Detailed Kinetic Modeling of Iron Nanoparticle ... - ACS Publications

J. Phys. Chem. C 2007, 111, 5677-5688

5677

Detailed Kinetic Modeling of Iron Nanoparticle Synthesis from the Decomposition of Fe(CO)5 John Z. Wen, C. Franklin Goldsmith, Robert W. Ashcraft, and William H. Green* Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts AVenue, Room 66-270, Cambridge, Massachusetts 02139 ReceiVed: October 6, 2006; In Final Form: January 24, 2007

A detailed chemical kinetic model for gas-phase synthesis of iron nanoparticles is presented in this work. The thermochemical data for Fen clusters (n g 2), iron carbonyls, and iron-cluster complexes with CO were computed using density functional theory at the B3PW91/6-311+G(d) level of theory. Chemically activated and fall-off reaction rates were estimated by the QRRK method and three-body reaction theory. Kinetic models were developed for two pressures (0.3 and 1.2 atm) and validated against literature shock-tube measurements of Fe concentrations and averaged nanoparticle diameters. The new model indicates that the nanoparticle formation chemistry is much more complex than that assumed in earlier studies. For the important temperature range near 800 K in a CO atmosphere, the Fe atom formation and consumption are largely controlled by the chemistry of Fe(CO)2, especially the reactions Fe(CO)2 h FeCO + CO, Fe + Fe(CO)2 h Fe2CO + CO, and Fe(CO)2 + Fe(CO)2 h Fe2(CO)3 + CO. The decomposition of Fe(CO)5 is restricted by the rate of the spinforbidden reaction, Fe(CO)5 h Fe(CO)4 + CO. This model facilitates the understanding of how the reaction conditions affect the yield and size distribution of iron nanoparticles, which will be a crucial aspect in the gas-phase synthesis of carbon nanotubes.

1. Introduction Iron pentacarbonyl (Fe(CO)5) has been used extensively as a gaseous precursor to generate iron catalyst nanoparticles when carbon nanotubes (CNT) were produced during flame synthesis,1 in the HiPco2 (high-pressure CO conversion) process, and in a laminar flow reactor.3 Because the diameter and yield of CNTs produced using these approaches directly correlate to the size and concentration of iron nanoparticles formed,4 precise control of iron nanoparticle formation processes is essential to improve the structure of CNTs and increase their production rates. In addition, it was found experimentally that the heating rate of a cold Fe(CO)5/CO mixture significantly affects the formation of CNTs.2 Therefore, the study of the chemical kinetics of Fe(CO)5 decomposition and iron nanoparticle formation is very important to facilitate further understanding of the CNT growth mechanism. This requires both reliable kinetic measurements for the thermal decomposition process and known thermochemical properties of related species. Recently, a few well-documented experimental datasets have been published describing iron particle formation from the decomposition of Fe(CO)5 behind shock waves,5 in a flow reactor,6 and in a premixed flame.7 Coupled with a simplified particle growth mechanism, Giesen et al.5 used a global reaction (Fe(CO)5 f Fe + 5CO) to describe the thermal decomposition of Fe(CO)5 without accounting for any intermediates or byproducts. They adjusted the association and dissociation rate constants of iron dimers (Fe2) in their model to make the predicted iron atom concentrations match their measurements. However, the rate constants inferred in this way appear to be unphysically fast, particularly because this two-atom association reaction must be deep in the fall-off regime at the experimental * Author to whom correspondence should be addressed. E-mail: [email protected].

pressures and temperatures. In an earlier study, Krestinin et al.8 proposed a reduced, two-stage mechanism (Fe(CO)5 f FeCO + 4CO and FeCO f Fe + CO) to describe the decomposition of Fe(CO)5. Unfortunately, the dissociation energy they used for D(Fe-CO), 20.5 kcal/mol, is much larger than the value determined experimentally using thermochemical cycles (8.1 ( 3.5 kcal/mol9). Giesen et al.5 showed that Krestinin’s model underpredicted the formation of Fe and CO behind a shock wave, where the temperature is 740 K, and suggested that FeCO is a minor species. Taking the relatively low dissociation energy of FeCO (8.1 kcal/mol in comparison with 36.7, 29.1, 27.9, and 41.5 kcal/mol for Fe(CO)2, Fe(CO)3, Fe(CO)4, and Fe(CO)5, respectively) into account, the dissociation of FeCO to Fe and CO happens very fast; other Fe(CO)n molecules, especially Fe(CO)2, are consequently much longer-lived and likely more important than FeCO. When modeling the inhibition effects of Fe(CO)5 on atmospheric flames, Rumminger et al.10 introduced a five-step Fe(CO)5 decomposition model, which described the dissociation of five iron carbonyls (Fe(CO)n f Fe(CO)n-1 + CO, n ) 1-5) and assumed that all of the iron became Fe atoms before forming particles. This mechanism was later implemented in modeling the formation of carbon nanotubes in HiPco processes11 under high pressure (30 atm). Indeed, when Fe(CO)5 is introduced with the fuel into diffusion flames10 or a HiPco reactor,11 the Fe(CO)5 decomposition occurs at lower temperatures than the fuel pyrolysis or CNT formation chemistry. It is therefore a reasonable first approximation to model the iron carbonyl chemistry separately as Rumminger et al. have done. However, as shown here, the conversion of Fe(CO)5 into iron nanoparticles involves several other important intermediates and reactions that should be included in a detailed kinetic model. The thermal decomposition of Fe(CO)5 and the subsequent iron nanoparticle formation involve iron clusters, Fen, n )

10.1021/jp066579q CCC: $37.00 © 2007 American Chemical Society Published on Web 03/28/2007

5678 J. Phys. Chem. C, Vol. 111, No. 15, 2007 1,2,3,..., iron carbonyls, Fe(CO)n, n ) 1-5, and other intermediates, Fem(CO)n. Among these species, only two, Fe and Fe(CO)5, can be found in the commonly used thermochemical table.12 The calculation of the thermochemical properties of other species is required. The geometries of Fen clusters have been experimentally investigated by different means.13,14 Nevertheless, identification of the ground-state structure of a cluster of even modest size using first-principles methods, for example, density functional theory (DFT), is a demanding computational task. Recently, DFT has been applied to study clusters with up to 32 atoms.15 However, research has shown that ab initio calculation results for iron dimers, Fe2,16 and trimers, Fe3,17 are still uncertain and vary depending on the DFT method used. Gutsev et al.18 recently performed DFT calculations of Fen, n ) 2-6, using different functionals, including a hybrid method (B3LYP) and several pure methods (BLYP, BP86, BPW91, and etc.). They found that both the hybrid method and aforementioned pure GGA methods predict the same ground-state structures and multiplicities for iron clusters. However, the pure methods, for example, BPW91, overestimated the fragmentation energies by about 1 eV (about 23 kcal/mol). The same authors also conducted studies on the effect of basis sets on DFT calculations of first-row transition-metal dimers.19 They found that the computed geometries of the Fe2 ground state are nearly the same for the recently developed triple-ζ (TZ) correlationconsistent basis set and the 6-311+G(d) basis set. Experimental studies of dissociation energies and reactivity of iron carbonyls date back to the late 1960s. Sunderlin et al.9 summarized the dissociation energies of iron carbonyls and showed that the measured variations are significant (e.g., the dissociation energy of Fe(CO)5 ranges from 16 to 58 kcal/mol, with the error bars up to 12 kcal/mol). While the thermochemical properties of iron pentacarbonyl are known,20 the structures and properties of other iron carbonyls were computed using ab initio methods. Recently, the geometries and dissociation energies of Fe(CO)n, n ) 1-4, were studied by Ricca et al.21,22 using correlation-consistent basis sets and the CCSD(T) approach. The computed dissociation energies were in good agreement with experimental data. Studies of other Fe/CO complexes are rare in the literature. Recently Gutsev et al.23 investigated the geometry and electronic structure of neutral and charged FenCO, n ) 1-6, and the energetics of their reactions with CO. By comparing results computed with many different functionals, they concluded that the hybrid functionals, for example, B3LYP and B3PW91, gave the most accurate bond dissociation energies. The lowest-energy unimolecular dissociation channel for FenCO was found to be the loss of CO. Very recently, Gutsev, Mochena, and Bauschlicher24 have studied the structure of Fe4 with different coverage by C and CO. The authors suggested that the energetics of the disproportionation reactions found in their work may be useful in attempting to model the iron-CO chemistry applicable in CNT formation.25 The aforementioned experimental and theoretical studies provide the necessary information to estimate thermochemical properties and rate constants for a large number of elementary reactions occurring in the thermal decomposition of Fe(CO)5 and in iron cluster formation. The objective of this work was to estimate the thermochemical properties of each important species by determining its geometry and vibrational frequencies and to develop kinetic models that can describe the nucleation and growth of iron nanoparticles under various reaction conditions. The computed thermochemical properties and kinetic models were compared with experimental data from the literature. The new thermochemical data and kinetic models

Wen et al.

