Single atom catalysts on amorphous supports: A quenched disorder perspective Baron Peters and Susannah L. Scott Citation: The Journal of Chemical Physics 142, 104708 (2015); doi: 10.1063/1.4914145 View online: http://dx.doi.org/10.1063/1.4914145 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamic structural disorder in supported nanoscale catalysts J. Chem. Phys. 140, 134701 (2014); 10.1063/1.4869178 Isolated catalyst sites on amorphous supports: A systematic algorithm for understanding heterogeneities in structure and reactivity J. Chem. Phys. 138, 204105 (2013); 10.1063/1.4807384 Morphological and chemical influences on alumina-supported palladium catalysts active for the gas phase hydrogenation of crotonaldehyde J. Chem. Phys. 134, 214704 (2011); 10.1063/1.3593472 Ultrathin ( 1 × 2 ) -Sn layer on GaAs(100) and InAs(100) substrates: A catalyst for removal of amorphous surface oxides Appl. Phys. Lett. 98, 231908 (2011); 10.1063/1.3596702 The CO oxidation kinetics on supported Pd model catalysts: A molecular beam/in situ time-resolved infrared reflection absorption spectroscopy study J. Chem. Phys. 114, 4669 (2001); 10.1063/1.1342240
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THE JOURNAL OF CHEMICAL PHYSICS 142, 104708 (2015)
Single atom catalysts on amorphous supports: A quenched disorder perspective Baron Peters1,2 and Susannah L. Scott1,2 1 2
Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, USA
(Received 30 November 2014; accepted 24 February 2015; published online 13 March 2015) Phenomenological models that invoke catalyst sites with different adsorption constants and rate constants are well-established, but computational and experimental methods are just beginning to provide atomically resolved details about amorphous surfaces and their active sites. This letter develops a statistical transformation from the quenched disorder distribution of site structures to the distribution of activation energies for sites on amorphous supports. We show that the overall kinetics are highly sensitive to the precise nature of the low energy tail in the activation energy distribution. Our analysis motivates further development of systematic methods to identify and understand the most reactive members of the active site distribution. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914145]
INTRODUCTION
Catalysts are critical in the efficient transformation of conventional and alternative hydrocarbon sources to fuels and chemical products, as well as in new energy technologies such as solar fuels and fuel cells. Many important and longstudied catalysts involve isolated metal centers dispersed on amorphous supports such as silica and silica-alumina. Examples include Mo(VI), W(VI), and Re(VII) sites supported on a variety of oxides for olefin metathesis,1–3 Cr(VI)/silica and Cr(VI)/AlPO4 catalysts for olefin polymerization,4,5 Ti(IV)/silica catalysts for olefin epoxidation,6,7 and oxidesupported V(V) catalysts for methanol partial oxidation.8,9 The active sites of these catalysts are often represented with skeletal formulas like those in Figure 1. For a homogeneous catalyst, the skeletal formula corresponds to the same molecular geometry for each catalyst site. For sites comprised of a metal atom grafted to an amorphous support, each site is slightly different from the others. Many catalysts made in this way are known to have very low fractions of active sites. Nevertheless, some closely related materials involving amorphous supports have recently been described as single-site catalysts.10–14 While this is appropriate in the sense that each site is independent of the others, amorphous supports present many slightly different grafting environments for metal ions, so the sites are unlikely to have identical reactivities. Taylor and others recognized very early that even small structural differences between sites can lead to important activity differences.15–17 In addition to their activity, the differences between sites may also influence adsorption, selectivity, and even cause rate-limiting steps to differ between sites.18 Similarly, Langmuir anticipated non-uniformity among adsorption sites and suggested that isotherms should be summed empirically over individual sites (i) with varying adsorption constants Ki to obtain a generalized coverage θ = Σi Ki P/(1 + Ki P).19 0021-9606/2015/142(10)/104708/5/$30.00
Under vacuum, desorption from a surface occurs once per site as the temperature increases, so the number of first-desorption events can be counted systematically at each temperature. Consequently, temperature-programmed desorption (TPD) can directly probe the distribution of desorption energies for an ensemble of non-uniform sites.20–23 Using a similar approach, van Bokhoven et al. characterized the activation energies for decomposition of pre-adsorbed sarin on γ-alumina.24,25 The study by van Bokhoven et al. was also made possible by the once-per-site nature of the decomposition process. One might anticipate that similar approaches could characterize the distribution of steady-state rates on an ensemble of sites. However, characterizing the distribution of reactivity is fundamentally more difficult because, at each temperature, the steady-state rate is a superposition of the activities for all sites. New imaging techniques can identify highly active sites (or patches of active sites) as sources of products, and these tools may become important for single atom catalysts on amorphous supports.26–28 Traditionally, the only direct probes of non-uniform activity were active-site counting techniques: selective poisoning, radiolabeling, stopped-flow techniques, and reaction-desorption studies like that of van Bokhoven et al.24,25,29 Even these techniques rely on uncertain (and sometimes questionable) assumptions, e.g., that poisons bind
