the aforementioned experimental con-ditions influence the C concentration in catalyst particles.

In this contribution, the first two steps for CNF produc-tion are investigated in detail from first principles in order tooffer an explanation for the variation of the C concentrationsin bulk metal, and hence the variation in the morphologiesof CNFs. The paper is structured as follows. First, two bulkmodels with different Ni to Fe molar ratios (2:1 and 1:1) areconstructed to represent the crystal structures of Ni/Fe alloys.Second, the adsorption behavior of the C1 derivatives in CH4

dissociation is depicted on the two Ni/Fe(111) surfaces. Third,the activation energy for each elementary step involved is ob-tained to gain the information of the overall potential energydiagram. Four, the effect of the Fe concentration on the rateof C diffusion through Ni/Fe particles are examined. With thecombination of DFT calculations and kinetic analysis, the fac-tors that determine the C concentration in metal particles areeventually revealed.

II. COMPUTATIONAL DETAILS

In this work, all the DFT calculations were performedusing the VASP code36–39 in which the wavefunctions at eachk-point are expanded in terms of a plane wave basis set witha kinetic energy cutoff of 400 eV. The interaction betweenvalence electrons and ion cores is treated by Blöchl’s all-electron-like PAW method40, 41 which regards the 4s 3d statesas the valence configuration for Ni and Fe, 2s 2p states forC and 1s state for H. Exchange and correlation of the Kohn–Sham theory was treated with the GGA-PBE functionalproposed by Perdew et al.42 Brillouin zone sampling wasperformed using a Monkhorst–Pack grid,43 and electronic oc-cupancies were determined according to a Methfessel–Paxtonscheme with an energy smearing of 0.2 eV.44 Because therewere two magnetic elements (Ni and Fe) involved in the targetsystem, spin-polarized effects were considered. Our previouscalculations indicated that surface magnetism is essential foran accurate quantitative description of total energy.28, 34

Before the construction of the crystal structure of al-loyed Ni/Fe, the segregation energy of Fe from bulk Ni toNi surfaces and the surface mixing energy between Ni andFe were first calculated and found to be 0.22 and 0.19 eV,respectively.45 Therefore, the Fe dissolution into bulk Ni wasenergetically favorable, and the phase separation of Ni/Fe al-loy was not preferred. This was in accordance with the previ-ous DFT results by Christensen et al.,46 and also agreed wellwith the experimental findings.47

Thus, two bulk models with different Ni to Fe molarratios, namely NiFe and Ni2Fe, were constructed in thefollowing way. As both experimental and theoretical workdemonstrated that the Ni/Fe alloy retains an fcc structurewhen the Fe content is no more than 60%,26, 47, 48 the NiFebulk alloy (Ni:Fe = 1:1) was simulated by replacing two Ni

134704-3 Toward CH4 dissociation and C diffusion J. Chem. Phys. 134, 134704 (2011)

FIG. 1. Comparison of XRD patterns between simulated results and exper-imental data; (a) and (c) are the experimental data of γ -Al2O3 supportedNi/Fe alloy, which are derived from (Ref. 26); (b) and (d) are the correspond-ing simulated XRD patterns of the bulk NiFe and Ni2Fe models, respectively.

atoms in a Ni4 conventional cell with two Fe atoms. Theequilibrium lattice constants of NiFe were optimized to bea = 3.589 Å, b = c = 3.546 Å. As for Ni2Fe (Ni:Fe = 2:1),a p(

√3 ×√

3) four-layer Ni(111) slab model with three Niatoms per layer was first built. With the removal of the vac-uum layer, a Ni9 cell with three close-packed layers was thenconstructed. As the calculated surface mixing energy dictatesthat the Fe atoms in bulk Ni be separated away from eachother, one Ni atom in each layer was replaced with one Featom, and therefore a Ni6Fe3 cell was attained. The equilib-rium lattice constants of Ni6Fe3 were optimized to be a = b= 4.366 Å, c = 6.133 Å.

To evaluate the reliability of the two bulk models, XRDsimulations were performed by employing the Reflex mod-ule in MATERIAL STUDIO software package of Accelrys Inc.An x-ray diffractometer of CuK α1 (λ1 = 1.5405 Å) wasused to obtain the Bragg reflection from the two bulk models.The simulated results, together with the experimental data,are shown in Fig. 1. It can be seen that the simulated resultsare in good agreement with the experimental data with nearlythe same 2θ positions of the three characteristic reflections[(111), (200), and (220)] of the fcc-structured Ni/Fe alloys.

Then, the two bulk models are utilized to cleave theNiFe(111) and Ni2Fe(111) surfaces. A rectangular (2

√3 × 2)

supercell with eight metal atoms per layer was used to rep-resent the NiFe(111) surface. For the Ni2Fe(111) surface, ap(3 × 3) supercell with nine metal atoms per layer was con-structed. Thus, the coverages of adsorbates on the two sur-faces are comparable. The two surface models, together withthe high symmetry threefold hollow sites, are shown in Fig. 2.Both of the slabs contain four close-packed layers, and theneighboring slabs were separated by vacuum layers as largeas 12 Å along the metal surface normal to avoid peri-odic interactions. The first Brillouin zones of the NiFe(111)and Ni2Fe(111) slab models were sampled with �-centered5 × 3 × 1 and 3 × 3 × 1 k-point grids, respectively, whichwere proven to be sufficient to converge the total energywithin 1 meV/atom.45 The bottom two layers in the slab mod-els were fixed and the top two layers as well as the adsor-bates were allowed to relax during geometry optimization andtransition state (TS) search.