Figure 1. Molecular structures of intermediates Fen(CO)m and their spin states.

allow more reliable predictions of iron nanoparticle formation under a variety of reaction conditions. 2. Methodology DFT calculations were conducted to compute the geometry and thermochemistry of iron clusters, carbonyls, and intermediate species. Statistical mechanics was employed to convert these quantum chemistry results into thermochemical properties. The reaction rates in the fall-off regime were estimated using the third-body theory and the QRRK method.26 Finally, the kinetic model was derived using the SPAMM27 approach to couple the gas-phase chemistry and reactions among nanoparticles. Each of these techniques is discussed below. 2.1. DFT Calculations. Ab initio calculations for iron clusters, carbonyls, and intermediates were carried out using the Gaussian 03 program.28 Because the pure GGA methods overestimate both the fragmentation energies of iron clusters and the dissociation energies of Fen-CO bonds, the hybrid B3PW91 method was used in this study to provide more accurate heats of formation. As will be mentioned later in this paper, heats of formation for Fen, n ) 2-4, and Fe(CO)n, n ) 1-4, were calculated based on their measured fragmentation energies and dissociation energies, respectively. For the atomic orbitals, we used the standard Gaussian version of the 6-311+G(d) basis set (Fe, [10s7p4d1f]; O, [5s4p1d]).29 The exponents and contraction coefficients of this basis set are listed in the Supporting Information. As mentioned previously, this basis set predicts similar properties for iron cluster ground states as the recently developed triple-ζ basis set. Geometry and frequency optimizations were performed for each possible spin multiplicity for species of interest until a ground state was found. The optimized geometry and spin state were compared with previous calculations from the literature, as shown later. 2.2. Thermochemical Properties. The entropies and heat capacities were calculated using statistical mechanics based on the vibrational frequencies and optimized structures obtained from the density functional study. The B3PW91/6-311+G(d) optimized geometries and harmonic vibrational frequencies (scaled by 0.98) were used to calculate the rotational and

Kinetic Modeling of Iron Nanoparticle Synthesis

J. Phys. Chem. C, Vol. 111, No. 15, 2007 5679

TABLE 1: Thermodynamic Data of Species Considered in the Present Work (Unit: ∆Hf0: kcal mol-1; S0 and Cp: cal mol-1 K-1, Zero-Point Energy from B3PW91/6-311+G(d): Hartrees)e species

∆Hf 0

S0

Cp,300K

Cp,400K

Cp,500K

Cp,600K

Cp,800K

Cp,1000K

Cp,1500K

literature multiplicity

zero-point energy

Fe(CO)5

-173.96a -109.08b -51.96b 5.16b 64.78b 99.30a 171.35d 226.67d 275.54d 319.37 351.48 389.52 422.02 115.93 173.33 217.32 261.80 302.83 347.78 63.81 130.40 21.22

106.30 103.25 90.22 76.41 61.27 43.14c 61.10 82.07 94.20 101.60 109.70 123.70 133.50 81.78 94.49 104.10 111.90 126.30 131.70 93.36 111.40 107.98

40.81 34.77 26.84 19.63 11.61 6.14c 8.38 13.33 19.16 24.77 29.76 36.01 41.53 16.41 22.22 27.71 33.44 38.98 44.81 24.15 30.82 32.59

45.07 37.67 28.94 21.00 12.20 6.10c 8.60 13.57 19.46 25.22 30.60 36.75 42.45 17.07 22.98 28.66 34.55 40.19 46.16 25.71 32.15 34.62

47.88 39.58 30.33 21.92 12.60 5.95c 8.72 13.69 19.60 25.44 31.01 37.11 42.90 17.50 23.47 29.24 35.23 40.89 46.94 26.73 33.03 35.93

50.01 41.08 31.43 22.64 12.93 5.79c 8.78 13.75 19.68 25.56 31.25 37.30 43.14 17.85 23.85 29.68 35.72 41.39 47.49 27.51 33.72 36.95

53.18 43.36 33.12 23.76 13.45 5.53c 8.85 13.82 19.77 25.68 31.48 37.50 43.39 18.39 24.41 30.30 36.37 42.08 48.21 28.69 34.80 38.51

55.35 44.96 34.30 24.54 13.82 5.38c 8.88 13.85 19.80 25.73 31.59 37.59 43.51 18.77 24.80 30.71 36.77 42.53 48.64 29.50 35.56 39.60

58.30 47.18 35.93 25.62 14.33 5.31c 8.92 13.88 19.84 25.79 31.71 37.68 43.63 19.29 25.30 31.24 37.26 43.10 49.16 30.59 36.60 41.08

148 348 348 348 343,48 518,21 718 1118 1518 1718 2118 2342 2542 743 943 1343 1743 1943 n/a n/a n/a n/a

-1830.278 -1716.936 -1603.598 -1490.257 -1376.908 -1263.590 -2527.204 -3790.851 -5054.528 -6318.222 -7581.916 -8845.602 -10109.27 -2640.567 -3904.221 -5167.897 -6431.572 -7695.253 -8958.928 -2753.908 -4017.549 -2867.235

1

3Fe(CO) 3

4

Fe(CO)3 Fe(CO)2 3 FeCO 5Fe 7 Fe2 11Fe 3 15Fe 4 17 Fe5 21Fe 6 23 Fe7 25Fe 8 7 Fe2CO 11Fe CO 3 15 Fe4CO 17Fe CO 5 21Fe CO 6 23Fe CO 7 7 Fe2(CO)2 11Fe (CO) 3 2 5Fe (CO) 2 3 3

a Reference 20 by Chase. b Determined from the measured dissociation energies reported in ref 9. c Reference 49 by Burcat. d Determined by the measured fragmentation energies reported in ref 31. e Note: the integers ahead of the species names are the ground-state spin multiplicities determined in this study.

vibrational partition functions, making use of the conventional rigid-rotor, harmonic oscillator approximation (RRHO).30 Although the simple RRHO approximation was valid for most molecules examined in this work, Fe2(CO)3 was an exception because of the presence of a hindered internal rotation about the Fe-Fe bond (see Figure 1). The thermochemistry of this molecule was obtained by the method outlined by Sumathi et al.30 The hindered rotor potential was obtained by scanning the dihedral angle (φ) about the Fe-Fe bond for 360° in increments of 30°, allowing for the relaxation of all other degrees of freedom. This potential energy scan was performed at the B3PW91/6-311+G(d) level of theory, and the corresponding values of the potential, V(φ), were fit to a fifth-order Fourier series of the following form, where a0, am, and bm were adjustable parameters:

V(φ) )

a0 2

5

+

∑ am cos(mφ) + bm sin(mφ) m)1

(1)