FIG. 1. Skeletal formula for a representative metal atom M with ligands L grafted via O atoms to A atoms in an amorphous support. For specific examples with these bonding topologies, see Vicente et al.3
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selectively to the most active sites. Nevertheless, carefully designed active-site counting experiments often suggest that low fractions, ca. 1%-10%, of sites are responsible for most of the catalytic activity.30–32 Catalysts with low fractions of active sites pose challenges for structural characterization methods, which report primarily on the structures of common sites instead of the rare and highly active sites.33 In the context of the Phillips catalyst, McDaniel noted that such spectra may be highly misleading in efforts to identify mechanisms.4 Ab initio computational studies of amorphous catalysts are also prohibitively difficult. In contrast to crystalline catalyst materials like zeolites, metals, and ordered oxides where computation has dramatically advanced our understanding of structure-activity relationships,34–41 computational advances for amorphous catalysts have been limited by a lack of detailed structural information. We and others have used small cluster models containing specific ring sizes to model amorphous silica and silica-alumina.3,42–47 Other investigators have melted, quenched, and cut silicas and then capped them with hydroxyl groups to model the hydroxyl-terminated surfaces of amorphous silica.48,49 Some investigators have used crystalline structures like zeolites44 or β-cristobalite50,51 to model amorphous silica-alumina and silica, respectively. To some degree, these models will always differ from real amorphous supports and from the processes by which they are made. Typically, one is not certain which reactions will be affected by such structural differences and which reactions will not. Recently, Goldsmith et al. developed an algorithm that quantifies the sensitivity of activation energies to structural differences between active sites without relying on any specific structural model.52,53 Rigid constraints on the peripheral atoms of a small cluster model were included to mimic connections to a solid matrix. Previous approaches also included rigid constraints, but the peripheral atom positions were based on the structures of crystalline materials or on ad hoc considerations. Real amorphous supports present a quenched distribution of peripheral atom positions corresponding to a multitude of possible site environments. Since exhaustively sampling the quenched distribution is not yet possible, the algorithm of Goldsmith et al. instead generates the structures of minimum energy active sites for each activation energy across the range of activation energies.52,53 The algorithm further assumes that site populations decrease with increasing site energy.52,53 The assumption seems reasonable at first inspection but raises an important question: which energies govern these site populations? Candidates include energies of the model sites before the catalyst was grafted, energies of the bare catalyst sites after grafting, energies of the sites in the most abundant surface intermediate (MASI) state, and energies of the sites at some critical point in the catalyst preparation history. This troublesome question could be avoided by beginning with an atomistically detailed model of the amorphous support. Fortunately, structural models are advancing in their complexity, in their fidelity to the actual preparation history, in the quality of the model chemistry, and in the diversity of structural features represented. Ugliengo,54 Johnson,55 and others56 are working toward realistic ab initio models of amorphous silica with variable hydroxyl contents. Adaptive
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kinetic Monte Carlo57 and event-cataloging techniques58,59 have the potential to simulate hydroxyl condensations and structural rearrangements during slow temperature quenches. On the experimental side, Freund and others60–63 have imaged thin layers of amorphous silica and aluminosilicate with atomic-scale resolution. These new imaging capabilities may soon enable direct comparisons between theory and catalysis experiments on surfaces with precisely known structural features. This letter provides a statistical framework for efficiently computing quenched averages64 of kinetic properties such as the overall rate, the activation energy, and the fraction of active sites. The formalism also provides an interpretation of the kinetics in terms of an effective synthesis temperature and a separate catalyst operation temperature. Finally, the statistical framework quantitatively prioritizes those aspects of future calculations that will most strongly influence the predictions.
