Monomer Formation Model versus Chain Growth Model of the Fischer

Article pubs.acs.org/JPCC

Monomer Formation Model versus Chain Growth Model of the Fischer−Tropsch Reaction Rutger A. van Santen,*,†,‡ Albert J. Markvoort,†,§ Minhaj M. Ghouri,†,‡ Peter A. J. Hilbers,†,§ and Emiel J. M. Hensen†,‡ †

Institute for Complex Molecular Systems, ‡Department of Chemical Engineering and Chemistry, and §Computational Biology Group, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands S Supporting Information *

ABSTRACT: One of the great challenges in molecular heterogeneous catalysis is to model selectivity of a heterogeneous catalytic reaction based on first principles. Molecular kinetics simulations of the Fischer−Tropsch reaction, which converts synthesis gas into linear hydrocarbons, demonstrate the need for microkinetics approaches that do not make a priori choices of rate controlling steps. A key question pertaining to this reaction, in which hydrocarbons are formed through consecutive insertion of adsorbed CHx monomers into adsorbed growing hydrocarbon chains, is whether the CO consumption rate depends on the rate of the CHx insertion polymerization process. Microkinetic theory of this heterogeneous catalytic reaction based on quantum-chemical data is used to deduce expressions for the CO consumption rate and chain growth parameter α in the two limiting cases where chain growth rate is fast compared to the formation of CHx (monomer formation limit) or where the reverse relation holds (chain growth limit). The conventional assumptions that CHx formation is rate controlling and that change in CO coverage due to reaction is negligible lead to substantial overestimation of the rate of CO consumption. It appears that intermediate reactivity of the catalytic reaction center, with neither too low nor too high activation energies for C−O bond cleavage, and low reagent gas pressure lead to such monomer formation limiting type behavior, whereas maximum rate of CO consumption is found when chain growth rate is limiting.

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INTRODUCTION The Fischer−Tropsch (FT) reaction is a heterogeneous catalytic reaction that is used to convert synthesis gas, which can be derived from natural gas, coal, or biomass, into diesel quality liquid fuels.1−4 Whereas the reaction has been discovered nearly a hundred years ago,5 there is still an increasing need especially to improve selectivity toward longer hydrocarbons over methane. Over the past decade, molecular understanding of the chemistry of this reaction at the catalyst surface has increased substantially mainly due to advances in computational catalysis,6−10 where one of the most interesting aspects is the deepening of our understanding of particle size dependent effects of this reaction at the nanometer level.11 In the engineering community an extensive modeling activity, based mainly on fitting of kinetic expressions to experimental data,4,12−25 has provided empirical data to compare with the first principle quantum-chemical results. Mechanistic and kinetic data has been reviewed especially well in a chapter in the recent book by Bartholomew and Farrauto.4 Many factors determine the performance of practical Fischer−Tropsch catalysts. Mass transport and product readsorption, as highlighted by the work of Iglesia et al.,12,26,27 as well as the complexity of metal−support interactions28−30 play an important role. As Schulz31 has emphasized the catalyst metal surfaces may self-organize in contact with the reactive gases, © 2013 American Chemical Society

which leads to surface reconstruction, as observed by Geerlings et al.32 and Wilson and de Groot.33 This may be the reason for particle size independent activity and selectivity reported widely in the literature, whereas also the reaction is recognized as being highly particle size dependent when particles vary in the nanometer regime. In view of this complexity, it is not surprising that on a molecular level many models have been proposed and no consensus exists on the details of the molecular reaction sequences that lead to the chain growth reaction.3,34−36 The availability of first principle quantum chemical mechanistic molecular models of FT reaction’s elementary steps,37−56 makes revisitation of the kinetics models of this reaction useful. It now appears that an important assumption, now more than 30 years old but still basic to the macro-kinetic models, has to be reassessed. Namely, it is generally assumed that the rate of CO consumption is independent of the rate of chain growth, that is, the rate of insertion of an adsorbed CHx monomer into the adsorbed growing hydrocarbon chain.16−19 This assumption is reasonable as long as one assumes that activation of CO to give the adsorbed CHx species is slow Received: October 16, 2012 Revised: February 5, 2013 Published: February 7, 2013 4488

dx.doi.org/10.1021/jp310245m | J. Phys. Chem. C 2013, 117, 4488−4504

The Journal of Physical Chemistry C

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bond cleavage reaction that proceeds through a formyl species formed by H attachment to CO, on Co overall barriers for CH formation are typically of the order of 130 kJ/mol.40,42,45,47,49,51 This short summary of computed data illustrates that so far there is no unequivocal computational result that enables to discriminate whether the monomer formation or chain growth is rate controlling. These data can be used for a microkinetics model of the Fischer−Tropsch reaction that limits itself to intrinsic kinetics of the reaction. Direct comparison with experimental data has to be done with care, because often the other factors mentioned before, which are not considered in the intrinsic kinetics studies, play a role. However, knowledge of the kinetic relations for effective chain growth kinetics from rate of CO conversion has important consequences for complex kinetics engineering models. Most of the early microkinetics models12,18,19,71−74 are based on the solution of Langmuir−Hinshelwood-type kinetic equations that are solved by assuming a particular rate limiting step. A common assumption is that CO dissociation is equilibrated and rate of CH formation is limiting. Some quantum-chemical studies agree with this view that CO dissociation is equilibrated,70 others suggest that CHx is formed through C−O bond dissociation of a CHO or COH intermediate.45,49,61,67,68 In the kinetic simulations justification of the assumption that CHx formation is limiting17,20 is based on empirical fits with resulting kinetics expressions and the spectroscopic observation that the surface at reaction conditions is mainly covered with CO. Empirical kinetic models21,75 that explicitly include the growth of hydrocarbons are also extensively used. The activation energies of the chain growth reactions are sometimes chosen to be zero. Model experiments on single crystal surfaces32,76,77 provide indications that instead of mainly CO covered, the reactive surface is covered with a high fraction of growing hydrocarbon chains, as has been proposed originally by Anderson.2 This can be reconciled with the studies on practical catalysts by assuming that on those catalysts there is only a limited amount of reactive centers. SSITKA data72,78−81 provide evidence that the number of active centers may be orders of magnitude less than exposed surface area. Recently, Storsaeter et al.82 published one of the first microkinetics studies for the production of methane and C2 products based on consideration of the complete set of elementary reaction steps that leads to these products. No assumption on a rate limiting step has been made. However, use has been made of the approximate Unity Bond Index− Quadratic Exponential Potential UBI-QEP83 method, which is elegant, but is not quantitative. The variation in CO activation energy with surface topology or composition implies that the rate of CHx formation can be lower or higher than the rates of other elementary steps of the FT reaction dependent on surface structure. Because a reactive site, in order not to produce only methane, has to satisfy the condition that activation free energy of methane formation from adsorbed CHx has to be high compared to that of CO transformation to CHx, there is an upper limit to the effective barrier of this latter reaction. No selective chain growth will occur when the latter is less than the apparent activation free energy for chain growth.63 Here, we study the Fischer−Tropsch reaction within the carbide mechanistic scheme. This scheme is currently most generally accepted.5,12,19,66,72,84−87 The influence of variations