The dimer method49 was used to locate the TSs for CH4

dissociation. The climbing image nudged elastic band (CI-NEB) method,50, 51 the improved version of the NEB method,was used to explore the minimum energy paths (MEPs) forC diffusion. A set of intermediate images were constructedalong the diffusion path between the energetically favor-able reactant and product. In all the calculations, a force-based conjugated-gradient method52 was used to optimizegeometries. Saddle points and minima were considered tobe converged when the maximum force in every degree offreedom was less than 0.03 eV/Å. To verify the configu-rations of adsorption and TSs, vibrational frequency calcu-lations were carried out by the numerical finite differencemethod.45

The adsorption energies of absorbates were calculatedby

Eads = Eadsorbate+sur − Esur − Eadsorbate, (1)

where the first term on the right hand was the total energyof an alloyed Ni/Fe surface with an adsorbate adsorbed, thesecond term was the total energy of the bare alloyed Ni/Fesurface, and the last term was the total energy of the isolated

FIG. 2. Schematic representations of the four different threefold hollow sites on (a) NiFe(111) and (b) Ni2Fe(111). The light blue balls denote Ni atoms; thedark blue balls denote Fe atoms.

134704-4 Fan et al. J. Chem. Phys. 134, 134704 (2011)

TABLE I. Calculated adsorption energies (eV) of the intermediates on the alloyed Ni/Fe(111) and Ni(111)surfaces. The energetics on Ni(111) are derived from Ref. 28. The energetics in bold are the adsorption energiesat the most stable adsorption sites.

Ni(111) Ni2Fe(111) NiFe(111)

Species hcp fcc hcp1 hcp2 fcc1 fcc2 hcp1 hcp2 fcc1 fcc2

H −2.79 −2.81 −2.78 −2.82 −2.80 −2.83 −2.86 −2.90 −2.88 −2.87C −6.81 −6.72 −6.76 −6.91 −6.67 −6.71 −6.90 −6.88 −6.83 −6.87CH −6.42 −6.43 −6.31 −6.45 −6.31 −6.40 −6.44 −6.44 −6.48 −6.47CH2 −3.95 −4.01 −3.94 −4.07 −4.05 −4.07 −4.07 −4.09 −4.12 −4.18CH3 1.86 −1.91 −1.88 −1.96 −1.93 −1.98 −1.93 −2.03 −1.98 −2.08

adsorbate. The first two terms were calculated with the sameparameters (k-point sampling, energy cutoff, etc.). The thirdterm was calculated by putting the isolated adsorbate in a boxwith dimensions of 15 Å × 15.5 Å × 16 Å and carrying outa spin-polarized �-point calculation. With this definition, amore negative value of adsorption energies denotes strongerbinding between adsorbates and surfaces.

III. RESULTS AND DISCUSSION

A. Adsorption of reaction intermediates

According to our previous calculations on Ni(111), thereaction intermediates in CH4 dissociation are predominantlyadsorbed at the hollow sites.28 Therefore, the threefold hol-low sites of the two alloyed Ni/Fe surfaces are highlightedin Fig. 2. In spite of the similar structural morphology, thealloyed Ni/Fe surfaces possess several new adsorption siteswhich are absent on Ni(111). These new sites are derived fromthe replacement of surface Ni atoms with Fe, which breaks thesurface homogeneity. As for NiFe(111), there are four differ-ent threefold hollow sites on the surface, namely hcp1, hcp2,fcc1, and fcc2 [see Fig. 2(a)]. The hcp1 and fcc1 sites arethe hollow sites that are surrounded by two Ni and one Featoms, while the hcp2 and fcc2 sites are surrounded by oneNi and two Fe atoms. On Ni2Fe(111), four threefold hollowsites are also found [see Fig. 2(b)], but all of the four hollowsites are surrounded by two Ni and one Fe atoms. The differ-ence among them is that a Ni (Fe) atom is located below thehcp1 (hcp2) site in the second layer and below the fcc1 (fcc2)site in the third layer.

1. Adsorption of H and surface electronic structure

The adsorption of atomic H on the alloyed Ni/Fe(111)surfaces is first investigated, and the adsorption energies atthe threefold hollow sites are listed in Table I. We have alsoexamined the adsorption of H at the bridge and atop sites andfound the adsorption energies of H at these sites are muchlower than those at the hollow sites.45 In some cases, H atomsmove from the bridge or atop site to the nearby hollow sitesafter geometry optimization. From Table I, it can be seen thatthe NiFe(111) surface is energetically more favorable than theNi2Fe(111) surface to adsorb H, which in turn is more favor-able than Ni(111). On each of the two alloyed surfaces, theH adsorption energies at the four threefold hollow sites are

rather close (The energy difference is within 0.05 eV.), indi-cating unapparent site-preference of H.

In order to provide a rational interpretation of the vari-ation in H adsorption energies with Fe concentrations, thedistributions of electronic charge within surface metal atomswere examined according to Bader’s theory.53 In this theory,space is divided up into atomic regions where the dividingsurfaces are at a minimum in the charge density, i.e., the gra-dient of the charge density is zero along the surface normal.Based on this algorithm, an efficient code for performing theBader’s analysis on a charge density grid was implementedby Henkelman et al.,54, 55 which was utilized to explore thecharge redistributions on Ni and Fe atoms.