THEORY
We consider only the simple case in which each active site on a catalyst has the same rate-limiting step and the same most abundant intermediate state, and does not deactivate. Suppose that each site in a quenched structural distribution of sites has a local geometry x and a rate law with power-law dependence on concentrations Ci .65,66 The turnover frequency at each site is k(x) i Ciα , i.e., a product of concentrations and a site-dependent (non-elementary) rate constant of Arrhenius form67 k(x) = A(x) exp[− β∆E(x)],
(1)
where β = 1/k BT, and where the prefactor A(x) may depend on geometry but only weakly depends on temperature. ∆E(x) is the activation energy for the site with structure x and not necessarily the site-averaged activation energy. All of the analysis below could be done instead for a transitionstate theory rate constant,68 but since catalytic reactions are never elementary, the overall rate does not precisely follow transition-state theory. The rate constant averaged over all sites is ⟨k⟩ = dxk(x)ρ(x), (2) where ρ(x) is the normalized probability density for sites of specific geometry x, and where the integration with unspecified bounds is an average ⟨ ⟩ over all site geometries. Wu et al. used Eq. (2) for equilibrium site averages in which different site activities emerge from different patterns of neighboring adsorbates on a Pt(111) surface.69 In this work, the catalyst structure ρ(x) is a quenched-in non-equilibrium distribution. It cannot be obtained with currently existing computational procedures. We assume that ρ(x) is irreversibly set by the temperature and other conditions during catalyst synthesis. We assume that, for high-melting materials like silica and silica-alumina, the distribution ρ(x) does not depend on the temperature T used for catalyst operation. The distribution ρ(x) can be formally projected onto ∆E to obtain
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ρ(∆E) ˜ ≡
dxρ(x)δ[∆E − ∆E(x)].
(3)
We can also define k(∆E) as a conditional average of the rate constant over all sites with activation energy ∆E, −1 k(∆E) ≡ ρ(∆E) ˜ dxρ(x)k(x)δ[∆E − ∆E(x)]. (4) From ρ(∆E) ˜ and k(∆E), all remaining kinetic properties are easily determined. For example, the site-averaged rate constant is ⟨k⟩ = d∆E k(∆E) ρ(∆E). ˜ (5) In this statistical framework, ρ(∆E) ˜ is a density of sites at activation energy ∆E, analogous to the density of states in equilibrium statistical mechanics. Not surprisingly then, the calculations of certain properties resemble familiar calculations from equilibrium statistical mechanics.70 For example, the Arrhenius activation energy is − ∂ ln⟨k⟩/∂ β = ⟨∆E⟩k ,
(6)
where the new, k-weighted average for any property Y is ⟨Y (∆E)⟩k ≡ ⟨k⟩−1 d∆E k(∆E) ρ(∆E)Y ˜ (∆E). (7) The observed (k-weighted) activation energy in (6) may be very different from the activation energy of a typical site. Additionally, the illustrative example below shows that the statistical framework can also quantify the fraction of active sites. ILLUSTRATIVE EXAMPLE
Suppose that a catalyst has a Gaussian (G) distribution of activation energies ρG (∆E) with mean ∆E0 and standard deviation σ. Equation (5) gives ⟨k⟩G = A exp[− β∆E0 + β 2σ 2/2]. The average rate is not the rate at the average activation energy. Instead, the apparent activation energy is − ∂ ln⟨k⟩G/∂ β = ∆E0 − βσ 2.