compared to the rate of hydrocarbon chain growth by insertion of this CHx surface species and rate of reaction is not limited by desorption of the hydrocarbon chains, as has also been proposed.57 However, current quantum-chemical data indicate that this may not always be the case. In the past decades, several extensive computational catalysis studies on elementary reaction steps of the Fischer−Tropsch reaction have appeared. On Co and Ru,40,41,49,58,59 as well as Fe,54 the decomposition of CO to give Cads and Oads and the subsequent hydrogenation of Cads to give CHx species has been studied. Also, hydrogen-assisted CO activation20,37,42,45,49,60,61 to give directly CHads and Oads or Cads and OHads has been studied. Extensive studies38,44,46,55 have been published also of the C−C bond formation reaction. The termination reaction varies dependent on the nature of the growing hydrocarbon chain intermediate considered and whether oxygenate or hydrocarbon formation is studied. Several different options have been explored.44,46,48,62 Apart from dependence on composition of the catalyst, these data indicate significant structure dependence of all elementary reaction steps.6,11,39−41,47,50,63 There is significant variation and no consensus on the reaction intermediates responsible for the chain growth reaction. Of relevance to the question of which C1 intermediate is inserted into the growing hydrocarbon chain are two essentially different proposals. In the first mechanism, that is, the carbide mechanism, a CHx,ads intermediate, which is generated by a CO bond cleavage and partial hydrogenation, is inserted. In the second mechanism, that is, the Pichler−Schulz mechanism,64 chain growth occurs through insertion of CO. Especially Hu et al.44 provide an extensive comparison of differences in activation energies for the chain growth reaction between different partially hydrogenated CHx species on a variety of surfaces and metals. On metals such as Co and Ru they conclude that the C−C bond formation reaction preferentially occurs on step edges with overall activation energies not less than 130 kJ/mol. On surface terraces substantially higher overall activation energies are predicted. The preferred reacting intermediates are different for terrace and step-edge. In agreement with ref 55, they also find that some recombination reactions such as CH2 with CH2 occur with substantially lower activation energies on step-edges (21 kJ/mol). This agrees with the Gaube mechanistic approach65 that favors chain growth through methylene and alkylidene recombination. However, several alternative recombination paths involving CH and alkylidene, related to the Maitlis mechanism, or CH3 and alkyl, the classical Brady and Pettit proposal,66 also occur with rather low barriers. The CHx intermediates have different stabilities38,39,46,67−69 and their relative stability affects overall C−C bond formation rates. For instance, predictions of the activation energies of CHads and CH3,ads recombination vary between 110 kJ/mol on a dense surface and 70 kJ/mol on step-edges. Sometimes overall C−C bond formation is exothermic, but often such steps may be thermodynamically neutral or even endothermic. Chain growth reaction steps that proceed through CO insertion and subsequent C−O bond cleavage have been reported to have overall activation energies of 180 kJ/mol48 or higher.46 The activation energy of CO dissociation shows also large changes with structure. For example, for different Ru surfaces, the activation energy for CO dissociation can vary between 220 kJ/mol for a terrace and 60 kJ/mol for an open step-edge-type surface.11,40,70 Typical values for subsequent Cads to CHx formation are between 70 and 100 kJ/mol. For the C−O 4489



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one type of growing hydrocarbon chain. The reaction sequence selected is shown in Scheme 1.

in rate constants of elementary reaction steps as a result of catalyst composition or structure are investigated with two types of kinetics simulations. Microkinetics simulations of this reaction, based on quantum chemical data,37−51 will be presented in which all elementary reactions that are part of the Fischer−Tropsch reaction are explicitly accounted for. The corresponding steady state solutions are found without assuming a rate controlling step, because we aim to follow the kinetics as a function of variation in rate controlling steps. Also, a lumped kinetics model88 is used. Using this model, new and useful expressions for the rate of CO consumption and chain growth rate parameter α are deduced in the two limiting cases where chain growth rate is fast compared to the formation of CHx (the monomer formation limit) or where the reverse relation holds (the chain growth limit). It is shown that the conventional assumptions that CH x formation is rate controlling and that change in CO coverage due to reaction is negligible leads to substantial overestimation of the rate of CO consumption. Also, the lumped kinetics model equations will be solved to analyze how kinetics performance parameters change as a function of changing catalyst adsorption strength. It appears that intermediate reactivity of the catalytic reaction center and low reagent gas pressure lead to such monomer formation limited type behavior, whereas maximum conversion is found when chain growth is limiting. In the Discussion and Conclusion sections, we will summarize the results and discuss their practical implications.

Scheme 1. List of Elementary Reaction Steps of Microkinetics Scheme up to the Formation of Propylenea

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a

Formation of the higher hydrocarbons is also included in the simulations through reactions steps analogous to that of propylene formation. Note the reversible and uni-directional reactions.

THEORETICAL METHODS In this article, the Fischer−Tropsch reaction is studied both using microkinetics simulations and using a lumped kinetics model. In both methods, kinetic Fischer−Tropsch parameters such as the Anderson−Schulz−Flory chain growth probability parameter α and the rate of CO consumption (RCO) will be calculated. The microkinetics simulations explicitly incorporate all elementary reaction steps. The primary aim of these microkinetics simulations is to analyze the dependence of Fischer−Tropsch kinetics parameters on the relative overall rate of CHx formation and the rate of methane formation. This will provide a direct kinetics prediction of the difference between the monomer formation limit and the chain growth limit. The lumped kinetics simulations are used to deduce analytical expressions of α and RCO valid in the two limits. The exact solution to the lumped kinetics expressions can be used to test the validity regimes of the expressions for the monomer formation limit and the chain growth limit, respectively. In order to do this, we will calculate α and RCO as a function of changes in the relative rates of primarily CHx formation and rate of chain growth termination. It will appear that a maximum in CO consumption arises when the rate of CHx formation is equal to the rate of chain growth. Because the elementary rates of the Fischer−Tropsch reaction will not vary independently when the interaction of reaction intermediates with surface changes, we use the Brønsted−Evans−Polanyi relations that are linear activation energy-reaction energy relations, to correlate changes of the lumped kinetics rate constants. Microkinetics Simulations. To design a reaction scheme of the Fischer−Tropsch reaction containing a complete set of elementary steps leading to the formation of olefins, we decided to select one type of CHx intermediate species to be incorporated into the growing hydrocarbon chain as well as

CHads is considered to be the C1 intermediate that is incorporated into the growing hydrocarbon chain. This CHads species is incorporated into an alkylidene-type adsorbed hydrocarbon intermediate. Addition of a hydrogen atom to the primary carbon atom of this chain will result in desorption of the corresponding olefin. Addition of a H atom to the β-C atom of the hydrocarbon chain will prepare it for chain growth. This CH intermediated chain growth mechanism is deduced from computational studies on the Ru(0001) surface37 as well Ru(112̅1).70 This reaction mechanism is close to that proposed by Maitlis,35,87,89,90 see also Inderwildi and Jenkins.45 The first C−C bond is formed between two CH species with a low activation barrier. Formation of CH occurs through direct dissociation of CO followed by the hydrogenation of C. Alternative ways of CH formation are intermediate formation of formyl or COH.42,45,49,51 To reduce complexity of the simulations, we do not include these reaction paths. This choice does not imply that we reject the idea that on some surfaces CH formation occurs through hydrogen activation of CO. When these additional steps are included into the simulations, it is observed that one path takes over from the other, dependent on relative activation barriers. Relevant is the overall activation barrier for CH formation from adsorbed CO. The choice of CH or CH2 as monomer for insertion into the growing chain will affect the hydrogen pressure dependence of the reaction. This we will discuss elsewhere.91 Methane formation from Cads or CHads has been studied by many authors.38,39,46,68,69 The details depend strongly on surface and metal. We decided to vary the rate of methane formation from CHads by changing the activation energy of the CHads to CH2,ads reaction step. One expects that recombination of Hads with 4490



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Scheme 2. Lumped Kinetics Model88 of Fischer−Tropsch Kineticsa