The one-dimensional Schro¨dinger equation was solved numerically using the Fourier series for the potential and a reduced moment of inertia based upon the equilibrium geometry of the molecule. The equation was solved with an increasing number of basis functions (sine and cosine) until the partition function of the hindered rotor (calculated by direct counting of the energy levels) converged. The thermodynamic quantities of interest were estimated by a combination of the standard RRHO approximation methodology (translational, rotational, and vibrational contributions with the vibration corresponding to the hindered rotor removed) and the contributions from the hindered rotor as described here. The contributions of the hindered rotor to the heat capacity and entropy were calculated using ensemble average energies, 〈E 〉 and 〈E2〉, as appropriate. For Fe(CO)5, the computed entropy, 106.3 cal/mol/K, agrees with the reported value,20 105.0 ( 3.0 cal/mol/K. The heats of formation for Fe and Fe(CO)5 were taken from experimental data.20 The previous work showed that the fragmentation energy

is not a smooth function of cluster size, with notable breaks at Fe5 and Fe8.13 The DFT calculations could not confirm the first break but predicted the smaller fragmentation energy for Fe8, which agrees with the measurement. On the basis of this comparison, the heats of formation of Fen, n ) 2-4, were calculated using the measured fragmentation energies,31 while the DFT-estimated heats of formation were used for Fen, n ) 5-8. The heats of formation for the simple iron carbonyls, Fe(CO)n, n ) 1-4, were calculated using the measured dissociation energies9 and the literature heats of formation for Fe and CO.20 For Fe(CO)2, the measured dissociation energy is 36.7 ( 2.5 kcal/mol. In this study, we took its dissociation energy near the lower limit of the measured error bar in order to obtain a heat of formation value (5.16 kcal/mol) that agrees with a recent DFT calculation (5.75 kcal/mol).22 The heats of formation of other intermediates were calculated from their DFT atomization energies. After the thermochemical properties were computed as functions of temperature, they were converted into a Chemkin32 format with the help of a computational code.33 2.3. Simultaneous Particle and Molecule Modeling. Gasphase synthesis of iron nanoparticles is a process with gas-phase reactions, particle nucleation, surface chemistry, and coagulation occurring in concert. To provide a detailed description of the chemistry occurring simultaneously in both the gaseous and condensed phases, the kinetic model should be able to describe not only the chemical pathways among small molecules, but also the coagulation of, and surface reactions on, condensed particles. An inaccurate description of the interactions between the two phases can be a source of error when predicting nanoparticle formation in combustion.34 An approach called simultaneous particle and molecule modeling (SPAMM) developed by Pope and Howard was employed here.27 As Pope and Howard showed, this method successfully predicts the formation of carbonaceous particles (or soot) under different combustion conditions.35,36 In this approach, sectional equations are applied for species larger than a certain mass. The surface

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Wen et al.

TABLE 2: Reaction Mechanism for Iron Nanoparticle Formation Behind Shock Wavesa k ) A × TB × exp(-EA/RT)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

LJ parameters

EA(cal/mol)

/k (K)

σ (Å)

thermodecomposition of Fe(CO)5 1.52E + 49 -12.3 1.42E + 46 -11.4 4.67E + 35 -7.6 2.46E + 32 -6.5 5.39E + 12 0.5 5.57E + 12 0.5 7.87E + 12 0.5 8.43E + 38 -8.9 1.50E + 37 -8.2 5.76E + 12 0.5 5.14E + 12 0.5 4.93E + 12 0.5 4.97E + 12 0.5 5.11E + 12 0.5 5.25E + 12 0.5 8.23E + 12 0.5 6.09E + 12 0.5 6.42E + 12 0.5 5.98E + 25 -5.3 5.05E + 25 -5.1 2.82E + 12 0.5 2.84E + 12 0.5 2.41E + 12 0.5 2.05E + 12 0.5 2.27E + 12 0.5 4.67E + 12 0.5 7.78E + 12 0.5 5.60E + 12 0.5 3.79E + 33 -7.1 2.17E + 33 -7.0 5.95E + 30 -5.8 5.91E + 31 -6.1 6.84E + 12 0.5 1.64E + 14 -2.1 6.45E + 14 -2.1 1.64E + 14 -2.1 5.55E + 12 0.5 2.68E + 12 0.5 2.57E + 12 0.5 2.58E + 12 0.5 2.69E + 12 0.5 2.78E + 12 0.5 2.90E + 12 0.5

9486 9243 4419 3976 0 0 0 4564 4559 0 0 0 0 0 0 0 0 0 0 67 0 0 0 0 0 0 0 0 3324 3348 2975 3625 0 630 628 630 7 0 0 0 0 0 0

1000

6.0

20884

6.0

n/a n/a n/a 14040

n/a n/a n/a 6.0

n/a n/a n/a n/a n/a n/a n/a n/a n/a 1763

n/a n/a n/a n/a n/a n/a n/a n/a n/a 8.6

n/a n/a n/a n/a n/a n/a n/a n/a 14946

n/a n/a n/a n/a n/a n/a n/a n/a 6.0

2019

4.3

n/a 18468

n/a 5.4

18468 n/a n/a n/a n/a n/a n/a n/a

5.4 n/a n/a n/a n/a n/a n/a n/a

reactions between CO and iron clusters Fe2 + CO h Fe2CO 5.30E + 25 -4.9 1.2 atm 2.00E + 26 -4.9 5.00E + 32 -7.4 Fe2CO + CO h Fe2(CO)2 1.2 atm 1.69E + 33 -7.4 Fe2CO + Fe h Fe3 + CO 3.07E + 12 0.5 Fe2CO + Fe2 h Fe4 + CO 2.94E + 12 0.5 Fe2CO + Fe3 h Fe5 + CO 1.01E + 13 0.5 1.19E + 13 0.5 Fe2CO + Fe4 h Fe6 + CO Fe2CO + Fe5 h Fe7 + CO 1.24E + 13 0.5 Fe2CO + Fe6 h B(1) + CO 1.33E + 13 0.5 1.42E + 13 0.5 Fe2CO + Fe7 f 0.875B(1) + 0.125B(2) + CO Fe2CO + FeCO h Fe3CO + CO 4.19E + 12 0.5 5.06E +12 0.5 Fe2CO + Fe2CO h Fe4CO + CO Fe3 + CO h Fe3CO 7.95E + 28 -6.3 1.2 atm 3.06E + 29 -6.3 6.41E + 24 -4.8 Fe3CO + CO h Fe3(CO)2 1.2 atm 2.79E + 25 -4.8 Fe3CO + Fe h Fe4 + CO 2.23E + 12 0.5 1.79E + 12 0.5 Fe3CO + Fe2 h Fe5 + CO Fe3CO + Fe3 h Fe6 + CO 1.45E + 12 0.5 Fe3CO + Fe4 h Fe7 + CO 1.22E + 12 0.5 1.21E + 12 0.5 Fe3CO + Fe5 h B(1) + CO Fe3CO + FeCO h Fe4CO + CO 3.52E + 12 0.5 Fe4 + CO h Fe4CO 1.50E + 36 -8.4 1.2 atm 2.61E + 36 -8.3 Fe4CO + Fe h Fe5 + CO 1.89E + 12 0.5 Fe4CO + Fe2 h Fe6 + CO 1.65E + 12 0.5 1.18E + 12 0.5 Fe4CO + Fe3 h Fe7 + CO Fe4CO + Fe4 h B(1) + CO 1.14E + 12 0.5

2127 2137 2543 2625 0 0 0 0 0 0 0 0 0 2553 2557 202 523 0 0 0 0 0 0 3212 3359 0 0 0 0

2019

4.3

1891

5.9

n/a n/a n/a n/a n/a n/a n/a n/a n/a 2019

n/a n/a n/a n/a n/a n/a n/a n/a n/a 4.6

1892

9.0

n/a n/a n/a n/a n/a n/a 2019

n/a n/a n/a n/a n/a n/a 4.7

reactions considered Fe(CO)4 + CO h Fe(CO)5 1.2 atm Fe(CO)3 + CO h Fe(CO)4 1.2 atm Fe(CO)3 + Fe h Fe2(CO)2 + CO Fe(CO)3 + Fe2 h Fe3(CO)2 + CO Fe(CO)3 + FeCO h Fe2(CO)3 + CO Fe(CO)2 + CO h Fe(CO)3 1.2 atm Fe(CO)2 + Fe h Fe2CO + CO Fe(CO)2 + Fe2 h Fe3CO + CO Fe(CO)2 + Fe3 h Fe4CO + CO Fe(CO)2 + Fe4 h Fe5CO + CO Fe(CO)2 + Fe5 h Fe6CO + CO Fe(CO)2 + Fe6 h Fe7CO + CO Fe(CO)2 + FeCO h Fe2(CO)2 + CO Fe(CO)2 + Fe2CO h Fe3(CO)2 + CO Fe(CO)2 + Fe(CO)2 h Fe2(CO)3 + CO Fe2(CO)2 + CO h Fe2(CO)3 1.2 atm Fe2(CO)2 + Fe h Fe3CO + CO Fe2(CO)2 + Fe2 h Fe4CO + CO Fe2(CO)2 + Fe3 h Fe5CO + CO Fe2(CO)2 + Fe4 h Fe6CO + CO Fe2(CO)2 + Fe5 h Fe7CO + CO Fe2(CO)2 + FeCO h Fe3(CO)2 + CO Fe3(CO)2 + Fe h Fe4CO + CO Fe2(CO)3 + Fe h Fe3(CO)2 + CO FeCO + CO h Fe(CO)2 1.2 atm FeCO + Fe h Fe2CO 1.2 atm FeCO + FeCO h Fe2CO + CO Fe + CO h FeCO 1.2 atm Fe + CO h FeCO Fe + FeCO h Fe2 + CO Fe2 + FeCO h Fe3 + CO Fe3 + FeCO h Fe4 + CO Fe4 + FeCO h Fe5 + CO Fe5 + FeCO h Fe6 + CO Fe6 + FeCO h Fe7 + CO Fe7 + FeCO h B01 + CO