(8)
Equation (8) highlights the importance of at least two temperatures: the temperature during catalyst synthesis, which influences ρ(∆E), ˜ and the temperature during catalyst operation which determines the activation barriers that can be surmounted. To illustrate realistic sizes of the ∆E0 and σ
parameters, we extract ρ(∆E) ˜ from the site energy vs. activation energy profile for the Mo-catalyzed metallacycle rotation examined by Goldsmith et al.53 If the synthesis temperature is 700 K, then a Boltzmann distribution applied to the previous results gives an activation energy distribution with mean ∆E µ = 53.1 kJ/mol and standard deviation σ = 11.5 kJ/mol. We stress that this is only an illustrative calculation. The synthesized catalyst need not correspond to an equilibrium structure at any temperature. Now, suppose the catalyst is operated near 450 K. Equation (8) predicts that the siteaveraged activation energy is only 20 kJ/mol. This example illustrates the degree to which sites on the low-activation energy tail of the distribution dominate the reactivity: in fact, it suggests that sites three standard deviations below the mean activation energy dominate the observed kinetics. More broadly, it suggests that the effects of quenched disorder might be even larger than the more widely appreciated effects of different density functionals and basis sets. The Gaussian example also illustrates the need to thoroughly characterize the low energy tail of the activation energy distribution. Goldsmith et al. did not actually identify any site with activation energies below 30 kJ/mol, yet the Gaussian approximation says they should be important. Do sites with activation energies three standard deviations below the average even exist? In general, the rate-weighted distribution ⟨k⟩−1 k(∆E) ρ(∆E) ˜ emphasizes low activation energy tails of the distribution, making this tail critical for accurate predictions. For example, consider a uniform block (B) distribution with the same mean and standard deviation as the Gaussian (G) distribution. √ block distribution√ is √ The corresponding ρ˜ B(∆E) = (2σ 3)−1 for ∆E µ − σ 3 < ∆E < ∆E µ + σ 3, and ρ˜ B(∆E) = 0 otherwise. The√site-averaged rate constant is √ ⟨k⟩ B = A exp[− β∆E µ ]sinh[ βσ 3]( βσ 3)−1, which depends very differently on the operating temperature compared to the rate that emerged from the Gaussian distribution. Figure 2 shows that the difference in shape causes a large difference in apparent activation energies for the Gaussian- and blockshaped models. The statistical framework can also be used to predict the results of active site counting experiments. The fraction of sites that are active can be expressed as the fraction of sites f X % which are responsible for X% of the activity. This quantity can be computed in two steps,
FIG. 2. (a) Showing a block-distribution (B) and a Gaussian distribution (G), each with the same mean activation energy ∆E µ and standard deviation σ. The corresponding k-weighted distributions, ⟨k⟩−1k ρ B and ⟨k⟩−1k ρ G , are also shown for the case βσ = 2. (b) The apparent activation energy E a for a block-distribution (B) exceeds that from a Gaussian (G) even though both distributions have the same first moments. The apparent activation energies are expressed in k BT units and thus represent the activation energies that would be extracted from an Arrhenius plot covering a narrow temperature interval.
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computationally. The statistical framework shows that the low activation energy tail in the activation energy distribution is particularly important for predicting kinetic properties. This letter should both motivate and guide the development of new computational algorithms and the modeling of catalysts with isolated sites on amorphous supports. FIG. 3. Illustrating how an activation energy distribution obtained from the distribution of active site structures identifies the fraction of sites responsible for 90% of the reaction.
X% = ⟨k⟩−1
∆E X %
d∆E ρ(∆E)k(∆E), ˜
(9)
−∞
which identifies the appropriate region ∆E < ∆E X % of the activation energy tail as shown in Figure 3. A second integration gives the corresponding fraction of active sites in the tail of the distribution ρ˜ B(∆E), ∆E X % f X% = d∆E ρ(∆E). ˜ (10) −∞
Figures 2 and 3 demonstrate the need for more powerful methods to probe the lowest reaches of the activation energy distribution computationally. Unless we can accurately predict how rapidly ρ˜ B(∆E) decays at low activation energies, we cannot predict overall kinetics, activation energies, or the results of active-site counting experiments. The framework developed here can potentially be generalized to predict averaged kinetics from rate expressions like those that emerge from Langmuir-Hinshelwood-Hougen-Watson analyses.66 Finally, the statistical framework in this letter can help reveal which aspects of a computational model are most critical for obtaining accurate predictions. For example, relative errors in the averaged rate constant, δ ln ⟨k⟩ = ⟨k⟩−1 δ ⟨k⟩, can be expanded (to first order) in the errors δ ρ(x), δ A(x), and δ∆E(x). As might be expected, ⟨k⟩ is most sensitive to errors in the mapping from x to ∆E(x): δ ln ⟨k⟩ = ⟨δ ln ρ(x)⟩k + ⟨δ ln A(x)⟩k − ⟨δ β∆E(x)⟩k . Perhaps less expected, the formalism also shows that all factors are primarily important near the peak in the k-weighted distribution, with errors in other locations being inconsequential. For appropriate systems, it may eventually be possible to relax the assumption that the distribution is quenched-in by the synthesis protocol. The actual distribution of site structures might evolve over time, likely affecting a type of catalyst deactivation. One could, in principle, augment the present framework with time-dependent population balance equations71 for time-dependent distributions ρ(x,t) or for ρ(∆E,t). However, the more immediate challenge, as emphasized in this work, is to understand the initial distributions ρ(x, 0) and ρ(∆E, 0). CONCLUSIONS
Atomistically detailed models of amorphous materials and surfaces are becoming available, and quantitative ab initio studies of catalysis on the quenched-distribution of surface sites will follow. We have outlined (1) how the distribution of activation energies can be constructed from a distribution of active site structures, (2) how apparent activation energies can be extracted from the activation energy distribution, and (3) how active site counting results can be predicted
ACKNOWLEDGMENTS
The authors thank the Department of Energy, Basic Energy Sciences for support under Grant No. DE-FG0203ER15467. 1J. C. Mol, “Olefin metathesis over supported rhenium oxide catalysts,” Catal.