CH3,ads is not very structure- or metal-dependent because it takes place over a single metal atom.92 Based on inspection of available quantum-chemical data,37−51 we have constructed the reaction energy diagrams of this reaction, as illustrated for methane and propene formation in Figure 1. The activation energies have been chosen to agree

a

Lumped reaction rate constants are indicated as well as the relation with surface coverage. θCO, θ1, and θi represent surface coverages of CO, CHx, and hydrocarbon chains of length i, respectively, while θv denotes vacant surface sites. PCO denotes the partial CO pressure and CO kCO ads and kdes , the CO adsorption and desorption rate constants, x respectively. Futhermore, kCH CO is the lumped rate constant for the CO transformation to CHx, kfCC for chain growth, kbCC for chain scission, kt for chain termination, i.e., desorption of product chains, and km t for the desorption of methane.

exothermic. Whereas in the microkinetics simulations all atomistic reactions are considered, as illustrated by Figure 1, in the lumped model of Scheme 2 we do not consider hydrogen transfer steps explicitly, but lump those and other reaction steps into rate parameters for the formation and conversion of the carbon species. We will use the lumped kinetics model to analyze, according to the carbide chain growth model, the chain growth probability α and the rate of CO consumption as a function of the relative rates of chain growth and CHx formation. We will see that also the relative rate of hydrocarbon chain termination versus the rate of CHx formation will be critically important. x In Scheme 2, kCH CO is the lumped rate constant for conversion of CO into the CHx monomer (C1) to be incorporated into the growing hydrocarbon chain. C1 is directly formed from CO and is the same for methane formation or incorporation into the growing hydrocarbon chain. θi is the surface concentration of growing hydrocarbon chains of length i (Ci). We use different rate constants for methane formation of C1 and termination of hydrocarbon chain Ci (i > 1). The chain growth probability parameter α is independent of hydrocarbon chain length, because all rate constants, except those for C1 formation, are independent of chain length. The gas phase is assumed to be unaltered by reaction, whereas experimentally H2O and olefins may readsorb. The rate of water formation is assumed so fast that the surface concentration of Oads becomes negligible. Whereas in the microkinetics simulations water formation is explicitly accounted for, in the lumped kinetics scheme the role of hydrogen is implicit to the kinetics rate parameters. Lumped Kinetics Model Rate Expressions. The kinetics of the lumped single reaction center model (Scheme 2) is described by the following set of coupled nonlinear differential equations:

Figure 1. Reaction energy diagrams of (red) methane formation from CO and H2 and (black) propene formation from CO and H2. TS denotes the transition states. Adsorbed species are indicated without a label; gas phase species are indicated with the label “(gas)”.

closely with those available mainly for Ru surfaces.37,70,93 We will vary the activation energies for CO dissociation, and the CH to CH2 transformation as well as for C−C bond formation to study the dependence of kinetics on the relative rate of CH formation, methane formation, and chain growth. Overall thermodynamics for methane and olefin formation agrees with available experimental thermodynamics data.94 Formation of all longer hydrocarbons is considered to be homologous. Activation entropies have been chosen to be only different from zero for reactions between gas phase and solid. These are the same as used elsewhere63 and also provided in the Supporting Information. Readsorption from the gas phase is ignored. The reaction energy diagrams of all of the hydrocarbons except for methane formation have the same elementary reaction rate parameters, so that the hydrocarbon chain growth distribution corresponds to the logarithmic Anderson−Schulz−Flory (ASF) distribution1 with the chain growth parameter α independent of hydrocarbon chain length. Solutions for overall rates are found by solving the complete set of reaction rate equations with proper initial conditions. We solved these equations using Matlab. Because of the higher adsorption energy of CO compared to that of the hydrogen atoms, competitive adsorption tends to suppress dissociative adsorption of H2. Lumped Kinetics Model. Scheme 2 schematically depicts the lumped single reaction center Fischer−Tropsch kinetics model that we introduced recently.88 It is very similar to conventionally used schemes of the Fischer−Tropsch reaction according to the carbide mechanism.5,12,19,66,72,84−87 The important difference is that reversibility of the chain growth steps is incorporated. In a previous paper88 we extensively discussed the conditions that reversibility and irreversibility of these reaction steps apply. The classical result of nonreversibility1,2 applies only when the chain growth reaction is

dθCO CHx CO CO = kads PCOθv − kdes θCO − k CO θCOθv dt

(1)

dθ1 CHx f b = k CO θvθCO − 2k CC θ12 − k tmθ1 + k CC θvθ2 dt ∞

b f + (k CC θv − k CC θ1) ∑ θi i=2

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(2)



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⎛ θ1 ⎞ CHx ⎟ θCO⎜1 − θCO − k CO ⎝ 1 − α⎠ ⎛ 1 ⎟⎞ f θ12⎜1 + − k tmθ1 − k CC =0 ⎝ 1 − α⎠

dθi f b f b = k CC θ1θi − 1 − k CC θvθi − k tθi − k CC θ1θi + k CC θvθi + 1 dt (3)

dP1 = k tmθ1 dt

which can be rewritten further into

(4)

θ1 =

dPi = k tθi dt

(5)

CHx f k CO θCO + (1 − α)k tm + k CC θ1(2 − α)

(6a)

The expression for θCO becomes

where θv denotes vacant catalyst sites, which are needed for some of the reactions in order to accommodate reaction products, and is given by the constraint

θCO =

∞

θCO + θv +

CHx (1 − α)k CO θCO(1 − θCO)

CO xKeq PCO CO 1 + xKeq PCO

(6b)

where

∑ θi = 1

CHx ⎛ ⎞−1 k CO θCO k CHx ⎟⎟ x = ⎜⎜1 + CO (1 − θ ) + CO CO f kdes θ1 + (1 − α)k tm ⎠ (2 − α)k CC ⎝ (6c)

i=1

that is, the total number of catalytic sites is constant. In these equations, the rate of CO dissociation not only depends on the concentration of adsorbed CO (θCO), but also on the concentration of vacant sites since such a vacant site is temporarily necessary to accommodate the adsorbed oxygen atom that is assumed to desorb rapidly as water. Vacant sites are also necessary for the adsorption of CO as well as for the reverse chain growth (C−C bond scission), where at the end of the chain a C−C bond is broken and a C1 fragment is generated. In eq 2 it is also taken into account that the formation of a chain of length 2 occurs at the expense of 2 monomers and, reversely, that the dissociation of such a θ2 yields again two θ1. The steady state is calculated by solving the mass balance, which implies that in steady state the surface concentrations should not change, that is, dθCO/dt = dθ1/dt = ∞ dθi/dt = 0 and that product formation (km t θ1 + kt∑i=2iθi)dt CHx should equal CO conversion kCO θCOθvdt. Because all the rate constants, except the rate of methane formation, are chain length independent, in steady state the ratio α = θi+1/θi is chain length independent,88 and all steady state surface concentrations for the chains can be written in terms of α and θ1, that is, θi = αi−1θ1. For the mass balance approach this has the advantage that all chain lengths can be taken into account as ∞ analytical expressions for θt = ∑∞ i=1θi and ∑i=1iθi can be 95,96 Determination of the steady state is then reduced derived. to solving a set of four equations with four unknowns, that is, α, θ1, θCO, and θv. General Expressions for RCO and α as a Function of θ1. In this subsection for convenience we will rederive two wellknown expressions for rate of CO consumption and chain growth parameter α. These expressions will be used to give the corresponding equations for within the monomer formation limited and chain growth limited kinetics Fischer−Tropsch models. We derive the limit expressions for the case that the reverse rate is negligible, that is, assuming kbCC = 0. In steady state, the change in surface coverage of θ1 should be 0, such that eq 2 can be rewritten as