The Bader’s analysis was first conducted on Ni(111) andFe(110) and the local valence charges on Ni and Fe atoms arecalculated to be 10.04 and 8.02 e, respectively. Taking thesedata as a reference, one can see that each of the surface Featoms “loses” ∼0.3 e on Ni2Fe(111), and the electrons re-moved from a Fe atom are shared equally by two Ni atoms,as shown in Table II. As the Fe content increases, about 0.2 eare “transferred” from a surface Fe atom to a surface Ni atomon NiFe(111). That is, while there are more electrons in thevalence shell of Ni, Fe acts as the electron donor on the Ni/Fealloyed surfaces. Given that Ni and Fe have electronegativi-ties of 1.91 and 1.83,56 respectively, the fact that Ni is moreelectronegative than Fe succeeds in accounting for the elec-tron transfer between them. Here the closest-packed surfaceof fcc Fe, Fe(111), was not applied because in the prelimi-nary calculations it was found reconstructed to a bcc Fe(110)structure upon H adsorption.45 Therefore, it is not feasible torelate the electronic structure of Fe(111) with its adsorptionproperties directly.

Then, the projected density of states (PDOS) onto the dstates of surface metal atoms on Ni/Fe(111) was calculated

TABLE II. Calculated local valence charges and d-band centers onmonometal and alloy surfaces.

Local charge (e) d-band center (εd eV)

Surface Ni Fe Ni Fe Weighted

Ni(111) 10.04 N/A −1.13 N/A −1.13Ni2Fe(111) 10.18 7.71 −1.20 −0.93 −1.09NiFe(111) 10.25 7.78 −1.29 −0.88 −0.99Fe(110) N/A 8.02 N/A −0.60 −0.60

134704-5 Toward CH4 dissociation and C diffusion J. Chem. Phys. 134, 134704 (2011)

FIG. 3. Density of states projected onto the d-band of different surface atoms. (a1)–(a3): d-band of surface Ni atoms; (b1)–(b3): d-band of surface Fe atoms.

and compared with those on Ni(111) and Fe(110), as shownin Fig. 3. The d-band centers defined within the frameworkof the Hammer–Nørskov model57 are given in Table II. FromFig. 3 and Table II, one can see that the d-band centers ofNi and Fe are shifted farther below the Fermi level as theirrespective contents decrease, which might be rationalized asfollows. Since the interatomic distance between the Ni atomsin the outermost and second layers is measured to be 2.47 Åon Ni(111), 2.49 Å on Ni2Fe(111), and 2.52 Å on NiFe(111),the strain effect induced by alloying on the electronic struc-ture of Ni is believed to be negligible. Thus, the shift in thed-band center is probably due to the so-called ligand effect.58

Because the coupling matrix element between the d-orbitalsof Ni and Fe increases with the Fe content, the overlap ofthe electronic states increases and the Ni d-band is progres-sively broadened from Ni(111) to Ni2Fe(111), and further toNiFe(111), which in turn leads to the downshift in the d-bandcenter. On the other hand, the Wigner–Seitz (WS) radii ofFe in the alloys are significantly reduced with respect to thatin bulk Fe, and therefore the stain effect predominates. Asthe embedding electron density from neighboring atoms in-creases with decreasing the WS radius of Fe, its d-band is alsobroadened. Consequently, the d-band center is shifted down-ward to preserve the d-band filling.59

On the basis of the calculations on the d-band centersof surface metal atoms, the weighted d-band centers at the

threefold hollow sites were derived according to the methodproposed by Pallassana et al.:60

εd−weighted =(V 2

NiεNid N Ni + V 2

FeεFed N Fe

)(V 2

Ni NNi + V 2

Fe N Fe) , (2)

where V 2Ni and V 2

Fe are the adsorbate (s or p)-metal d cou-pling matrix elements squared for surface Ni and Fe atomswith respect to Cu,59 and N Niand N Fe are the numbers of theproduced Ni–H and Fe–H bonds, respectively. From Fig. 4,it was found that the farther the weighted d-band center isshifted below the Fermi level, the weaker the interaction of Hwith metal surfaces.

As proposed by Hammer et al.,61 the changes inchemisorption energies can be linearly correlated to thechanges in d-band centers, according to the followingequation:

δEchem = V ′2

(ε)2δεd , (3)

where V ′2 = V 2/

r2(la+ld+1). Here, r is the metal–H distance(1.79 Å for Fe–H bond and 1.71 Å for Ni–H bond) and la

and ld are the angular moment quantum numbers of the ad-sorbate state and metal d-state, respectively. In Eq. (3) ε

= |εd − εa|, where εd and εa are the d-band center and the H1s orbital state in energy, respectively. Through Eq. (3), it canbe expected that the plot of the H adsorption energies against

134704-6 Fan et al. J. Chem. Phys. 134, 134704 (2011)

FIG. 4. H chemisorption energy as a function of the calculated weighted d-band center.

the d-band centers on Ni/Fe metal and alloy surfaces wouldgive a straight line only if the term V ′2/ (ε)2 for Ni and Feis equal. As the H 1s state is centered at about −5 eV belowthe Fermi level, one can obtain

V ′2Ni

(εNi)2

V ′2Fe

(εFe)2

= 1.24, (4)

which is found to be much greater than unity. This provides arational interpretation of the fact that there is no strong linearrelationship between the H chemisorption energies and theweighted d-band centers in Fig. 4.