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van der Waals complexes at surface sites in comparison with the experiment,” Chem. Rev. 94, 2095-2160 (1994). 47B. Civalleri, E. Garrone, and P. Ugliengo, “Cage-like clusters as models for the hydroxyls of silica: Ab initio calculation of 1H and 29Si NMR chemical shifts,” Chem. Phys. Lett. 299, 443-450 (1999). 48B. P. Feuston and J. B. Higgins, “Model structures for MCM-41 materials: A molecular dynamics simulation,” J. Phys. Chem. 98, 4459-4462 (1994). 49F. Tielens, C. Gervais, J. Franc, F. Mauri, and D. Costa, “Ab initio study of the hydroxylated surface of amorphous silica: A representative model,” Chem. Mater. 3, 3336-3344 (2008). 50J. Handzlik, “Theoretical investigations of isolated Mo(VI) and Mo(IV) centers of a molybdena-silica catalyst for olefin metathesis,” J. Phys. Chem. C 111, 9337-9348 (2007). 51J. Handzlik and J. Ogonowski, “Structure of isolated molybdenum(VI) and molybdenum(IV) oxide species on silica: periodic and cluster DFT studies,” J. Phys. Chem. C 116, 5571-5584 (2012). 52B. R. Goldsmith, A. M. Fong, and B. Peters, “Understanding reactivity with reduced potential energy landscapes: Recent advances and new directions,” in Reaction Rate Computations: Theories and Applications, edited by K.-L. Han and T.-S. Chu (RSC Publishing, 2013). 53B. R. Goldsmith, E. D. Sanderson, D. Bean, and B. Peters, “Isolated catalyst sites on amorphous supports: A systematic algorithm for understanding heterogeneities in structure and reactivity,” J. Chem. Phys. 138, 204105 (2013). 54P. Ugliengo, M. Sodupe, F. Musso, I. J. Bush, R. Orlando, and R. Dovesi, “Realistic models of hydroxylated amorphous silica surfaces and MCM41 mesoporous material simulated by large-scale periodic B3LYP calculations,” Adv. Mater. 20, 4579-4583 (2008). 55C. S. Ewing, S. Bhavsar, G. Veser, J. J. McCarthy, and J. K. Johnson, “Accurate amorphous silica surface models from first-principles thermodynamics of surface dehydroxylation,” Langmuir 30, 5133-5141 (2014). 56C. Schumacher, J. Gonzalez, P. A. Wright, and N. A. Seaton, “Generation of atomistic models of periodic mesoporous silica by kinetic Monte Carlo simulation of the synthesis of the material,” J. Phys. Chem. B 110, 319-333 (2006). 57G. Henkelman and H. Jonsson, “Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table,” J. Chem. Phys. 115, 9657-9666 (2001). 58L. K. Beland, P. Brommer, F. El-Mellouhi, J.-F. Joly, and N. Mousseau, “Kinetic activation-relaxation technique,” Phys. Rev. E 84, 046704 (2011). 59J.-F. Joly, L. K. Beland, P. Brommer, and N. Mousseau, “Contribution of vacancies to relaxation in amorphous materials: A kinetic activationrelaxation technique study,” Phys. Rev. E 87, 144204 (2013). 60J. A. Boscoboinik, X. Yu, B. Yang, F. D. Fischer, R. Włodarczyk, M. Sierka, S. Shaikhutdinov, J. Sauer, and H.-J. Freund, “Modeling zeolites with metalsupported two-dimensional aluminosilicate films,” Angew. Chem., Int. Ed. 51, 6005-6008 (2012). 61L. Lichtenstein, C. Buchner, B. Yang, S. Shaikhutdinov, M. Heyde, M. Sierka, R. Wlodarczyk, J. Sauer, and H. -J. Freund, “The atomic structure of a metal-supported vitreous thin silica film,” Angew. Chem., Int. Ed. 51, 404-407 (2012). 62L. Lichtenstein, M. Heyde, and H.-J. Freund, “Crystalline-vitreous interface in two dimensional silica,” Phys. Rev. Lett. 109, 106101 (2012). 63P. Y. Huang, S. Kurasch, A. Srivastava, V. Skakalova, J. Kotakoski, A. V. Krasheninnikov, R. Hovden, Q. Mao, J. C. Meyer, J. Smet, D. A. Muller, and U. Kaiser, “Direct imaging of a two-dimensional silica glass on graphene,” Nano Lett. 12, 1081-1086 (2012). 64D. P. Landau and K. Binder, A Guide to Monte-Carlo Simulations in Statistical Physics, 3rd ed. (Cambridge University Press, New York, 2009). 65G. F. Froment and K. B. Bischoff, Chemical Reactor Analysis and Design (Wiley, New York, 1990). 66C. G. Hill and T. W. Root, Introduction to Chemical Engineering Kinetics and Reactor Design (Wiley, Hoboken, NJ, 2014). 67S. Arrhenius, “On the reaction velocity of inversion of cane sugar by acids,” Z. Phys. Chem. 4, 226 (1889). 68H. Eyring, “The activated complex in chemical reactions,” J. Chem. Phys. 3, 107-115 (1935). 69C. Wu, D. J. Schmidt, C. Wolverton, and W. F. Schneider, “Accurate coverage dependence incorporated into first-principles kinetic models: Catalytic NO oxidation on Pt(111),” J. Catal. 286, 88-94 (2012). 70D. Chandler, Introduction to Modern Statistical Mechanics (Oxford Press, New York, 1997). 71N. G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, 1992).
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THE JOURNAL OF CHEMICAL PHYSICS 142, 104708 (2015)