A(mole‚cm‚sec‚K)

B

n/a n/a n/a n/a

n/a n/a n/a n/a



TABLE 2 (Continued) k ) A × TB × exp(-EA/RT) reactions considered 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 -237 238 -427

Fe5 + CO h Fe5CO 1.2 atm Fe5CO + Fe h Fe6 + CO Fe5CO + Fe2 h Fe7 + CO Fe5CO + Fe3 h B(1) + CO Fe6 + CO h Fe6CO 1.2 atm Fe6CO + Fe h Fe7 + CO Fe6CO + Fe2 h B(1) + CO Fe7 + CO h Fe7CO 1.2 atm Fe7CO + Fe h B(1) + CO Fe + Fe + M h Fe2 + M Fe + Fe2 h Fe3 1.2 atm Fe + Fe3 h Fe4 1.2 atm Fe + Fe4 h Fe5 1.2 atm Fe + Fe5 h Fe6 1.2 atm Fe + Fe6 h Fe7 1.2 atm Fe + Fe7 h B(1) 1.2 atm Fe2 + Fe2 h Fe4 1.2 atm Fe2 + Fe2 h Fe + Fe3 Fe2 + Fe3 h Fe5 1.2 atm Fe2 + Fe3 h Fe + Fe4 Fe2 + Fe4 h Fe6 1.2 atm Fe2 + Fe4 h Fe3 + Fe3 Fe2 + Fe4 h Fe + Fe5 Fe2 + Fe5 h Fe7 1.2 atm Fe2 + Fe5 h Fe + Fe6 Fe2 + Fe5 h Fe3 + Fe4 Fe2 + Fe6 h B(1) 1.2 atm Fe2 + Fe7 f 0.875B(1) + 0.125B(2) Fe3 + Fe3 h Fe6 1.2 atm Fe3 + Fe3 h Fe + Fe5 Fe3 + Fe4 h Fe7 1.2 atm Fe3 + Fe4 h Fe + Fe6 Fe3 + Fe5 h B(1) 1.2 atm Fe3 + Fe6 f 0.875B(1) + 0.125B(2) Fe3 + Fe7 f 0.750B(1) + 0.250B(2) Fe4 + Fe4 h B(1) 1.2 atm Fe4 + Fe5 f 0.875B(1) + 0.125B(2) Fe4 + Fe6 f 0.750B(1) + 0.250B(2) Fe4 + Fe7 f 0.625B(1) + 0.375B(2) Fe5 + Fe5 f 0.750B(1) + 0.250B(2) Fe5 + Fe6 f 0.625B(1) + 0.375B(2) Fe5 + Fe7 f 0.500B(1) + 0.500B(2) Fe6 + Fe6 f 0.500B(1) + 0.500B(2) Fe6 + Fe7 f 0.375B(1) + 0.625B(2) Fe7 + Fe7 f 0.250B(1) + 0.750B(2) Fe(i) + B(j) f xB(j) + yB(j + 1) i ) 1,2,...,7; j ) 1,2,...,19 B(m) + B(n) f xB(n) + yB(n + 1) m ) 1,2,...,19; n ) 1,2,...,19

A(mole‚cm‚sec‚K)

B

reactions between CO and iron clusters 3.87E + 39 -9.4 5.36E + 39 -9.3 1.97E + 12 0.5 1.48E + 12 0.5 1.04E + 12 0.5 1.45E + 33 -7.2 2.43E + 33 -7.1 1.94E + 12 0.5 1.48E + 12 0.5 5.10E + 21 -4.0 5.78E + 21 -4.0 1.92E + 12 0.5 growth of iron clusters 7.89E + 17 4.60E + 23 2.20E + 24 1.94E + 30 7.25E + 30 1.58E + 38 1.54E + 37 2.81E + 40 4.57E + 37 1.49E + 34 2.71E + 34 1.62E + 37 2.01E + 33 1.09E + 28 1.83E + 27 3.41E + 12 4.71E + 33 1.03E + 33 6.91E + 12 4.72E +37 4.45E + 35 4.65E + 20 1.00E + 13 2.43E + 27 1.69E + 25 3.45E + 12 7.79E + 25 9.63E + 23 1.22E + 21 1.38E + 13 5.27E + 38 7.97E + 37 6.63E + 15 8.84E + 37 1.88E + 35 8.98E + 14 7.07E + 30 5.05E + 27 1.28E + 13 1.32E + 13 3.40E + 34 2.44E + 31 1.24E + 13 1.27E + 13 1.31E + 13 1.26E + 13 1.28E + 13 1.31E + 13 1.30E + 13 1.32E + 13 1.33E + 13

-0.52 -4.1 -4.2 -6.2 -6.2 -8.6 -8.1 -8.8 -7.9 -6.6 -6.7 -7.6 -6.3 -5.0 -4.6 0.5 -7.1 -6.7 0.4 -8.1 -7.3 -1.8 0.3 -4.7 -3.7 0.5 -3.1 -3.3 -2.3 0.5 -8.8 -8.4 -0.7 -8.1 -7.2 -0.4 -5.6 -4.5 0.5 0.5 -6.9 -5.8 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

growth of iron nanoparticles γβi,j

0.5 0.5

γβm,n

LJ parameters EA(cal/mol) 3608 3876 0 0 0 2024 2447 0 0 0 0 0 14.81 1717 1796 2898 3059 4846 4927 5965 5522 5008 5094 5725 4888 2814 2808 0 3759 3810 97 5132 4893 7751 145 3733 3043 59 15404 2804 2129 0 4749 4794 780 5540 5084 653 4340 3657 0 0 5123 4471 0 0 0 0 0 0 0 0 0

/k (K)

σ (Å)

2019

4.7

n/a n/a n/a 2019

n/a n/a n/a 4.9

n/a n/a n/a n/a n/a

n/a n/a n/a n/a n/a

n/a 20308

n/a 6.2

20424

4.6

21584

4.6

13693

4.9

13693

5.0

13693

5.0

20424

4.6

n/a 21584

n/a 4.6

n/a 13693

n/a 4.9

n/a n/a 13693

n/a n/a 5.0

n/a n/a 13693

n/a n/a 5.0

n/a 13693

n/a 4.9

n/a 13693

n/a 5.0

n/a 13693

n/a 5.0

13693

5.0

n/a n/a n/a n/a n/a n/a n/a n/a n/a

n/a n/a n/a n/a n/a n/a n/a n/a n/a

0

n/a

n/a

0

n/a

n/a

a Note: pressure-dependent rates were calculated for 0.3 and 1.2 atm; the A factor for R69 has units of cm6/mol2‚sec‚KB; The coefficients, x and y, were calculated using eqs 2 and 3, respectively; the determination of γ and βi,j are shown in the text.