2. Adsorption of C and CH

The calculated C and CH adsorption energies at thethreefold hollow sites are given in Table I, and the adsorptionconfigurations on the two alloyed surfaces are shown inFigs. 5(a), 5(b), 5(e), and 5(f). We have also examined theadsorption of C and CH at the bridge and atop sites, and thesituation is quite similar to that for the H adsorption.45 FromTable I, it can be seen that the adsorption of C and CH showsno apparent site-preference on NiFe(111). On one hand, one

FIG. 5. The most stable adsorption configurations and corresponding ad-sorption energies (eV) of the CHx (x = 0–3) on the NiFe(111) [(a)–(d)] andNi2Fe(111) [(e)–(h)]. The light blue balls denote Ni atoms; the dark blueballs denote Fe atoms; the white balls denote H atoms; the black balls denoteC atoms.

C–Ni bond is replaced with one C–Fe bond at the hcp2 andfcc2 sites with respect to the hcp1 and fcc1 sites, and the C–Fe interaction is expected to be energetically more favorablethan the C–Ni interaction. On the other hand, the Fe atom inthe second (third) layer below the hcp1 (fcc1) site binds theC and CH fragments more tightly than the Ni atom belowthe hcp2 (fcc2) site. It is these two opposite effects that leadto the close adsorption energies at the four hollow sites. Thehcp2 and fcc2 sites on Ni2Fe(111), however, are much morefavorable than the hcp1 and fcc1 sites to adsorb C and CH.This is probably because the outermost surface is uniform andonly the metal atoms as well as the depressions in the second(third) layer affect the adsorption properties of Ni2Fe(111).

3. Adsorption of CH2 and CH3

The most stable adsorption configurations of CH2 andCH3 on the two alloyed Ni/Fe(111) surfaces are shown inFig. 5(c), 5(d), 5(g), and 5(h). As for CH2, one of the twoH atoms is bonded to a surface metal atom to form the so-called C–H–M (M = Ni or Fe) three-center bond to lower thesystem energy,62, 63 while the other C–H bond is tilted awayfrom the alloyed surfaces. Upon adsorption, the asymmetricC–H stretching frequency occurs at 3007 cm−1 on NiFe(111)and at 3013 cm−1 on Ni2Fe(111), which is close to the gas-phase C–H stretching frequency (∼3000 cm−1). In contrast,a redshift of ∼400 cm−1 is observed for the symmetric C–H stretching frequency on the two alloyed surfaces, arisingfrom the stretched C–H bonds. In the CH3 adsorption config-urations, all of the three H atoms point to the surface atoms toform three C–H–M three-center bonds. Similarly, a redshiftof ∼200 cm−1 for the C–H stretching frequencies is observedbecause the C–H bonds are weakened by the formation of theC–H–M three-center bonds.

The binding of CH2 and CH3 on the alloyed surfacesshows apparent site preference, compared to the species men-tioned above. From Table I, one can see that the adsorptionenergies of CH2 and CH3 at the fcc sites are higher than thoseat the hcp sites, which resembles the scenario on Ni(111).28

Furthermore, the fcc2 site is preferred over the fcc1 site onNiFe(111) because both the C atom and the C–H bonds inCH2 (CH3) prefers surface Fe atoms to Ni atoms, which in-dicates that the adsorption of CH2 (CH3) is also sensitive tothe local catalyst composition. However, the adsorption ener-gies of CH2 (CH3) at the fcc1 and fcc2 sites are rather closeon Ni2Fe(111) because the metal atom in the third layer has aminor effect on their adsorption. With the comparison of theadsorption energy of CH2 (CH3) on the three surfaces, it canbe deduced that the introduction of Fe in Ni catalysts gener-ally promotes the adsorption of the C1 derivatives.

4. Adsorption of CH4

CH4 is found to be physisorbed on the two alloyed sur-faces with the small adsorption energies of about −0.02 eVand shows no site preference, indicating a weak van derwaals-type interaction between CH4 and metal atoms. This

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FIG. 6. Potential energy diagram of CH4 dissociation on Ni(111),Ni2Fe(111), and NiFe(111). The chemisorbed H needed to balance the ad-sorption states have been omitted to simplify the notation.

has already been verified by the previous calculations and ex-perimental observations.18, 64

B. Successive dissociation of CH4

Now that the adsorption behavior of the reaction in-termediates on the alloyed Ni/Fe surfaces is explored, thesuccessive dissociation of CH4 to form surface C and H2 areinvestigated in detail.