Single atom catalysts on amorphous supports: A quenched disorder perspective Baron Peters1,2 and Susannah L. Scott1,2 1 2
Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, USA
(Received 30 November 2014; accepted 24 February 2015; published online 13 March 2015) Phenomenological models that invoke catalyst sites with different adsorption constants and rate constants are well-established, but computational and experimental methods are just beginning to provide atomically resolved details about amorphous surfaces and their active sites. This letter develops a statistical transformation from the quenched disorder distribution of site structures to the distribution of activation energies for sites on amorphous supports. We show that the overall kinetics are highly sensitive to the precise nature of the low energy tail in the activation energy distribution. Our analysis motivates further development of systematic methods to identify and understand the most reactive members of the active site distribution. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914145]
INTRODUCTION
Catalysts are critical in the efficient transformation of conventional and alternative hydrocarbon sources to fuels and chemical products, as well as in new energy technologies such as solar fuels and fuel cells. Many important and longstudied catalysts involve isolated metal centers dispersed on amorphous supports such as silica and silica-alumina. Examples include Mo(VI), W(VI), and Re(VII) sites supported on a variety of oxides for olefin metathesis,1–3 Cr(VI)/silica and Cr(VI)/AlPO4 catalysts for olefin polymerization,4,5 Ti(IV)/silica catalysts for olefin epoxidation,6,7 and oxidesupported V(V) catalysts for methanol partial oxidation.8,9 The active sites of these catalysts are often represented with skeletal formulas like those in Figure 1. For a homogeneous catalyst, the skeletal formula corresponds to the same molecular geometry for each catalyst site. For sites comprised of a metal atom grafted to an amorphous support, each site is slightly different from the others. Many catalysts made in this way are known to have very low fractions of active sites. Nevertheless, some closely related materials involving amorphous supports have recently been described as single-site catalysts.10–14 While this is appropriate in the sense that each site is independent of the others, amorphous supports present many slightly different grafting environments for metal ions, so the sites are unlikely to have identical reactivities. Taylor and others recognized very early that even small structural differences between sites can lead to important activity differences.15–17 In addition to their activity, the differences between sites may also influence adsorption, selectivity, and even cause rate-limiting steps to differ between sites.18 Similarly, Langmuir anticipated non-uniformity among adsorption sites and suggested that isotherms should be summed empirically over individual sites (i) with varying adsorption constants Ki to obtain a generalized coverage θ = Σi Ki P/(1 + Ki P).19 0021-9606/2015/142(10)/104708/5/$30.00
Under vacuum, desorption from a surface occurs once per site as the temperature increases, so the number of first-desorption events can be counted systematically at each temperature. Consequently, temperature-programmed desorption (TPD) can directly probe the distribution of desorption energies for an ensemble of non-uniform sites.20–23 Using a similar approach, van Bokhoven et al. characterized the activation energies for decomposition of pre-adsorbed sarin on γ-alumina.24,25 The study by van Bokhoven et al. was also made possible by the once-per-site nature of the decomposition process. One might anticipate that similar approaches could characterize the distribution of steady-state rates on an ensemble of sites. However, characterizing the distribution of reactivity is fundamentally more difficult because, at each temperature, the steady-state rate is a superposition of the activities for all sites. New imaging techniques can identify highly active sites (or patches of active sites) as sources of products, and these tools may become important for single atom catalysts on amorphous supports.26–28 Traditionally, the only direct probes of non-uniform activity were active-site counting techniques: selective poisoning, radiolabeling, stopped-flow techniques, and reaction-desorption studies like that of van Bokhoven et al.24,25,29 Even these techniques rely on uncertain (and sometimes questionable) assumptions, e.g., that poisons bind
FIG. 1. Skeletal formula for a representative metal atom M with ligands L grafted via O atoms to A atoms in an amorphous support. For specific examples with these bonding topologies, see Vicente et al.3