θCO is seen to decrease with pressure and when kfCC becomes CH smaller than kCC x. Furthermore, for the case where kbCC = 0 that we are considering, α is simply given by the classical ASF expression of chain growth parameter α19,97

α=

(7)

∞

R CO = k tmθ1 + k t ∑ iθi i=2

as each chain of length i contains i carbons. Because it still holds i 2 that θi=1 = αθi and ∑∞ i=1ix = (1/(1 − x) ), this latter equation can be written again as a function of the CHx monomers only, that is, R CO = (k tm − k t)θ1 +

k tθ1 (1 − α)2

(8)

With expression eq 8 a classical result of Fischer−Tropsch kinetics is rederived.2 Monomer Formation Limit Expressions. In the monomer x f formation limit, that is, kfCCθ1 ≫ kCH CO θCO and kCCθ1 ≫ m max(kt,kt ), eq 6a can be approximated by θ1 =

CHx f f k CO θvθCO − k CC θ12 − k tmθ1 − k CC θ1 ∑ θi = 0

CHx (1 − α)k CO θCO(1 − θCO) f k CC θ1(2 − α)

When α is close to 1, one finds the approximate solution

i=1

Because θi+1 = αθi, θv can be substituted, and x), this can again be rewritten as

kt

This is a well-known expression for the chain growth parameter α. Its importance is that it shows explicit dependence on θ1. When used within the carbide mechanism θ1 is the surface concentration of CHx species that is incorporated into the growing hydrocarbon chain. Through this concentration at particular conditions α will be shown as a function of the rate of CO activation. The total CO consumption can be calculated from

∞

i ∑∞ i=1x

f k CC θ1 f k CCθ1 +

⎛ k k CHxθ (1 − θ ) ⎞1/3 θ1 = ⎜⎜ t CO COf 2 CO ⎟⎟ (k CC) ⎝ ⎠

= (1/1 − 4492

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also, the validity regime of this expression will be tested against exact kinetics modeling results. Brønsted−Evans−Polanyi Deduced Relations between the Lumped Rate Constants. To assess the validity regimes of monomer formation limited versus chain growth limited kinetics one needs a full solution of the lumped kinetics expression to test against. Such a solution can be found by a comparison of the reactivity of different surfaces using the Brønsted−Evans−Polanyi6−8,11 relations. They provide a mathematical relationship between the elementary rate constants that will not vary independently when the reactivity of a surface changes. In the Results section we will calculate α and RCO as a function of the adsorption energy of a C atom that is used as metal surface reactivity parameter. An analogous approach has been used before by Nørskov et al.98 in their study of the steam reforming reaction. The Brønsted−Evans−Polanyi relations relate, for surface elementary reaction steps, activation energy changes to reaction energy changes.

Substituting this in eq 7 yields as monomer formation limit expression for α −1 ⎛ ⎛ ⎞1/3⎞ kt2 ⎟ ⎜ ⎟⎟ α = 1 + ⎜⎜ f CH ⎜ ⎝ k CCk CO xθCO(1 − θCO) ⎠ ⎟⎠ ⎝

(11)

Substitution of eqs 10 and 11 in eq 8 yields as monomer formation limited model expression for the CO consumption ⎛ k k CHxθ (1 − θ ) ⎞1/3 R CO = (k tm − k t)⎜⎜ t CO COf 2 CO ⎟⎟ (k CC) ⎝ ⎠ CHx + k CO θCO(1 − θCO)

(12)

In the case of km t = kt, the rate of CO consumption is simply the rate of formation of CHx intermediate incorporated into the growing chain. This is the conventionally used assumption of monomer formation limited kinetics in Fischer−Tropsch.19 We have demonstrated here that this is a valid assumption as long as the chain growth rate is fast compared to the rate of CO to CHx conversion. It has to be noted that the concentrations of θ1 and that of the growing hydrocarbon chains have dropped out of the equations. In the Results section we will test the validity regime of expressions eq 11 and eq 12. The important new result we deduced is the expression of α, that is to be used consistent with the CO consumption rate expression eq 12. This chain growth parameter α explicitly depends on the rate of CO consumption and cannot be considered an independent parameter. Chain Growth Limit Expressions. In the chain growth limit, CH that is, kCOxθCO ≫ kfCCθ1 ≫ max(kt,km t ), eq 6a can be approximated by θ1 = (1 − α)(1 − θCO)

ΔEact = αBEPΔEreaction

kt f k CC

(13a)

(1 − θCO) (13b)

Substituting this in eq 7 yields as chain growth limit expression for α ⎛ α = ⎜⎜1 + ⎝

⎞−1 ⎟⎟ f k CC (1 − θCO) ⎠ kt

(14)

and as chain growth limit expression for the CO consumption: R CO = (k tm − k t)

kt f k CC

(1 − θCO) +

(16)

ΔEact is the change in activation energy, ΔEreaction is the change in reaction energy, and αBEP is the proportionality parameter. The expression applies when the composition of a catalytically reactive surface changes, as long as surface topology and reaction paths remain similar. For a dissociation reaction as CO bond cleavage, values of αBEP are typically close to 1. For a recombination reaction, the corresponding value has to be close to zero because of microscopic reversibility.92 The lumped rate constants in Scheme 2 are not true elementary rate constants in terms of their microkinetics definition. Because we are interested in model results of the lumped kinetics equations that provide variations in rate performance as a function of changes in relative rates, we decided to apply the BEP relations also to these kinetics constants. It essentially implies that we ignore any changes in the activation energies of the hydrogen transfer reactions. The reaction energies refer to the energy differences between intermediates with the molecular bond present and the corresponding molecular fragments that often are adsorbed similar as a Cads atom. Whereas chemisorption of molecules varies relatively weakly with metal composition, that of C or O atoms varies strongly. We therefore assumed the adsorption energy of CO also to be unchanged when different metals are compared. One realizes that within this approximate scheme of parameter changes, the reactivity of a metal is mainly determined by the changes in the adsorption energies of dissociated molecular fragment intermediates. Using the scaling law concept as introduced by Nørskov11,99 the essential surface reactivity parameter is the adsorption energy of a surface adatom. The two surface adatoms important to the Fischer− Tropsch reaction are C and O. Within the lumped kinetics scheme water formation is fast. Changes in Oads affect the activation barrier of CO. This necessitates the use of two dimensions for correlation according to the changes in the adsorption energies C and O.100 This can be reduced to one dimension, as we do in our simulations, when it is assumed that changes in Cads and Oads are parallel. This is not generally true, but approximately for Fe, Co, and Ru such that it will effect the change in CO activation energies only slightly. The expressions to be used in the model studies with BEPtype relations of the rate constants are as in

This implies no surface vacancies. Assuming nonreversibility of the chain growth reaction, this yields as CHx coverage in the chain growth limit

θ1 =

0 ≤ αBEP ≤ 1

f k tk CC (1 − θCO)3

(15)

Interesting in eq 14 of α is the square root relation with kt and kfCC, that we deduced earlier in a more approximate way in ref 50. It is the expression to be used within Anderson’s2 assumption that the rate of product desorption is rate limiting. Expression eq 15 has not been applied before in Fischer− Tropsch kinetics. As we discussed in ref 88, the increase in CO consumption rate with rate of termination has to imply that now the rate of CO consumption is controlled by the rate of hydrocarbon removal from the surface. In the Results section,

0

kx = Ax e−(Ex + βxEads(C))/ RT 4493

(17)