1. CH4 dissociation

For the purpose of elucidating the effect of the Fe contenton the kinetics of CH4 dissociation, the potential energy dia-grams and the activation energies for CH4 dissociation on thealloyed Ni/Fe surfaces are compared with those on Ni(111),28

as shown in Fig. 6 and Table III.For the first step for CH4 dissociation, the most stable

configurations of the TS on the two Ni/Fe surfaces are givenin Fig. 7 (TS1 and TS5). In the two configurations, the CH3

fragment is bound at the atop site while the detached H atomis located at the hollow site nearby. This geometry resem-bles those of the activated complexes for CH4 dissociation onNi(111) and other close-packed transition metal surfaces.21, 28

The energy barriers are calculated to be 0.89 eV on NiFe(111)and 0.94 eV on Ni2Fe(111), respectively, and the lengths ofthe elongated C–H bonds are 1.58 and 1.60 Å. Furthermore,

TABLE III. Calculated activation energies (Ea) and reaction heats(E) for the elementary steps involved in CH4 dissociation on Ni(111),Ni2Fe(111), and NiFe(111).

Ni(111) Ni2Fe(111) NiFe(111)

Ea E Ea E Ea E

Elementary reaction (eV) (eV) (eV) (eV) (eV) (eV)

CH∗4 → CH∗

3 + H∗ 0.91 0.01 0.94 − 0.06 0.89 − 0.23CH3

∗ → CH2∗ + H∗ 0.70 0.07 0.68 0.04 0.65 − 0.03

CH2∗ → CH∗ + H∗ 0.35 − 0.34 0.34 − 0.32 0.36 − 0.32

CH∗ → C∗ + H∗ 1.33 0.52 1.18 0.39 1.11 0.36

FIG. 7. Schematic representations of the most stable transition states for CH4dissociation on the NiFe(111) (TS1–TS4) and Ni2Fe(111) (TS5–TS8). Thelight blue balls denote Ni atoms; the dark blue balls denote Fe atoms; theblack balls denote C atoms; the white balls denote H atoms; the distances ofthe detached C–H bonds (Å) are also shown.

some other configurations have also been screened to estimatethe possibilities to act as the TSs for CH4 dissociation,45 buttheir total energies are generally higher.

It should be noted that while the reaction heat (E) forCH4 dissociation increases with the Fe content, the activa-tion energies on the three surfaces are rather close (the energydifference is within 0.05 eV). This is in conflict with the well-known Brønsted–Evans–Polanyi (BEP) relationship65 whichclaims that the activation energies for elementary steps scalelinearly with the reaction heats if entropy effects are ne-glected. To provide a rational interpretation, the activation en-ergy for CH4 dissociation (Ea) is decomposed as follows:

Ea = Eads,TS − Ebond(H−CH3), (5)

where Ebond(H−CH3) is the H–CH3 bond energy of the gas-phase CH4 molecule and Eads,TS is the chemisorption en-ergy of the activated complex with respect to the gas-phaseCH3 and H fragments. As the catalyst composition has no ef-fect on Ebond(H−CH3), the variation in Ea arises only from thechange in Eads,TS which can be further decomposed into threeterms:66

Eads,TS = Eads,TS(CH3) + Eads,TS(H) + Eint, (6)

where Eads,TS(CH3) [Eads,TS(H)] is the CH3 (H) chemisorptionenergy in the geometry of the activated complex without theH (CH3) fragment. Here, Eint is positive and defined as theinteraction energy between CH3 and H. Combining Eqs. (7)and (8), one can see that a more negative Eads,TS(CH3)

[Eads,TS(H)] and a lower Eint lead to a more negative Eads,TS,and hence a lower Ea .

The decomposition of the activation energy for CH4 dis-sociation on the three surfaces is summarized in Table IV.From the table, it can be seen that Eads,TS(CH3) and Eads,TS(H)

turn more negative as the Fe content increases, which is what

TABLE IV. Decomposition of the activation energy for CH4 dissociationon Ni(111), Ni2Fe(111), and NiFe(111).

Surface Eads,TS(CH3) Eads,TS(H) Eint Eads,TS

Ni(111) −1.37 −2.59 0.14 −3.82Ni2Fe(111) −1.38 −2.62 0.20 −3.80NiFe(111) −1.44 −2.67 0.26 −3.85

134704-8 Fan et al. J. Chem. Phys. 134, 134704 (2011)

one might expect on the basis of the examination of the ad-sorption properties of the alloyed surfaces. On the other hand,Eint become higher from Ni(111) to Ni2Fe(111), and furtherto NiFe(111). In general, Eint is determined in a large partby two factors: (i) the bonding competition effect,66 arisingfrom the sharing of one surface atom by CH3 and H and(ii) the direct Pauli repulsion, which is related to the dis-tance between CH3 and H. First, the CH3–metal and H–metalinteractions are strengthened as the Fe content increases,which would enhance the bonding competition effect. Sec-ond, the distance between CH3 and H is 1.61 Å on Ni(111),1.60 Å on Ni2Fe(111), and 1.58 Å on NiFe(111), leading tothe enhancement of the direct Pauli repulsion between them.Therefore, as the Fe content increases, the more negativeEads,TS(CH3) [Eads,TS(H)] and higher Eint result in the overallsmall change in Eads,TS, and hence the close Ea for CH4 dis-sociation on the three surfaces, which also helps explain theaforementioned deviation from the BEP relationship.

2. CH3 and CH2 dissociation

As for the second and third dehydrogenation steps, themost stable TSs for CH3 and CH2 dissociation on the two al-loyed surfaces are shown in Fig. 7 (TS2, TS3, TS5, and TS6).The TS configurations are similar to those on Ni(111) andother transition metal surfaces,21, 28, 67 in which the carbon-containing fragments are located at their favorite adsorptionsites and the detached H atoms are positioned at the adjacentatop sites. As the incorporation of Fe in Ni stabilizes the initialsate (IS), TS, and FS to a similar extent, the energy barriersas well as the reaction heats on the three surfaces are ratherclose.