Wen et al. around 8 nm37). The collision coefficient for bins of size i and j, βi,j, in cm3 mol-1 s-1 K-0.5 in Table 2, is calculated by

βi,j ) di,j2

( ) 8πkB µ

1/2

NA

(4)

where kB is the Boltzmann constant, NA is Avogadro’s number, and di,j and µ are the collision diameter and the reduced mass, respectively Figure 2. Boundary of the high-pressure-limit region calculated for Fe + Fe7 h Fe8. The curve corresponds to the assumed average energy transfer values: ∆E ) 400 cm-1. The horizontal lines represent the pressure and temperatures used in three experiments: the HiPco,2 shock tube,5 and C2H2 flame.1

reaction and particle-particle coagulation processes are converted into equivalent elementary-step reactions between the sections (called “bins”). These reactions can be incorporated into a gas-phase reaction mechanism and allow for the simultaneous modeling of the gas-phase chemistry and the nanoparticle formation processes. In this study, the entire size range of condensed iron particles is divided into 20 bins (represented by B(1), B(2), ..., B(20) in the mechanism). The first sectional bin (denoted ′B(1)′), which is equivalent to the size of the largest cluster whose reactions are treated in detail, is formed through the reactions between the explicitly treated molecules. Each subsequent bin is increased in size by a factor of 2. The number of Fe atoms in the bin B(i) is 8 × 2i-1. The reactions between bins are expressed in such a way as to guarantee conservation of mass. When a new particle forms with a size located between two defined bins, its mass will be proportionally divided in two neighboring bins according to the ratio of their sizes. The following equations show how x and y in Table 2 were defined for the reaction

B(m) + B(n) f xB(n) + yB(n + 1) x)1y)

2m-1 2n-1

2m-1 2n-1

(2)

(3)

where m e n. The largest size of an iron nanoparticle in the present kinetic model is Fe4194304 (B(20)), which has a diameter of around 80 nm. Giesen et al.6 have estimated the critical cluster size needed to nucleate the formation of Fe nanoparticles. They found that the size changes in a range from Fe to Fe10 when the temperature is increased. It has been shown (in Figure 2 of the previous work13) that for iron clusters larger than Fe8 the fragmentation energies D(Fen-Fe) start to increase and finally plateau to a large number. In other words, the fragmentation process (e.g., the evaporation of iron atoms from clusters) becomes much more difficult for larger iron clusters. Our SPAMM model assumes that the second bin, B(2), is already above the critical cluster size for the conditions studied (i.e., we only consider the formation, rather than dissociation, for nanoparticles larger than Fe8). These bins grow through the particle-particle coagulation and the surface condensation of gaseous iron atoms. Both reactions were treated as being irreversible, and the corresponding rates were determined by their collision coefficients and a sticking coefficient γ (γ ) 0.3 was assumed in this study and is consistent with the value measured for soot particles with a diameter of

di,j )

di + dj 2

(5)

where di is the hard-sphere diameter of the ith nanoparticle.

µ)

mi mj m i + mj

(6)

where mi is the mass of the ith nanoparticle. The effect of the formation of iron nanoparticles on the temperature field in shock tubes is neglected because their concentrations are relatively small. 2.4. High-Pressure-Limit Reaction Rates. Experimental studies of the kinetics of Fe(CO)5 decomposition is limited in the literature. Seder et al. measured the rate constants for reactions of Fe(CO)2-4 with CO using the transient infrared absorption spectra generated via excimer laser photolysis of Fe(CO)5.38 They found that near room temperature the rate constants for reactions of Fe(CO)4, Fe(CO)3, and Fe(CO)2 with CO are (3.5 ( 0.9) × 1010, (1.3 ( 0.2) × 1013, and (1.8 ( 0.3) × 1013 cm3 mol-1 s-1, respectively. The large difference in the magnitude of the rate constants is due to the spin flip required to transition from Fe(CO)5 to Fe(CO)4. Most recently, Tsuchiya and Roos39 studied this spin-forbidden reaction using multiconfigurational quantum chemical methods (CASSCF/CASPT2). They found that, at this level of theory, the reaction path for Fe(CO)4 + CO f Fe(CO)5 has an effective activation energy barrier of 4.8 kcal/mol to reach the crossing between the two potential energy surfaces of different spins. Different levels of theory yield different effective activation energies, so this number is rather uncertain. In this study, we used a preexponential factor of 3.3 × 1012 cm3 mol-1 s-1 with an energy barrier of 2.5 kcal/mol. This expression gives a rate of 5 × 1010 cm3 mol-1 s-1 at room temperature, which agrees with the measured value. Another spin flip is involved in the dissociation of the transient intermediate FeCO (S ) 1) to Fe (S ) 2) and CO. Taking into account the relatively small dissociation energy of FeCO, here we assume that this spin flip is fast enough that it does not impede the kinetics. We used the measured reaction rate constants38 as the high-pressure-limit rates for two other reactions, that is, Fe(CO) 3 + CO f Fe(CO)4 and Fe(CO)2 + CO f Fe(CO)3. Because the dissociation processes of other iron carbonyls and intermediates only involve barrierless or low-barrier bondbreaking reactions that lack rigid transition states, and because the entire surface of an iron cluster is essentially reactive (minimal steric hindrance), the high-pressure-limit rates for bimolecular association reactions were estimated based on the hard-sphere collision rates. The rate constant was calculated as a product of the collision rate and a spin statistical factor. The spin statistical factor accounts for the fact that only some of the spin states of the reactants correlate with the spin states of stable products. We used the following equation to calculate the spin statistical factor:



spin statistical factor ) product of spin multiplicities of stable products product of spin multiplicities of reactants 2.5. Pressure-Dependent Reaction Rates. As will be shown in the results, the reactions among smaller iron clusters and iron carbonyls are located in the fall-off regime, making the calculation of pressure-dependent rates necessary. Depending on the nature of individual reactions, the pressure-dependent rate constants were calculated in several ways in this study. 2.5.1. Fe + Fe + M f Fe2 + M. The previous studies of iron particle formation5 have suggested that the following reaction could be rate-determining: kf,kb

Fe + Fe + M 798 Fe2 + M

(R69)

The Arrhenius expressions of the forward and reverse rate constants have been proposed as

kf ) 1.0 × 10 cm mol 19

6

-2 -1

s

1019.63(0.40 exp(-17800 ( 700 K/T) cm3 mol -1s-1 (8) where T is the temperature in Kelvin. The above rate constants were determined empirically without taking potential fall-off effects into consideration, and they are unexpectedly large. To more accurately calculate the rate constant for this reaction, a simple third-body theory was implemented. The overall reaction R69 is the result of three elementary steps: (a) two Fe atoms collide to form an activated intermediate Fe*2 (with rate constant k1); the activated intermediate either (b) dissociates back to the reactants (with rate constant k2), or (c) it collides with the bath gas, M, to form a stable product Fe2 (with rate constant ks). k1,k2

Fe + Fe 798 Fe*2

(R69a,b)

ks

Fe*2 + M 98 Fe2 + M

(R69c)

The two forward rate constants, k1 and ks, are assumed to be barrierless and are calculated using classical collision theory

x2 Eµ d

ks (T) )

x

8π

2

(9)

kBT 2 d µ

(10)

where the E in eq 9 is the relative translational energy, and d and µ are calculated using eqs 5 and 6, respectively. The dissociation rate constant, k2, is more complicated to compute. This first-order rate constant is the inverse of τ(E), the period during an Fe + Fe collision for which the Fe-Fe bond distance is small enough that a Franck-Condon collision could stabilize the complex. To estimate τ(E), the potential energy surface of Fe2 is assumed to be a Morse potential, with the zero of energy chosen so V(r) f 0 for well-separated Fe atoms. Substituting this potential into the total energy equation and solving for the velocity as a function of the position and energy yields

V(r,E) )

(m2 [E + D - D (1 - e e

e

τ(E) ) 2

-β(r-r0) 2

)]

)

(11)

∫rr

+

-

1 dr V(r,E)

(12)

The outer point r+ is the largest distance r(Fe-Fe) for which a collision is expected to stabilize Fe2* to Fe2. Here the limits were chosen to guarantee that the Fe2 formed should have a potential energy at least kBT below the dissociation limit, that is, Vstabilized e -kBT. Consequently, the limits of integration are given by

r - ) r0 -

1 ln(1 + x1 - kBT/De) β

(13a)

r + ) r0 -

1 ln(1 - x1 - kBT/De) β

(13b)

(7)

kb ) 1.0 ×

k1 (E) ) 2π

where V is the velocity, E is the total energy, r0, β, and De are the Morse potential parameters, and r is the distance. The value for the Morse potential parameters (calculated from a potential energy scan using 0.05 Å bond-length increments and B3PW91/ 6-311+G(d)) are r0 ) 2.02 Å, β ) 1.39 Å-1, and De ) 5.38 eV. The time during which the Fe-Fe distance is less than r+ is