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selectively to the most active sites. Nevertheless, carefully designed active-site counting experiments often suggest that low fractions, ca. 1%-10%, of sites are responsible for most of the catalytic activity.30–32 Catalysts with low fractions of active sites pose challenges for structural characterization methods, which report primarily on the structures of common sites instead of the rare and highly active sites.33 In the context of the Phillips catalyst, McDaniel noted that such spectra may be highly misleading in efforts to identify mechanisms.4 Ab initio computational studies of amorphous catalysts are also prohibitively difficult. In contrast to crystalline catalyst materials like zeolites, metals, and ordered oxides where computation has dramatically advanced our understanding of structure-activity relationships,34–41 computational advances for amorphous catalysts have been limited by a lack of detailed structural information. We and others have used small cluster models containing specific ring sizes to model amorphous silica and silica-alumina.3,42–47 Other investigators have melted, quenched, and cut silicas and then capped them with hydroxyl groups to model the hydroxyl-terminated surfaces of amorphous silica.48,49 Some investigators have used crystalline structures like zeolites44 or β-cristobalite50,51 to model amorphous silica-alumina and silica, respectively. To some degree, these models will always differ from real amorphous supports and from the processes by which they are made. Typically, one is not certain which reactions will be affected by such structural differences and which reactions will not. Recently, Goldsmith et al. developed an algorithm that quantifies the sensitivity of activation energies to structural differences between active sites without relying on any specific structural model.52,53 Rigid constraints on the peripheral atoms of a small cluster model were included to mimic connections to a solid matrix. Previous approaches also included rigid constraints, but the peripheral atom positions were based on the structures of crystalline materials or on ad hoc considerations. Real amorphous supports present a quenched distribution of peripheral atom positions corresponding to a multitude of possible site environments. Since exhaustively sampling the quenched distribution is not yet possible, the algorithm of Goldsmith et al. instead generates the structures of minimum energy active sites for each activation energy across the range of activation energies.52,53 The algorithm further assumes that site populations decrease with increasing site energy.52,53 The assumption seems reasonable at first inspection but raises an important question: which energies govern these site populations? Candidates include energies of the model sites before the catalyst was grafted, energies of the bare catalyst sites after grafting, energies of the sites in the most abundant surface intermediate (MASI) state, and energies of the sites at some critical point in the catalyst preparation history. This troublesome question could be avoided by beginning with an atomistically detailed model of the amorphous support. Fortunately, structural models are advancing in their complexity, in their fidelity to the actual preparation history, in the quality of the model chemistry, and in the diversity of structural features represented. Ugliengo,54 Johnson,55 and others56 are working toward realistic ab initio models of amorphous silica with variable hydroxyl contents. Adaptive
J. Chem. Phys. 142, 104708 (2015)
kinetic Monte Carlo57 and event-cataloging techniques58,59 have the potential to simulate hydroxyl condensations and structural rearrangements during slow temperature quenches. On the experimental side, Freund and others60–63 have imaged thin layers of amorphous silica and aluminosilicate with atomic-scale resolution. These new imaging capabilities may soon enable direct comparisons between theory and catalysis experiments on surfaces with precisely known structural features. This letter provides a statistical framework for efficiently computing quenched averages64 of kinetic properties such as the overall rate, the activation energy, and the fraction of active sites. The formalism also provides an interpretation of the kinetics in terms of an effective synthesis temperature and a separate catalyst operation temperature. Finally, the statistical framework quantitatively prioritizes those aspects of future calculations that will most strongly influence the predictions.