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where R is the gas constant and parameters Ax, βx, and Ex0, as given in Table 1. Different from the microkinetics simulations,

the rate of CO consumption RCO as a function of temperature. The latter we present as Turn Over Frequencies (TOFs): number of CO molecules consumed per second per site. The simulations have been done at pressures PCO = 5 bar and PH2 = 15 bar. The purpose of these simulations is to establish the dependence of catalyst performance as a function of the changes of four lumped elementary reaction steps: the rate of CO activation to give CHx, the rate of methane formation from CHx, the rate of the chain growth reaction and its rate of termination. In the simulations, the rate of CO dissociation determines largely the rate at which the CH intermediates are formed that are inserted into the growing hydrocarbon chain. As mentioned, the activation energy of CO dissociation is highly structure sensitive and may vary from approximately 200 kJ/mol for dense surfaces to 60 kJ/mol for step-edge B5-type surfaces.92 In the simulations we will vary the CO activation energies between 70 and 110 kJ/mol. As we will see, the simulations indicate that kinetics then change from the monomer formation limited case to the chain growth limit. The rate of methane formation from CHx also will vary when surface structure changes. In the microkinetics simulations, we change the apparent rate or methane formation by increasing the barrier to form CH2 from CH from 70 to 90 kJ/mol. These values are within the regime of computationally found values for Ru.67,102 Consistent with the case of a low barrier of CO activation and CH formation and a higher barrier for CH2 formation, a combined theoretical and experimental study103 concluded that CH formation is fast but CH2 formation is slow for Co Fischer−Tropsch catalysts. To determine the maximum chain length to be considered in our simulations, we first studied the effect of truncating the differential equations at certain chain lengths. Figure 2 shows results of simulations using elementary rate constant parame-

Table 1. Default Parameters Used in the Brønsted−Evans− Polanyi Relations rate constant kt CH kCOx kfCC kbCC

Ax 17

−1

At = 10 s Ad = 1013 s−1 Af = 1013 s−1 Ab = 0 s−1

βx

Ex0

βt = −0.3 βd = 1.2 βf = 0.0

Et0 = 70 kJ/mol Ed0 = 270 kJ/mol Ef0 = 70 kJ/mol

the lumped kinetics model simulations will be done at the same CO temperature of 500 K. kCO ads PCO and kdes have been taken independent of Eads(C), with default values of 107 and 2 × 106, respectively, such that the CO surface coverage in absence of reaction would be 0.83. In the Results section we will also study the effect of varying adsorption conditions and changes in the rate of CO desorption. Default values used in eq 17 are given in Table 1. The preexponents are derived from the microkinetics activation entropy choices (Table S1 in Supporting Information). The β parameters are proportional or equal to αBEP selected using the principles discussed above. The rate of termination is chosen to be weakly M−C dependent because it is the combination of hydrogen atom transfer and M−C cleavage but with already a carbon atom and hydrogen atom attached to it. The default values of E0x have been chosen to correspond to estimated activation energies for a weakly reactive surface as one of the Nickel surfaces.42,101

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RESULTS Microkinetics Simulations. Here we present results of the microkinetics simulations of the Fischer−Tropsch reaction. We will calculate the chain growth parameter α, the C2+ yield and

Figure 2. Chain growth probability α as a function of temperature calculated with microkinetics and different chain length cut offs N. Parameters used are given in Table S1 of Supporting Information. α is calculated as the ratio of produced Cn/Cn−1 (n > 2). Pressures used are PCO = 5 bar, PH2 = 15 bar. 4494



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Figure 3. Microkinetics simulations of chain growth parameter α, C2+ yield, RC+2 and rate of CO consumption RCO as a function of temperature. Total pressure PCO = 5 bar and PH2 = 15 bar. The rates are expressed as turn over frequencies (TOFs) of CO, where the unit is the number of CO molecules consumed per site per second. Default elementary rate parameter values as in Table S1 of SI. (a) α, RC+2, and RCO for three different values CO→CH2 CO + of ECO is now increased from 70 to 90 kJ/mol. (c) α, RC+2 , and RCO are compared for act . (b) α, RC2 , and RCO for three values of Eact , where Eact CO→CH2 m three cases. The case kt high corresponds to the default values of Table S1 from SI. In case km has been changed from 70 to 90 kJ/mol t low Eact low and k low, the latter rate constant is decreased by increasing the activation energy that corresponds to the step in which a as in (b). In case km t t hydrogen atom is added to the primary carbon atom of the adsorbed alkenyl chain from 70 to 90 kJ/mol.

Figure 4. Microkinetics simulations with variation of the rate of chain growth. α, RC+2, and RCO are calculated as a function of temperature (PCO = 5 bar and PH2 = 15 bar). Default values of SI, Table S1 are used. (a) The case of default values is compared with that of decreased chain growth rate by an increase of the activation energy for the incorporation of CH monomer into growing chain (ECC act ) from 50 to 90 kJ/mol. (b) The case with CC increased ECO act = 110 kJ/mol is studied for the two values of Eact .

simulations, a maximum chain length N = 100 is used, where cutoff effects are negligible. The higher than 1 values of α of Figure 2 for low chain length cutoff parameters will only appear when chain growth formation steps are reversible, that is, endothermic or thermodynamically neutral. Then only in the limit of an

ters of Table S1 in the Supporting Information. As shown in Figure 2 for small cutoff lengths N, chain growth parameter α becomes much larger than 1 in the low temperature range, whereas only when sufficiently long hydrocarbon chain growth is simulated, the value of α converges to a physically sensible value, that is, a value below one. Therefore, for all further 4495



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infinite chain, α values of 1 or less will be found, because of the need to maintain mass balance. In a previous paper,63 we applied dynamic Monte Carlo simulations to the solution of Fischer−Tropsch microkinetics using formation of hydrocarbons of limited chain length that showed related cutoff effects. In Figure 3, we compare calculated values of chain growth parameter α, the production rate of hydrocarbons other than methane (RC+2 ) and the rate of CO consumption (RCO) as a function of temperature for different values of the CO activation barrier. The general observation to be made of these results is that the C2+ production rate has a maximum at a substantially lower temperature than that of methane formation. This agrees with the experimental observation3 that low selectivity of methane production requires low temperatures. Temperature maxima in overall rates of heterogeneous catalytic reactions are a general phenomenon.104 At low temperature the rate is limited by desorption (in this case CO desorption), whereas at the higher temperature rate becomes limited by the rate of CO or H2 adsorption. The initial temperature independence of α and decrease with temperature also corresponds well to experimental observation.3 The computed TOFs depend sensitively on temperature and rate parameters. We will comment on these in the Discussion section. We note from Figure 3a that over a significant range of CO activation energies, that is, up to 90 kJ/mol, the selectivity and rates of reaction are not affected. In the low activation barrier regime of CO activation CHads formation is fast compared to the slower apparent rate of C−C bond formation and α as well as CO consumption rate RCO are independent of CO activation. This is the chain growth limited regime of eq 15. When the CO activation barrier for dissociation increases to 110 kJ/mol, there is a large decrease in CO consumption rate as well as chain growth parameter α. Then the apparent rate of chain growth has become the faster rate. This corresponds to the monomer formation limit. As is illustrated in Figure 3b, α, and C2+ yield increase when the rate of CH4 formation is decreased. This increases the fraction of CHads species to be incorporated into the growing hydrocarbon chains. Overall CO consumption rate RCO again only changes when the activation energy for CO dissociation increases to 110 kJ/mol. For the chain growth limited case in column c we find that a decrease of the rate of chain growth termination further increases α, but now overall rate of C2+ production (RC+2 ) as well as total rate of CO consumption (RCO) decreases in agreement with the predictions of eq 15. In Figure 4, the simulation results complementary to those of Figure 3 are given for the case that the rate of chain growth is changed. A comparison is made for the chain growth limited case low activation energy of CO dissociation (Figure 4a) and the monomer formation limited case, with a high activation energy of CO (Figure 4b). One observes that in both cases α and RC2+ are decreased when the rate of chain growth termination decreases. However, in agreement with the expressions eqs 12 and 15 RCO only changes in the chain growth limited case (Figure 4a) but not in the monomer limited case (Figure 4b). Changes in surface coverage are summarized in Tables 2 and 3. As is illustrated in Table 2, that applies to the chain growth limited case with low activation energy of CO (ECO act = 70 kJ/ mol) as long as the rate of chain growth termination remains