To derive an effective three-body rate constant for the production of Fe2, the pseudo steady-state hypothesis is assumed for the concentration of the excited intermediate, and appropriately Boltzmann-averaged

∫0∞ keff (T) )

k1(E)F(E) k2(E) ks(T)[M]

exp

+1

( )

-E dE kBT

( )

dE ∫0∞ F(E) exp k-E BT

(14)

where F(E) ∼ E -1/2 is the density of state for 1D translational motion. Note that this effective rate constant provides an upper bound on the rate of the reaction R69 because it assumes that every collision (R69c) stabilizes Fe2. As shown below, even with this overestimate, the reaction R69 is so slow that it hardly plays a role under the experimental conditions. Equation 14 was solved for a range of temperatures and pressures, and the results were fitted to a modified Arrhenius form for termolecular reactions. Reaction R69 is in the low-pressure limit (i.e., a simple termolecular reaction) for P < 10 atm. 2.5.2. Estimating k(T,P) for Larger Species. To predict pressure-dependent rates for the reactions involving larger species, we used a modified version of CHEMDIS,26 which employs a three-frequency quantum Rice-Ramsperger-Kassel (QRRK) calculation for microcanonical rate constants and the modified strong-collision approach for treating collisional activation and deactivation. The Lennard-Jones parameters used in our calculation are listed in Table 2. The errors introduced by the QRRK approximation have been found to be acceptable for most gas-phase reactions, typically less than the uncertainties in determining the equilibrium constants.40 In addition to a reaction network that describes the formation and growth of pure Fe clusters (Fen, n ) 1-7), pathways describing the reactions among intermediates, that is, Fem(CO)n, were also included by adding chemically activated Fe-CO exchange reactions. The pressure-dependent rates were calculated for 0.3 and 1.2 atm of argon bath gas and in a temperature range from 200 to 3000 K. They were then fit to Arrhenius expressions with fitting errors less than 20% (e.g., 16% for FeCO formation at 1400 K). This fitting procedure is


Wen et al.

the source of the high pre-exponential factors and negative temperature exponents in the kinetic model shown in Table 2; the high-pressure-limit Arrhenius factors all lie in the normal ranges. A constant-volume, homogeneous reactor model included in Chemkin32 was then used to compute the species concentrations behind shock waves under experimentally measured temperatures and pressures. 3. Results and Discussion 3.1. Ground-State Geometries and Thermochemical Properties. Total energies and zero-point vibrational energies (ZPE) were computed for a variety of iron clusters, carbonyls, and intermediates at the B3PW91/6-311+G(d) level. The Gaussian calculations show that while iron carbonyls usually have open structures, iron clusters are more stable in a caged arrangement. The bond lengths of FeCO calculated in this study (RFe-C ) 1.75 Å and RC-O ) 1.18 Å) are in good agreement with the measured data (RFe-C ) 1.73 Å and RC-O ) 1.16 Å).41 The ground-state geometries of other carbonyls, Fe(CO)n, n ) 2-4, are in agreement with a previous DFT study.22 The structures and vibrational frequencies of iron clusters, Fen, n ) 2-6, with ground-state spin multiplicities of 7, 11, 15, 17, and 21, respectively, are in agreement with those found in another DFT study.18 The spin multiplicities found for the ground states of Fe7 and Fe8 in this study agree with the values computed previously.42 This study calculated the ground-state properties of several intermediates (Fen(CO)m) with moderate molecular sizes as well; these species have not been studied previously. For FenCO, n ) 2-6, the spin multiplicities of the groundstate structures are similar to the values of the corresponding iron clusters, Fen. These multiplicities are in reasonable agreement with the previous work.43 In some cases, two spin-states are nearly degenerate and the B3PW91 functional gives a different ordering from the BPW91 functional. On the basis of the prior work,23 we expect the B3PW91 enthalpies to be more accurate. The computed vibrational frequencies and geometries are given in the Supporting Information. The ground-state structures and spin multiplicities for intermediates are shown in Figure 1. Table 1 lists the entropies, heats of formation, heat capacities, zero-point energies, and spin multiplicities of all species involved in the gas-phase reactions. The heats of formation were taken from experimental data where available. The comparison of the literature20 heat capacity of Fe(CO)5 with that calculated by DFT suggests uncertainties of several cal mol-1 K-1. 3.2. Cluster Size and the Fall-off Regime. Following the thermal decomposition of Fe(CO)5, gaseous iron atoms add to dimers and larger clusters through bimolecular association reactions. To investigate the maximum cluster size to which these gas-phase reactions are still in the fall-off regime, we used a recently proposed relationship between temperature and the high-pressure limit.44 The high-pressure-limit boundary (which separates the regions where the high-pressure-limit approximation is valid from regions where fall-off and chemical-activation dominate) was calculated for the formation processes of gaseous iron clusters. Figure 2 shows the computed boundary corresponding to the formation pathways of gaseous Fe8 clusters. The boundary was calculated by assuming the following average absolute energy transfer per collision: ∆E ) 400 cm-1. Reasonable variations in ∆E do not significantly change the range of conditions where the high-pressure-limit approximation is valid. Figure 2 also presents the pressure ranges of three typical iron nanoparticle formation processes, that is, in the shock wave case of this study, HiPco,2 and flame synthesis.1

Figure 3. Prediction of the Fe concentration in a mixture of 5 ppm Fe(CO)5 in Ar at 705 K and 0.35 atm. The empty circles refer to the measurement,45 the solid line refers to the DFT-based model in this study, and the dash-dot line shows the linear formation rate of Fe atoms based on the measurement.

Figure 2 suggests that for the largest cluster considered in detail in the present work, Fe8, it may be safe to assume the highpressure limit for the HiPco process, but this approximation is probably not accurate for shock tube or low-pressure flame experiments. However, it is reasonable to use the high-pressurelimit approximation for the larger bin species (B(2), B(3), etc.) that correspond to Fe16 and larger. This is due in part to the very large density of states of these large clusters and because the binding energies are expected to be larger. 3.3. Analysis of the Kinetic Model and Its Validation. The reaction mechanism for iron nanoparticle formation during the thermal decomposition of Fe(CO)5 is described in Table 2. The reaction mechanism and thermodynamic data are available in the Supporting Information in Chemkin format. In Table 2, the reaction rates calculated for 0.3 and 1.2 atm are shown separately. The following sections highlight the important technical issues and findings from this study. 3.3.1. Thermal Decomposition Pathway of Fe(CO)5. The description of thermal decomposition of Fe(CO)5 is the first part of the kinetic model. With the help of Gaussian-calculated thermochemical data and assuming there are no barriers to the reverse reactions (except for the spin-forbidden reaction, Fe(CO)5 h Fe(CO)4 + CO, which has EA ) 2.5 kcal/mol), this study calculated the dissociation reaction rate for each iron carbonyl, Fe(CO)n, n ) 1-5, using the QRRK method. When compared to the other iron carbonyls in this study, Fe(CO)2 is a relatively stable molecule. Therefore, a set of reaction pathways were added to address the role of Fe(CO)2 and its adducts in the formation of iron clusters. These pathways will be presented separately in the next section. To validate the decomposition pathways of Fe(CO)5, the induction period and global formation rate (defined as the slope of the number density curve) in the iron atom formation process were studied in a dilute Fe(CO)5 mixture (5 ppm Fe(CO)5 in argon). The temperature (705 K) and pressure (0.35 atm) are the same as those reported in the literature,45 where the Fe atom concentration was measured by atomic resonance absorption. Figure 3 shows the comparison between the calculated Fe number concentration and the measured data. Without any adjustment, the model predicts the Fe atom formation rate very well. The rise time and absolute concentration of Fe atoms are predicted within an error of 50%. 3.3.2. Pathways through Fe(CO)2. Under certain reaction conditions, the Fe(CO)n intermediates live long enough to participate in bimolecular reactions. Because of its thermochemistry, Fe(CO)2 is the longest-lived among the Fe(CO)n intermediates, making its bimolecular reactions the most important. Some of the key reactions involving Fe(CO)2 are listed here:



Fe(CO)2 + Fe h Fe2CO + CO Fe(CO)2 + Fen h Fe2CO + CO

(R07)

n ) 2-6 (R08-R12)

Fe(CO)2 + FenCO h Fen+1(CO)2 + CO

n ) 1,2 (R13-R14)

Fe(CO)2 + Fe(CO)2 h Fe2(CO)3 + CO

(R15)

The major reactions among the intermediates produced by the aforementioned reactions are the following:

Fe2(CO)2 + Fen h Fen+2CO + CO

n)1-5 (R17-R21)

Fe3(CO)2 + Fe h Fe4CO + CO

(R23)

Fe2(CO)3 + Fe h Fe3(CO)2 + CO

(R24)

Fe2CO h Fe2 + CO Fe2CO + Fen h Fen+2 + CO

(36) n)1-7 (R38-R44)

The predictions of species concentrations show that at lower temperatures (from 400 to 800 K), Fe(CO)2 survives for a relatively long time while FeCO decomposes almost immediately into Fe and CO. The Fe(CO)2 forms Fe2CO and Fe2(CO)3 when it reacts with Fe (R07) and through its self-reaction (R15). The secondary reactions include Fe(CO)2 reacting with other iron clusters and iron carbonyls (R08-R14). The significant concentration of Fe2CO subsequently results in the formation of larger iron clusters, that is, Fe2-7, through reactions R36 and R38-R44. The formation of Fe2(CO)3 contributes to the formation of larger iron clusters as well. Because Fe(CO)2 is the precursor of both Fe and larger iron clusters, the formation of Fe atoms is a slower process in the earlier stages at low temperatures. The reaction kinetics are also slowed by high CO concentrations, because the reaction, Fe(CO)5 h Fe(CO)4 + CO, is significantly reversible. The concentration of Fe atoms declines during the latter stages of the process, when they are consumed by the reactions with larger iron clusters. 3.3.3. OVerView of Nanoparticle Formation Kinetics. The nucleation and growth of iron clusters follow the decomposition of Fe(CO)5. At high temperatures, this decomposition runs rather rapidly to form a supersaturated concentration of Fe atoms, but the atoms do not have any fast pathway to condense into nanoparticles. However, once large nanoparticles are formed, they will grow rapidly by adsorbing the free Fe atoms. These large nanoparticles have such high heat capacities that they can absorb the chemical bond energy; therefore, their rate constants are in the high-pressure limit. As will be shown later, the size distribution of the iron nanoparticles formed depends on the competition between nucleation and particle growth. The rate of the nucleation process is controlled by the chemistry of species containing three or fewer Fe atoms. Under shock tube or combustion-relevant conditions, the smallest pure Fe species (Fe1-3) agglomerate very slowly through processes such as Fe + Fe + M h Fe2 + M and Fe + Fe2 h Fe3 due to fall-off effects and/or fast reverse-reactions. This is contrary to a previously proposed model46 that reproduced the experimental trend of Fe formation at lower temperatures by overestimating the significance of the pathway from Fe atoms to the Fe2 dimer. The current model suggests that, at lower temperatures near 800 K, the decomposition of Fe(CO)5 is slow and the Fe(CO)n

Figure 4. Validation of model-predicted Fe concentrations for 30 ppm Fe(CO)5 in Ar at a pressure of 0.3 atm and various temperatures. (a) T ) 740 K; (b) T ) 815 K; (c) T ) 900 K. The line with symbols refers to the measurement,5,46 and the lines refer to the DFT-based model in this study: solid line for the pure a priori predictions and dashed line for the adjusted model.

intermediates live long enough to undergo bimolecular reactions, which efficiently seed nucleation. Therefore, the most important chemistry needed to reproduce the trend of Fe formation is the decomposition, self-reaction, and reaction with Fe of Fe(CO)2 through R25, R15, and R07, respectively. The nucleation rate of iron nanoparticles is largely controlled by the chemistry of these three reactions. At higher temperatures around 900 K, the unimolecular decomposition reactions of Fe(CO)n are so fast that they cannot react bimolecularly, meaning that Fe(CO)5 decomposes into Fe atoms and CO very rapidly, and the formation of intermediates, Fem(CO)n, may be neglected. In other words, following the fast decomposition of Fe(CO)5 and little formation of intermediates, the consumption of Fe atoms arises mainly from the formation of larger clusters through the association of Fe atoms to form Fe2, and the condensation of Fe atoms on larger clusters. As will be shown later in this study, the total number density of iron nanoparticles formed at the largest residence time is smaller at high temperatures. This may be explained by the fact that when the Fe(CO)2 pathways are negligible at high temperatures, the agglomeration reactions of Fe, Fe2, and Fe3 are very slow due to fall-off effects. Also, some of the reaction steps have significant reverse rates at high temperatures. Hence, the nucleation rate of iron nanoparticles is very small at high temperatures because there is no fast consumption pathway for the Fe atoms formed by the fast decomposition of Fe(CO)5. 3.3.4. Kinetic Model Validation. Figure 4 shows the validation of the proposed kinetic model based on Fe atom concentrations measured in shock tube studies.5,46 The model simulates a 30 ppm Fe(CO)5/argon mixture at various temperatures and 0.3 atm. The measurements and predictions show that at lower temperatures (740 and 815 K in the figure) the concentration of Fe atoms increases slowly to a maximum and then drops when the residence time increases. However, at higher temperatures (900 K), the concentration of Fe atoms quickly rises to its maximum and remains nearly constant. The uncertainties in the estimated rate constants are at least a factor of 3, leading to similar uncertainties in the model predictions. In Figure 4a and


Figure 5. Sensitivity analysis for reactions affecting the concentration of Fe atoms for 30 ppm Fe(CO)5 in Ar at 740 K and 0.3 atm. The expressions for bins and their coefficients are shown in Table 2.

Figure 6. Sensitivity analysis for reactions affecting the concentration of Fe atoms for 30 ppm Fe(CO)5 in Ar at 0.3 atm and 815 K.

b, we show the pure predictions (solid lines) and a fit where two of the rate constants for the most sensitive reactions (R07 and R25) were adjusted by a factor of 3 (dashed line). The quality of the fit illustrates that the present physically realistic model is consistent with the experimental data within the modeling uncertainties. (Note that there is not sufficient information content in the data to unambiguously determine the many parameters in the model experimentally.) 3.3.5. SensitiVity Analysis. Figures 5 and 6 show the sensitivity analysis of formation pathways of Fe atoms at two different temperatures (740 and 815 K, respectively). Only the reactions with large effects are shown. The effect of the Fe atom association reaction, R69, does not show up in either figure because the reaction rate of R69 plays a minor role during the formation and consumption of Fe atoms. Instead, the decomposition and self-reaction of Fe(CO)2 (R25 and R15) and the Fe2CO pathways (including its formation R07) significantly affect Fe formation and consumption, especially at short residence times. In Figure 5a, the influence of the three largest iron carbonyls, Fe(CO)3-5, are significant at very small residence times (less than 8 × 10-5 s). However, the decomposition rate