3. CH dissociation

CH dissociation is of particular importance in steammethane reforming because the relative rate between this stepand CH oxidation determines the rate of carbon deposition oncatalyst particles.28, 68 The most stable TS configurations onthe two alloyed surfaces are shown in Fig. 7 (TS4 and TS8),where the positions of the detached H atoms are found to bedifferent. H is positioned on top of a surface Fe atom in TS4while it is located at the nearby hollow site in TS8. The energybarriers are calculated to be 1.11 and 1.18 eV, much lowerthan that on the Ni(111).

It should be noted that in our previous calculations the TSon Ni(111) where H is located at the hollow site was found tobe 0.09 eV lower in energy than that where H is located atthe atop site.28 In the present work, one can see on NiFe(111)the TS with the H atom at the nearby hollow site is 0.16 eVhigher in energy than TS4, while on Ni2Fe(111) the TS withthe H atom at the atop site is 0.01 eV higher in energy thanTS8.45 The reason for the different site preference of the de-tached H atoms in the TSs for CH dissociation might be ra-tionalized as follows. On one hand, because the hollow site isthermodynamically more favorable to adsorb H than the atopsite, the potential energy of the system is lowered if H is posi-tioned at the adjacent hollow site. On the other hand, however,the C and H atoms are placed too closely to release the local

FIG. 8. Schematic representations of C dissolution in the Ni16Fe16 supercell.(a) O1 site; (b) T site; (c) O2 site; The light blue balls denote Ni atoms; thedark blue balls denote Fe atoms; and the red balls denote C atoms.

strain induced by their sharing of the two surface atoms. Thus,the balance of these two effects eventually determines the sitewhere H is positioned in the TSs.

C. C diffusion in bulk Ni/Fe alloys

Through DFT calculations, Abild-Pedersen et al. claimedthat the C surface or subsurface diffusion is dominant duringCNF growth,30 and the bulk diffusion is unlikely to occur ow-ing to the considerably high energy barrier. Under the real-istic catalytic conditions, however, the surface sites might beblocked by the surface C or other reaction intermediates and,therefore, the C diffusion through the bulk of catalyst particlesbecomes feasible.33 For instance, Hongtao et al. found that thelattice expansion of catalyst particles is induced by C dissolu-tion through high resolution transmission electron microscope(HRTEM) (Ref. 69) and Avdeeva et al. also proved the avail-ability of C in bulk metal by XRD measurements where thediffraction lines of Ni(220) and Ni(311) are shifted to somesmaller angles.70

1. C diffusion in Ni16Fe16

The dissolution of C in the Ni16Fe16 supercell is first in-vestigated. The typical interstitial sites in the bulk of NiFeare the octahedral (O-site) and tetrahedral (T-site) sites, asshown in Fig. 8. According to the number of the formed C–Fe covalent bonds, two different O sites have been considered,namely O1-site and O2-site.

To calculate the dissolution heat, both the internal coor-dinates and cell parameters are allowed to relax during ge-ometry optimization. The previous calculation revealed thatincluding the cell relaxation is important to obtain the reason-able dissolution heat.34, 71, 72 The dissolution heats of C at theO1, O2, and T sites with respect to the isolated C atom areshown in Table V. From the table, one can see that the O1 siteis the most stable site for C to occupy in the Ni16Fe16 super-cell. The percentages of change in cell volume (V ) are alsogiven in Table V, which are defined as

V = V (NimFenC) − V (NimFen)

V (NimFen), (7)

where V (NimFenC) is the relaxed cell volume with a C atomdissolved, and V (NimFen) is the relaxed volume of the al-loy cell. Considering that the adsorption energy of C on theNiFe(111) surface is calculated to be −6.90 eV, the dissolu-tion of C atoms at the O sites of the alloyed metal bulk is

134704-9 Toward CH4 dissociation and C diffusion J. Chem. Phys. 134, 134704 (2011)

FIG. 9. Potential energy diagram of C diffusion in the bulk of the Ni16Fe16and Ni24Fe12 supercell along the direct O1 and O2 pathway.

energetically more favorable. The occupation of C at the Tsite is much less stable than those at the O sites and on thesurface, and thus energetically unfavorable.

The C dissolution at the O1 site is taken as the IS andFS to explore the C diffusion in the Ni16Fe16 supercell andthe C dissolution at the O2 site is taken as an intermediatestate. Using the CI-NEB method, the MEP for C diffusionhas been located, as shown in Fig. 9. One can see that theMEP shows a symmetric pattern with two saddle points bothat 1.81 eV. The image 00, 03, 05, 07, 10 [00, 03, 05, 07, 10are the sequence numbers of the calculation points in Fig. 9]corresponds to the IS, first TS, intermediate state, second TS,and FS, respectively. In the TSs, the C atom is located nearlyat the middle point of the Ni–Fe bond. The C–Ni and C–Fedistances are measured to be 1.72 and 1.70 Å, respectively,and the Ni–Fe distance is dramatically stretched from 2.52 to3.42 Å to release the lattice strain.