Table 2. Complementary Results of Microkinetics Simulations Corresponding to Figure 3c (the Chain Growth Limited Case), with Default Parameters of Table S1, Except a for km t and kt CO→CH

high km; CO→CH2 t Eact = 70 kJ/mol temp RCO (s−1) RC+2 (s−1) Sel. CH4 α θCO θref CO θV θC θt

500 K (Tmax(C2+)) 41

CO→CH2

low km t ; Eact kJ/mol 500 K

= 90

2 low km = 90 t ; Eact kJ/mol and low k ; CHCH R→CH2CH2tR Eact 2 = 90 kJ/mol

40

540 K (Tmax(C2+)) 203

500 K 0.9

550 K (Tmax(C2+)) 92

17

40

203

0.9

91

50%

0%

0%

0%

2%

0.99 91% 97% 2% 0% 7%

1.0 88% 97% 2% 0% 10%

0.96 58% 80% 15% 7% 20%

1.0 13% 97% 1% 0% 86%

1.0 45% 68% 20% 13% 22%

a

Rates are compared with surface compositions at different temperatures. Tmax(C2+) is the temperature of maximum C2+ production. θref CO is the surface coverage of CO in absence of reaction at a particular temperature. θt is the total coverage of growing hydrocarbon chains.

relatively fast, at the temperatures of RC+2 maxima the surface coverage remains dominated by CO. This shows that in contrast with some suggestions in the literature105 within the carbide mechanism high α values can be consistent with high CO coverage. The CO surface coverage will decrease with decreasing rate of chain growth termination, and then the surface becomes covered with growing hydrocarbon chains. Table 3 gives a similar comparison as in Table 2, but now for the case of decreased rate of CO dissociation. Monomer limited kinetics applies as long as CO coverage remains high. Again, we observe that high CO coverage and high α values can coexist. Compared to the TOFs in Table 2, when rate of CO dissociation decreases, the TOFs decrease significantly. When the rate of chain growth termination becomes very low even for the high CO activation case, we observe from the drop in CO coverage that monomer formation limited kinetics changes into chain growth limited kinetics. Lumped Kinetics Simulations. The conditions of the validity of the monomer formation model versus the chain growth model can be scrutinized in more detail using Brønsted−Evans−Polanyi (BEP) principle based constructions of the elementary rate constants that constitute the Fischer− Tropsch reaction. It enables to study selectivity and rate of CO consumption of the simulated Fischer−Tropsch reaction as a function of the interaction energy of Cads with the metal surface Eads(C).88 Eads(C) is a measure of the interaction of adsorbed intermediates to the metal surface and hence a measure of the reactivity of a metals surface. The Brønsted−Evans−Polanyi relations only apply when reactions following similar reaction paths on the same surface are compared. In Figure 5, we compare a simulation with the default parameters of Table S1 in the Supporting Information, with another simulation where for CO dissociation a lower value has been used. The latter represents the case of a more reactive surface. In Figures 5−7, we expressed the rate of CO consumption with units of number of molecules of CO per site per second. This relates with the 4496



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Table 3. Simulated Coverages under the Initial Kinetic Condition of the Monomer Formation Limit (ECO act = 110 kJ/mol) at Different Temperaturesa CO→CH

CH→CH

2 Eact CHCH =R→CH 60 (kJ/mol) 2CH2R and Eact 2 = 70

2 =CH 90R Eact R→CH CHCH 2 2 and Eact 2 = 70

CH→CH

2 =CH90R Eact R→CH CHCH 2 2 and Eact 2 = 120

temp RCO (s−1) RC+2 (s−1)

500 K 4 2 × 10−3

530 K (Tmax(C2+)) 89 4 × 10−2

500 K 4 4

530 K (Tmax(C2+)) 71 52

530 K (Tmax(C2+)) 9 1 × 10−1

Sel. CH4 α θCO θref CO θV θC θt

100% 0.06 97% 97% 2% 0% 1%

100% 0.04 85% 85% 12% 0% 3%

3% 0.9 96% 97% 2% 0% 2%

26% 0.78 80% 85% 11% 3% 6%

33% (1.00) 29% 85% 4% 1% 66%

CO→CH2 2R→CH2CH2R Tmax(C2+) is temperature of maximum C2+ yield. Default parameters of Table S1 except for ECO , ECHCH , also mentioned in the act , Eact act table headings. Meanings of symbols same as in Table 2. a

and selectivity to longer hydrocarbons is favored by the systems with lower activation energies to produce CHx. Methane production occurs on the surface with the high activation energy of CO, a dense surface, whereas the more open surface with B5 sites produces longer hydrocarbons. Within Fischer−Tropsch catalysis a classical assumption is that excess methane formation occurs at a different site than the Fischer−Tropsch chain growth reaction. This is the dual site assumption.16−18 Here we present a model for the difference between the two sites. They distinguish themselves as sites with high activation energy for CO bond activation and CHx hydrogenation (surface terraces) and sites with low barriers of CO activation energies and strong metal−carbon bond energies (step-edge type sites) In Figure 6a,b, the exact solutions of the lumped kinetics equations are compared with those when we apply the catalysis performance expressions of the two limiting cases, that is, eqs 11 and 12 for the monomer formation limit and eqs 14 and 15 for the chain growth limit, respectively. We varied kt and kfCC to illustrate the dependence of the two limiting cases on ratios of chain growth and rate of CHx formation. As Figure 6c indicates, we see indeed crossover from monomer formation limited to chain growth limited kinetics when these rates cross. The simulated curves of RCO have a bell-shape dependence with increasing value of Eads(C). At the right of the maximum x the rate of CO consumption is initially limited by kCH CO . To the left of the maximum, when the rate of consumption reduces because of increasing concentration of growing hydrocarbon chains, the expression for RCO becomes (ktkfCC)1/2, which we denote the simple chain growth limit. The overall rate of chain x growth has become small compared to kCH CO . α starts to increase when kt becomes small enough that at the right of the RCO x maximum it crosses the increasing rate of kCH CO . In the figure, three cases are compared. In one case (black curve, with Et0 = 70 kJ/mol and Ef0 = 70 kJ/mol) the monomer formation limit is never reached. The red curves (decreased Ef0) as well as the blue curves (increased Et0) represent cases where below a particular value of Eads(C) monomer formation limit behavior is found. With increased values of kfCC (i.e., lower Ef0, red curves), α increases overall and there is a larger Eads(C) interval where the monomer formation limit expression reasonably approximates RCO. Decreased values of kt (i.e., higher Et0, blue curves) also increases chain growth parameter α, as it increases the surface coverage of CHx, although it decreases overall consumption RCO. As expected, the expression for α in eq 11

Figure 5. Influence of the CO activation energy Ed0 on CO consumption rate RCO (top) and RCO chain growth parameter α (middle) as a function of the adsorption energy of carbon Eads(C). (bottom) The rate constants (sec−1) result from the BEP relations as a function of the adsorption energy of carbon Eads(C). Red lines correspond to an initial value (i.e., at Eads(C) = 0) of the CO activation energy Ed0 of 220 kJ/mol and black lines to an initial value of 270 kJ/ mol. Other parameters as in Table 1, except that Et0 = 85 kJ/mol and Ef0 = 55 kJ/mol.