Wen et al. of Fe(CO)5 through the spin-forbidden reaction R01 determines the Fe atom rise-time, as shown in Figure 4a where an increase in the reaction rate of R01 leads to the earlier formation of Fe atoms. Figure 5 also shows that the yield of Fe atoms is quite sensitive to both the dissociation and the bimolecular reactions of Fe(CO)2. The rate of reaction R25 significantly affects the peak of Fe atoms at 740 K. Further analysis shows that the slow formation of Fe atoms at smaller residence times is attributed to the reactions R07 and R15, through which a larger amount of iron-containing intermediates form, causing a reduction in the production rates of Fe atoms. When the residence time increases, Fe2CO pathways (i.e., R36 in Figure 5a, and R40 and R41 in Figure 5b) become important. These reactions further contribute to the formation of larger iron clusters by bypassing the formation of Fe atoms. Figure 5b shows that at larger residence times (greater than 1.6 × 10-4 s) the consumption of Fe atoms results mainly from the surface condensation of Fe on larger iron nanoparticles (e.g, through R105 and R106). The formation of the smallest iron nanoparticles, B(1), does not consume Fe atoms. Instead, the reverse of reaction R75 increases the concentration of Fe atoms through the fragmentation of nanoparticles. Figure 6 shows that at a higher temperature (815 K) and short residence times the major reactions contributing to the formation and consumption of Fe atoms are the decomposition, selfreaction, and reaction with Fe of Fe(CO)2 through R25, R15, and R07. Reactions involving other intermediates become insignificant due to the faster decomposition of Fe(CO)5 and less formation of FenCO. At larger residence times, the surface condensation of Fe atoms on iron nanoparticles contributes to the consumption of Fe atoms (e.g., through R105). Note that the dissociation of Fe2CO (i.e., the backward reaction of R36) contributes to the increase in the concentration of Fe2 but to the decrease in the concentration of Fe atoms. On the basis of the aforementioned sensitivity analysis, we conclude that in a temperature range near 800 K, in the early stages of iron nanoparticle formation when the dissociation of Fe(CO)5 and the formation of Fe atoms occur, the reactions involving Fe(CO)2 and Fe2CO are critical for understanding the system dynamics. These findings are supported by the sensitivity analysis and DFT calculations but are contradictory to the previous study that suggested that the association of Fe atoms is critical during iron cluster formation. 3.3.6. Modeling of Condensed Iron Nanoparticle Growth. Through the aforementioned SPAMM approach, the wellvalidated kinetics of the thermodecomposition of Fe(CO)5 and the formation/consumption of gaseous Fe clusters are coupled with the mechanism that describes the nucleation, coagulation, and surface growth of condensed iron nanoparticles. As shown earlier in Figures 5 and 6, the reactions of large condensed iron nanoparticles have an influence on the Fe concentration at larger residence times. This agrees with experimental findings. In this study, a particle diameter measurement conducted in a 5000 ppm Fe(CO)5/argon mixture47 was used to validate the reactions among condensed nanoparticles. In that experimental study, the particle formation was monitored by cw-laser extinction and laser-induced incandescence (LII) at 1100 K and 1.2 atm. The prediction of iron nanoparticle formation was conducted using the new kinetic model with parameter values applicable at 1.2 atm. As before, a temperature-constrained homogeneous reactor model was used. Figure 7 shows the comparison of averaged particle diameters. In this study, the averaged diameters were represented by their volume equivalent values. The quoted error bars for experimental data are (15%. The assumption that 30%


Figure 7. Validation of model-predicted particle diameters formed in a 5000 ppm Fe(CO)5 in Ar mixture at 1100 K and 1.2 atm. The symbols refer to the measurement47 with the error bar of (15%; and the solid line refers to the model prediction in this study.

Figure 8. Total number density of iron nanoparticles formed in the mixture of 30 ppm Fe(CO)5 in Ar at 0.3 atm as predicted by the current model: solid line, T ) 745 K; dashed line, T ) 815 K; dotted line, T ) 900 K.

of the collisions between condensed iron clusters with more than eight Fe atoms lead to reactive coagulations yields the curve shown in Figure 7. The model predictions are within the error bars of experimental data. In this case, where a high concentration of Fe(CO)5 was used, larger nanoparticles (around 9 nm) form at a smaller residence time (1 millisecond). Because the diameter of carbon nanotubes (1 to 2 nm) produced in aerosol reactors directly correlates to the size of iron nanoparticles,4 smaller iron nanoparticles are preferred and a low concentration of Fe(CO)5 is often used. It is also known that a smaller total particle number density will result in a reduced particle-particle coagulation rate. To show the temperature dependence of nanoparticle formation under a condition that closely mimics a carbon nanotube reactor, the kinetic model was implemented at three different temperatures for a 30 ppm Fe(CO)5/argon mixture at 0.3 atm. The total number density of iron particles is plotted in Figure 8. It shows that the mixture at 740 K produces the largest number of iron nanoparticles. This is mainly attributed to the Fe(CO)2 pathways developed in this study. At a specific residence time, the iron nanoparticles formed at 740 K will be smaller than ones formed at 900 K. Therefore, for this mixture of 30 ppm Fe(CO)5 in argon at 0.3 atm, the reactor operated at 740 K would be the better choice to produce the small iron nanoparticles needed for the synthesis of carbon nanotubes. 4. Summary A detailed kinetic model has been developed to predict iron nanoparticle formation behind shock waves based on ab initio thermochemistry estimates for Fem(CO)n and other intermediates. In contrast to prior models for this process, all the parameters have physically reasonable values that satisfy thermodynamic constraints, and the effects of fall-off and chemically activated

J. Phys. Chem. C, Vol. 111, No. 15, 2007 5687 reactions have been included. The structure and thermodynamic data for iron carbonyls, iron clusters, and Fem(CO)n intermediates were calculated using density functional theory. In the fall-off regime, the pressure-dependent reaction rates were calculated using the third-body theory and the QRRK method. The size distribution of nanoparticles is predicted to depend very strongly on both temperature and the concentrations of intermediates. It was found that the major reason for the decrease in Fe atom concentrations at lower temperatures is the stability of the intermediates Fe(CO)2 and Fe2CO. These intermediates live long enough to react bimolecularly via pathways (R07, R15, and R25 in text) through which Fe2 and other larger iron clusters form. This can result in relatively high nucleation rates for the nanoparticles. At high temperatures, the iron pentacarbonyl decomposes rapidly into Fe atoms and CO through a sequence of fast unimolecular reactions; those decomposition reactions are so fast that the bimolecular reactions cannot compete. Once the system has evolved completely to Fe atoms at high temperatures, there are no fast reaction paths leading to iron nanoparticles because all of the bimolecular reactions of atoms and diatoms will be significantly in the fall-off regime under these conditions. In this low-nucleation-rate limit, only a small number of relatively large iron nanoparticles will be formed. The kinetic model was validated based on experimental measurements of the concentrations of Fe atoms at various temperatures and against measured particle diameters. It is expected that this model will be useful in predicting and understanding the formation of catalytic iron nanoparticles during the synthesis of carbon nanotubes in aerosol reactors. Acknowledgment. This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a PDF scholarship for J.Z.W. C.F.G. gratefully acknowledges support from a U.S. DOD fellowship, and R.W.A. acknowledges support from an U.S. NSF fellowship. We thank Dr. H. Richter, Dr. J. Yu, Dr. D. M. Matheu, and Dr. G. Beran for helpful discussions. Supporting Information Available: The exponents and contraction coefficients of Gaussian version 6-311+G(d) basis for Fe, C, and O; the DFT results including the multiplicities and Cartesian coordinates of ground states, the calculated vibrational frequencies, and zero-point energies; the thermodynamic data in CHEMKIN format and the mechanism for iron nanoparticle formation. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Height, M. J.; Howard, J. B.; Tester, J. W.; Sande, J. B. V. Carbon 2004, 42, 2295. (2) Nikolaev, P.; Bronikowski, M. J.; Bradley, R. K.; Rohmund, F.; Colbert, D. T.; Smith, K. A.; Smalley, R. E. Chem. Phys. Lett. 1999, 313, 91. (3) Moisala, A.; Nasibulin, A. G.; Brown, D. P.; Jiang, H.; Khriachtchev, L.; Kauppinen, E. I. Chem. Eng. Sci. 2006, 61, 4393. (4) Nasibulin, A. G.; Pikhitsa, P. V.; Jiang, H.; Kauppinen, E. I. Carbon 2005, 43, 2251. (5) Giesen, A.; Herzler, J.; Roth, P. J. Phys. Chem. A 2003, 107, 5202. (6) Giesen, B.; Orthner, H. R.; Kowalik, A.; Roth, P. Chem. Eng. Sci. 2004, 59, 2201. (7) Janzen, C.; Roth, P. Combust. Flame 2001, 125, 1150. (8) Krestinin, A. V.; Smirnov, V. N.; Zaslonko, I. S. SoV. J. Chem. Phys. 1991, 8, 689. (9) Sunderlin, L. S.; Wang, D. N.; Squires, R. R. J. Am. Chem. Soc. 1992, 114, 2788. (10) Rumminger, M. D.; Reinelt, D.; Babushok, V.; Linteris, G. T. Combust. Flame 1999, 116, 207. (11) Dateo, C. E.; Gokcen, T.; Meyyappan, M. J. Nanosci. Nanotechnol. 2002, 2, 523.

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