2. C diffusion in Ni24Fe12

Two different O sites have been considered in bulk Ni2Fe,namely O1-site and O2-site. The dissolution heats of C at theO1 and O2 sites and the corresponding percentages of changein cell volume are given in Table V. It can be seen that the O1site is the most stable site for the C dissolution in the Ni24Fe12

supercell. Considering that the adsorption energy of C on theNi2Fe(111) surface was calculated to be −6.91 eV, the dis-

TABLE V. Calculated heats of solution (Esol) and percentages of changein cell volume (V ).

Site

Cell O1 O2 T

Esol(eV) Ni16Fe16 − 7.27 − 7.10 −5.79Ni12Fe24 − 7.31 − 7.12 Not stable

Ni32 − 7.25 − 5.65V (%) Ni16Fe16 1.52 1.80 2.23

Ni12Fe24 1.49 2.04 N/ANi32 0.07 0.09

solution of C from the surface to the O sites is energeticallyfavorable. The occupation of C at the T site is found to be un-stable, which is relaxed to the nearby O site during geometryoptimization.

The C dissolution at the O1 site is, therefore, taken as theIS and FS to explore the C diffusion in Ni24Fe12, and the Cdissolution at the O2 site is taken as an intermediate state. Us-ing the CI-NEB method, the MEP of C diffusion has beenlocated and illustrated in Fig. 9. Similarly, the MEP alsoshows a symmetric pattern with two saddle points both at 1.73eV. In Ni32C (Ref. 34) and Ni16Fe16, the energy barriers werefound to be 1.64 and 1.81 eV, respectively, implying that the Cdiffusion in bulk metal becomes more difficult as the Fe con-tent increases. The image 00, 03, 05, 07, and 10 correspondsto the IS, first TS, intermediate state, second TS, and FS, re-spectively. In this case, the C atom is located exactly at themiddle point of the Ni–Fe bond in the TSs, and the C–Ni andC–Fe distances are both calculated to be 1.72 Å.

It should be noted that the energy barriers for C diffusionwas calculated at low C concentrations in this work. However,the diffusion barrier in bulk metal was previously found to bevery sensitive to the C concentration in catalyst particles.34

As the C concentration is increased, the diffusion barrier isexpected to be lowered. Thus, a kinetic study to estimate the Cconcentration in bulk metal under experimental conditions ishighly demanding, which will be addressed in the following.

D. Kinetic analysis

Using the technique of controlled atmosphere electronmicroscopy, Baker et al. measured the activation energiesfor CNF growth, which show excellent agreement with thosefor the C diffusion through catalyst particles.17 It is there-fore generally accepted that the C diffusion through thebulk of catalyst particles is the rate-limiting step for CNFgrowth.73 That is, it is reasonable to assume the dissociationof CH4 to produce surface C atoms and H2 molecules is inquasiequilibrium.

CH4 dissociation can be described using the followingequations:

CH4(g) + 2∗ = CH3∗ + H∗, (8)

CH3∗+∗ = CH2

∗ + H∗, (9)

CH2∗+∗ = CH∗ + H∗, (10)

CH∗+∗ = C∗ + H∗, (11)

2H∗ = H2(g). (12)

Considering that steps (8)–(12) are all in equilibrium, theoverall reaction can be written as

CH4(g)+∗ = C∗ + 2H2(g). (13)

134704-10 Fan et al. J. Chem. Phys. 134, 134704 (2011)

Then, we have

K 0i = exp

(−G0i

RT

), (14)

PCH4

P0θ2∗ exp

(−G01

RT

)= θCH3θH, (15)

θCH3θ∗ exp

(−G02

RT

)= θCH2θH, (16)

θCH2θ∗ exp

(−G03

RT

)= θCHθH, (17)

θCHθ∗ exp

(−G04

RT

)= θCθH, (18)

θ2H exp

(−G05

RT

)= PH2

P0θ2∗ , (19)

PCH4

P0θ∗ exp

(−G06

RT

)=

(PH2

P0

)2

θC, (20)

where the K 0i (i = 1–6) is the standard equilibrium constant

for steps (8)–(13) and G0i is the corresponding standard

Gibbs free energy change. The calculated Gibbs free energychanges at 873.15 K are listed in Table VI. The details onhow to derive the Gibbs free energy at finite temperature andpressure can be found in our previous work.28 If it is assumedthat the partial pressure of CH4 is kept at 1 bar and the mo-lar ratio of CH4 to H2 is 4:1, the coverage of the CHx specieson Ni(111), Ni2Fe(111), and NiFe(111) can be quantitativelyestimated, which is also given in Table VI. From the table, itcan be seen that the adsorbed H is the most abundant reac-tion intermediate in CH4 dissociation, and the surface C cov-erage, ranked in descending order, is as follows: NiFe(111)

TABLE VI. Calculated standard Gibbs free energy changes (eV) at 873.15K and the surface coverages of the CHx and H species on Ni(111),Ni2Fe(111), and NiFe(111).