TOF when it is divided by the number of CO collisions, which 5 is equal to kCO ads PCO. In case RCO is of the order of 10 per site per second, as in Figures 5 and 6, this gives a TOF of 10−2. In Figure 5, parameter values of kt and kfCC have been slightly adjusted for clarity of presentation. Obviously, the rate of CO consumption maximum in the latter case is at lower absolute value of Eads(C) than for the less reactive surface. In Figure 5, one observes that the maxima in rate of CO consumption CH correspond with the Eads(C) values where kCOx and kfCC cross. The figure also illustrates nicely that on a surface with the same composition (Eads(C) the same) the rate of CO consumption 4497



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Figure 6. Catalyst performance parameters RCO (a) and α (b) as a function of surface reactivity with variation in elementary lumped rate parameters based on BEP type expressions (see Theoretical Methods for details). Apart from the simulated curves also the monomer formation eqs 11 and 12 and chain growth eqs 14 and 15 limits are shown. (c) Comparison of the effective rates (r) determining monomer formation (right of crossings) or chain growth limit (left of crossings). Black curves correspond to default activation energies for termination, Et0 = 70 kJ/mol, and chain growth, Ef0 = 70 kJ/mol, while for the red curves Ef0 is decreased by 15 kJ/mol, whereas for the blue curves, Et0 is increased by 15 kJ/mol. Other parameters again as in Table 1.

gives an excellent fit with the exact simulation when the monomer formation limit applies and α is larger than 0.7. In this regime, the approximate expression for α in the chain growth limit (eq 14) performs poorly. The optimum Fischer−Tropsch performance is located within the chain growth limit kinetics regime. Maximum C2+ yield is thus favored by a value of kt high enough that poisoning by growing hydrocarbon chains does not yet occur, but low x compared to kfCC. Then kCH CO has to be fast compared to kt, as well as for overall rate of chain growth, to be consistent with the chain growth model limit. Then θCO will remain to dominate the surface coverage and it will not yet have been blocked by growing hydrocarbon chains. In Figure 7, that studies the dependence of Fischer−Tropsch kinetics as a function of CO pressure, such changes in surface composition are illustrated in Figure 7c and d, respectively. Figure 7a provides information on the pressure dependence of the CO conversion optimum at a constant value of the CO adsorption equilibrium, but when adsorption and desorption rates vary. The partial CO pressure used in the microkinetics 7 simulations (PCO = 5 bar) corresponds to kCO ads PCO = 10 CO collisions per second per site (black curves). The solid lines represent the model results, while the broken lines indicate RCO

limits for the two limiting cases, that is, the monomer formation limit eq 12 and the chain growth limit eq 15, respectively. For the latter the CO coverage is ignored resulting in the simple chain growth limit expression (ktkfCC)1/2. For this reason, the simple chain growth limits do not depend on CO pressure. Also the monomer formation limit lines overlap except for the low CO pressure case when the rate of CO adsorption becomes rate controlling. One notes that to the right of the CO consumption rate maximum this rate increases with decreasing pressure. This agrees with the generally observed negative order of rate of consumption with CO pressure. Because of this increased rate the fit with monomer formation limit expression becomes better when the pressure becomes lower. As expected from the chain growth limited rate expression, to the left of the CO consumption rate maximum, the rates appear to be independent of CO pressure, except when the pressure becomes too low as at this low pressure the rate of CO adsorption becomes the rate controlling step. Figure 7b illustrates that for these parameters the chain growth parameter α is independent of the CO pressure because in the pressure sensitive regime its value is almost 1. Figure 7c,d illustrates the changes in surface concentration for the different situations. One notes the decrease in CO 4498



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5 6 Figure 7. Catalyst performance as a function of surface reactivity for different partial pressures of CO, that is, kCO ads PCO equals 10 (blue), 10 (pink), 107 (black), and 108 (red) CO collisions per site per second, respectively. (a) Comparison of CO consumption RCO. (b) Comparison of chain growth parameter α. (c) Surface occupancies with CHx (θ1) and CO (θCO). (d) Surface occupancies with hydrocarbons (θt) and vacant sites (θv). (e) Comparison of exact results with monomer formation limit results shows that monomer formation limit fits best for low pressures as well as low Et0 and Ef0. Parameters again as in Table 1.

a function of some rate parameters and CO pressure (indicated as the number of CO collisions per site per second). It can be seen that at a pressure 2 orders of magnitude less than the value used in the microkinetics expressions, when activation energy of kt is low and activation energy for C−C bond formation is also low there is nearly perfect agreement. At this low pressure, there is a cutoff in maximum simulated rate when the rate of CO consumption approaches that of CO adsorption.

coverage and increase in C1 concentration with increasing surface interaction energies. The C1 concentration goes through a maximum and for strongly binding surfaces the surface becomes covered with growing chains. When pressures decrease, θCO starts to reduce at lower interaction energies. At very low pressures, strongly interacting surfaces show coexistence of surface vacancies and growing hydrocarbon chains. Figure 7e shows in detail the difference between monomer formation limit and exact rate of CO consumption as 4499



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DISCUSSION Within the carbide mechanism we have analyzed the validity of the kinetic relations that sustain high activity and selectivity, as sketched in Figure 8.

is fast and CO surface concentration remains high. On the other hand, when the rate of chain growth termination decreases and the rate of CO activation is slow, at the same temperatures, TOFs may decrease by 104 to give values of the order 10−1 s−1. At the temperatures and pressures we apply, experimental values are typically 10−2 s−14,12,106,107 (see also Tables 6.28 and 6.25 in ref 4). As SSITKA data72,78,80 on the working catalyst indicate that the number of reactive centers is low, the simulated TOFs are consistent with the assumption of the presence of a low number of reactive centers. This also implies that the reactivity of these catalysts is far from their potential maximum. In the theoretical part based on the solution of lumped kinetics equations we have given the basic expressions for RCO and α and show how to deduce the two limiting cases of monomer formation limit and chain growth limit. These have been tested against BEP constructed solutions of the lumped kinetics equations. The lumped kinetics expression studies serve to identify which kinetics limits actually apply experimentally. Because they confirm that the ratio of CO activation to form CHx versus the rate of methane formation is key to the selectivity toward methane formation, they also support the proposition that reaction in the nanometer particle size regime is highly structure sensitive, as now has been observed for many systems.108−112 This is the dual site model as usually applied in Fischer−Tropsch kinetics.16,18,20 For a particular surface, the Brønsted−Evans−Polanyi rules enable to estimate the changes in elementary rate constant when composition changes the reactivity of the surface. This provides a theoretical approach to determine the rate maximum of the Fischer−Tropsch reaction as a function of surface reactivity. The interesting result is that at the CO consumption rate maximum, the rate of CO to CHx transformation is equal to the rate of chain growth. The absolute value of this maximum depends also on the values of the rate of termination for this surface reactivity. To the right of the CO consumption rate maximum, the elementary rate of CO activation is limiting and the elementary rate of chain growth is relatively fast (monomer formation limited model). To the left of this maximum, the elementary rate of chain growth is slow compared to the elementary rate of CO activation. The rate of CO consumption becomes limited by the rate of chain growth termination (chain growth limited model). The CO coverage remains high when monomer formation is rate limiting, while the surface coverage becomes dominated by growing hydrocarbon chains when the chain growth rate becomes limiting. The chain growth parameter α can be high in both cases and is controlled by the ratio of the rate of chain growth and the rate of chain growth termination. The rate of CO consumption and α computed using the BEP relations has been used to test the validity regimes of the two related limiting expressions in the monomer formation model versus the chain growth model. The maximum in rate of CO consumption can be reasonably well reproduced using the expressions that correspond to the chain growth limit. It is essential to properly calculate the changes in CO coverage from expression eq 6b. The monomer formation model is most commonly used in Fischer−Tropsch kinetics. We find that it only applies over a limited reactivity interval for relatively weakly interacting surfaces. The elementary rate of chain growth has to be fast, so that a relatively high rate of chain growth termination can be

Figure 8. Schematic representation of the necessary relations between elementary rate constants in the Fischer−Tropsch reaction to give high chain growth.

A second objective has been the deduction of analytical expressions for rate of CO consumption and chain growth parameter α. These expressions appear to depend strongly on the relative rate of C−C bond formation versus rate of CHx formation through activation of CO. The microkinetics simulations have demonstrated the general validity of the rate relations for high chain growth, as schematically illustrated in Figure 8. As observed from Figures 3 and 4, they lead to a very different temperature maximum of rate of methane formation versus that of C2+ yield. At the low temperature end of the reaction, high chain growth dominates, and at the higher temperature, methane formation dominates. This is in line with the general experimental observation that relative methane yield increases with temperature.3 Methane yield is especially high when the rate of CO transformation to CHx is slow compared to the rate of CHx hydrogenation and the rate of chain growth termination. This sensitivity of methane formation on relative rate of CO activation supports the idea that excess methane is formed on different sites than that of the chain growth reaction, the twosite model.16,18,20 Because the rate of formation of CHx from CO compared to CHx hydrogenation is slow on terraces, we suggest that excess methane formation primarily occurs on the terraces. High chain growth requires step-edge type of sites, where rates of CO activation are fast. The microkinetics simulations have also demonstrated that within the carbide mechanism high chain growth may occur on sites with high CO coverage as long as the rate of chain growth is fast compared to the rate of CHx formation. This is the monomer formation kinetics limit. A high α value can also be found when the rate of propagation is slow compared to the rate of CHx formation. Then the rate of termination is small and surface becomes covered with growing chains. This is the chain growth limit. Both kinetics limits are consistent with data obtained from quantum-chemical calculations. We have demonstrated that rather small parameter changes may cause a tip-over from the one kinetics regime to the other. Quite sensitive to parameter choices of the elementary rate constants are predicted values of TOF. At the temperatures of maximum overall rates TOFs for C2+ formation can be as high as 203 s−1 when CO activation has a low barrier, chain growth 4500



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accommodated and α remains high. On the other hand, the rate of CO activation has to be fast compared to the rate at which methane is formed from adsorbed CHx intermediates. A lower rate of CO desorption and corresponding rate of adsorption, determined by CO pressure, improves the fit of the monomer formation limit model and BEP calculated overall rate, because the rate of CO activation now competes more favorably with the rate of CO desorption and the CO coverage decreases. When the interaction energy with the surface increases, a substantial error between the monomer formation limit model and BEP calculated rate develops. The monomer limited expression tends to overestimate the CO consumption rate. In this regime to the right of the rate of CO consumption rate maximum, the expression for chain growth limited kinetics leads to a better fit as long as the change in CO coverage is properly calculated. One is still in the regime of high CO coverage, but an increasing coverage with CHx and growing hydrocarbon chains appears. The order in CO in the rate of CO consumption decreases. The higher temperature kinetics modeling results of Visconti et al.75 illustrate the appearance of such a surface state when the rate of CO activation increases with respect to the rate of chain growth. Surfaces of higher reactivity have an increased rate of CO activation and decreased rates of chain growth termination. Kinetics will shift to the chain growth limited regime and higher temperatures will be required for a high CO consumption rate. The recent experiments by de Jong et al.113 demonstrate hightemperature catalysis for supported Fe catalysts. The reduced value of α is used to selectively produce lower olefins. The monomer formation limited kinetics assumption can be used to simplify microkinetics studies of the rate of CO consumption. As long as the rate of chain growth is fast, one can assume that surface is mainly covered with CO. Then the Fischer−Tropsch high chain growth condition is satisfied as long as the rate of CHx to methane transformation is slow compared to the other reaction steps. The rate of CHx formation calculated without explicitly including the chain growth then agrees with the rate of CO consumption at Fischer−Tropsch condition. This is the approach taken by van Steen and Schulz.18 Whereas such a simulation properly predicts the rate of CO consumption, it cannot predict the chain growth parameter α. In case chain growth is explicitly included, the total CO consumption rate will not be affected, but in the simulation chain growth will occur at the cost of methane formation. This is illustrated by the microkinetics simulations of Figure 4b. The selectivity for chain growth can be estimated independently from expression eq 11. The surface coverage θCO to be used in this equation can be obtained from a simulation including only C1 formation. Knowledge of published activation energies can be used to estimate FT selectivities. Ojeda et al.49 indicate for the Fe(110) surface an activation energy of CH formation from adsorbed CO through the formyl intermediate of 153 kJ/mol. As activation energy for the CH to CH2 transformation, they find 35 kJ/mol. On the Co(0001) surface, these numbers are 192 and 36 kJ/mol, respectively. Because the activation free energies of the consecutive reaction toward methane are less than 100 kJ/mol,46,56 these surfaces are predicted to have high selectivity for methane production. On the other hand, Shetty et al.93 find a direct activation energy of CO to CH transformation of 60 kJ/mol on the open Ru(112̅1) surface and an activation energy of CH2 formation of 120 kJ/mol.70

Therefore, this surface is an excellent candidate for high selectivity of growing chains.

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CONCLUSION Intrinsic Fischer−Tropsch kinetics in which CO is converted to long chain hydrocarbons is a strong function of the relative ratios of four lumped elementary reaction rates: the rate of CO to CHx transformation, the rate of C−C bond formation, the rate of chain growth termination and the rate of methane formation, respectively. Because this dependence is nonlinear it causes high selectivity (α) of the reaction to be not uniquely determined by one combination of these elementary reaction rates, although these combinations lead to very different CO consumption rates. It appears that maximum yield cannot be properly modeled using the monomer formation limited assumption, but it is well modeled within the chain growth limit as long as the change in CO surface coverage is properly modeled. The elementary rate of chain growth tends to be much less sensitive to metal or site structure than C−O bond activation, rate of methane formation, and rate of chain growth termination. Intermediate reactivity of the catalytic reaction center and low reagent gas pressure lead to monomer limited type behavior. However, the conventionally used assumption that CHx formation is rate controlling and change in CO coverage due to reaction is negligible leads to a substantial overestimation of the CO consumption rate. Whereas a low x activation energy value of kCH CO compared to that of the chain growth reaction rate will bias the chain growth model condition, intermediate values of CO activation that are higher than that of the chain growth reaction will lead to the monomer formation limited condition and even higher values will lead to low chain growth and predominantly methane formation. Because kinetics modeling indicates that practical Fischer− Tropsch catalysis operates in the monomer limited kinetics region far from the CO consumption rate maximum, there is a significant opportunity to increase yield of Fischer−Tropsch catalysts. It also appears that the number of chain growth selective sites is low. Maximum rate of CO consumption with high chain growth requires a catalyst with low activation energy of CHx formation from CO, but with weak enough M−C bond energy so that the reaction center does not become poisoned by growing hydrocarbon chains. Excess methane formation will be suppressed when the presence of terrace sites can be suppressed. These conditions are consistent with the dual center single step-edge site model we proposed before70,88 with activation of the C−O bond on a different location than that of the chain growth reaction.

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ASSOCIATED CONTENT

S Supporting Information *

Description of microkinetics equations and parameters and list of symbols and their meanings. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 4501



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ACKNOWLEDGMENTS We thank the ICMS animation studio for the rendered images. REFERENCES

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Monomer Formation Model versus Chain Growth Model of the Fischer

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