Surface Ni(111) Ni2Fe(111) NiFe(111)

G01 1.24 1.12 0.98

G02 0.01 −0.03 −0.25

G03 −0.33 −0.29 −0.16

G04 0.45 0.37 0.20

G05 −0.14 −0.10 0.03

G06 1.09 0.96 0.83

G0dis −0.31 −0.26 −0.23

G06 + G0

dis 0.78 0.70 0.60θCH3 2.99 × 10−7 1.10 × 10−6 2.18 × 10−6

θCH2 1.32 × 10−6 6.08 × 10−6 9.42 × 10−5

θCH 5.23 × 10−4 1.25 × 10−3 1.36 × 10−3

θC 6.76 × 10−6 3.68 × 10−5 1.64 × 10−4

θH 1.67 × 10−1 2.04 × 10−1 3.81 × 10−1

θ∗ 8.33 × 10−1 7.95 × 10−1 6.18 × 10−1

xb 4.99 × 10−4 1.47 × 10−3 5.60 × 10−2

> Ni2Fe(111) > Ni(111), which is in accordance with theprediction of the potential energy diagram in the Sec. III B.Thus, it can be concluded that the incorporation of Fe in Nicatalysts would facilitate the formation of surface C.

It has been demonstrated that a segregation/dissolutionequilibrium exists at the gas side of catalyst particles betweenthe adsorbed C and C dissolved in particles just below theselvedge.74, 75 Therefore, the C concentration in bulk metalis in equilibrium with the surface C coverage, which can bedescribed by a Langmuir-type equation:76

θC(1 − xb) exp

(−G0

dis

RT

)= xbθ∗, (21)

which can be rearranged to

xb =θC exp

(−G0

disRT

)θC exp

(−G0

disRT

)+ θ∗

, (22)

where the G0dis is the Gibbs free energy change of a C atom

dissolution into the bulk of catalyst particles and xb is the Cconcentration in the bulk of catalyst particles. The G0

dis forone C atom dissolution in the Ni16Fe16, Ni24Fe12, and Ni32

supercells as well as the calculated xb are given in Table VI.From the table, one trend is found: The higher the Fe content,the higher the C concentration in bulk metal.

In order to correlate the C concentration in catalyst par-ticles (xb) with other experimental conditions, Eq. (22) issimplified by considering that θC exp(−G0

dis

/RT ) is much

smaller than θ∗ under experimental conditions (see Table VI):

xb = θC

θ∗exp

(−G0

dis

RT

). (23)

Combining Eqs. (20) and (23), one can obtain

xb =PCH4P0(

PH2P0

)2 exp

(− (

G0tot + G0

dis

)RT

). (24)

From this equation, it is apparent that (PCH4/P0)/(PH2/P0)2

is dependent on the feedstock partial pressure. This is inaccordance with the conclusion by Snoeck et al. who foundthat the solubility of C in the catalyst particle depends on theaffinity for C formation: The higher the affinity for C forma-tion (high PCH4 or low PH2 ), the higher the solubility of C inthe particle.74 On the other hand, G0

6 is determined by thecomposition of feedstock and reaction temperature. There-fore, all the experimental conditions including feedstockcomposition, the partial pressures of CH4 and H2, reactiontemperature, and catalyst composition have effects on the Cconcentration in the catalyst particle, which in turn play a keyrole in determining the morphologies of produced CNFs.

IV. CONCLUSIONS

Density functional theory calculations have been per-formed to investigate CH4 dissociation and C diffusion dur-ing the Ni/Fe-catalyzed growth of CNFs. Two bulk modelswith different Ni to Fe molar ratios (NiFe and Ni2Fe) are con-structed, and XRD simulations are conducted to evaluate their

134704-11 Toward CH4 dissociation and C diffusion J. Chem. Phys. 134, 134704 (2011)

reliability. As the simulated XRD patterns are in good agree-ment with the experimental data with nearly the same 2θ po-sitions of the three characteristic reflections [(111), (200), and(220)], these two models are well suited to reproduce the crys-talline structures of the fcc Ni/Fe alloys.

Through the calculations on the electronic structure, itis found that while there are more electrons in the valenceshell of Ni, Fe acts as the electron donor on the Ni/Fe al-loyed surfaces. The d-band centers of Ni and Fe are shiftedfarther below the Fermi level as their respective contents de-crease, and the weighted d-band centers at the threefold hol-low sites are shifted up as the Fe content increases, whichaccounts for the strengthened binding of the C1 derivatives tothe alloyed Ni/Fe(111) surfaces compared to that on Ni(111).The introduction of Fe in Ni catalysts has a negligible effecton the energy barriers for the first three elementary steps forCH4 dissociation, while the activation energy for CH disso-ciation is substantially lowered as the Fe content increases,which would facilitate the formation of surface C.

The C dissolution from catalyst surfaces into bulk metalis energetically favorable for Ni and Ni/Fe catalysts, and thediffusion of C through catalyst particles is hindered by the Fecomponent. With the combination of DFT calculations and ki-netic analysis, the C concentration in catalyst particles is pre-dicted to increase with the Fe content and is also dependenton other experimental conditions, including catalyst composi-tion, the partial pressures of feedstock, and reaction tempera-ture. Our results provide a rational interpretation of the effectsof various experimental conditions on the C concentration inbulk metal, and hence the microstructures of produced CNFs.

ACKNOWLEDGMENTS

This work is supported by Doctoral Fund of Min-istry of Education of China (No. 200802511007), Na-tional Science Foundation (NSF) of China (No. 21003046),and Fundamental Research Funds for Central Universities(No. WA1014027). The computational time provided by theNotur project is highly acknowledged (No. nn4685k).

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