THEORY
We consider only the simple case in which each active site on a catalyst has the same rate-limiting step and the same most abundant intermediate state, and does not deactivate. Suppose that each site in a quenched structural distribution of sites has a local geometry x and a rate law with power-law dependence on concentrations Ci .65,66 The turnover frequency at each site is k(x) i Ciα , i.e., a product of concentrations and a site-dependent (non-elementary) rate constant of Arrhenius form67 k(x) = A(x) exp[− β∆E(x)],
(1)
where β = 1/k BT, and where the prefactor A(x) may depend on geometry but only weakly depends on temperature. ∆E(x) is the activation energy for the site with structure x and not necessarily the site-averaged activation energy. All of the analysis below could be done instead for a transitionstate theory rate constant,68 but since catalytic reactions are never elementary, the overall rate does not precisely follow transition-state theory. The rate constant averaged over all sites is ⟨k⟩ = dxk(x)ρ(x), (2) where ρ(x) is the normalized probability density for sites of specific geometry x, and where the integration with unspecified bounds is an average ⟨ ⟩ over all site geometries. Wu et al. used Eq. (2) for equilibrium site averages in which different site activities emerge from different patterns of neighboring adsorbates on a Pt(111) surface.69 In this work, the catalyst structure ρ(x) is a quenched-in non-equilibrium distribution. It cannot be obtained with currently existing computational procedures. We assume that ρ(x) is irreversibly set by the temperature and other conditions during catalyst synthesis. We assume that, for high-melting materials like silica and silica-alumina, the distribution ρ(x) does not depend on the temperature T used for catalyst operation. The distribution ρ(x) can be formally projected onto ∆E to obtain
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J. Chem. Phys. 142, 104708 (2015)
ρ(∆E) ˜ ≡
dxρ(x)δ[∆E − ∆E(x)].
(3)
We can also define k(∆E) as a conditional average of the rate constant over all sites with activation energy ∆E, −1 k(∆E) ≡ ρ(∆E) ˜ dxρ(x)k(x)δ[∆E − ∆E(x)]. (4) From ρ(∆E) ˜ and k(∆E), all remaining kinetic properties are easily determined. For example, the site-averaged rate constant is ⟨k⟩ = d∆E k(∆E) ρ(∆E). ˜ (5) In this statistical framework, ρ(∆E) ˜ is a density of sites at activation energy ∆E, analogous to the density of states in equilibrium statistical mechanics. Not surprisingly then, the calculations of certain properties resemble familiar calculations from equilibrium statistical mechanics.70 For example, the Arrhenius activation energy is − ∂ ln⟨k⟩/∂ β = ⟨∆E⟩k ,
(6)
where the new, k-weighted average for any property Y is ⟨Y (∆E)⟩k ≡ ⟨k⟩−1 d∆E k(∆E) ρ(∆E)Y ˜ (∆E). (7) The observed (k-weighted) activation energy in (6) may be very different from the activation energy of a typical site. Additionally, the illustrative example below shows that the statistical framework can also quantify the fraction of active sites. ILLUSTRATIVE EXAMPLE
Suppose that a catalyst has a Gaussian (G) distribution of activation energies ρG (∆E) with mean ∆E0 and standard deviation σ. Equation (5) gives ⟨k⟩G = A exp[− β∆E0 + β 2σ 2/2]. The average rate is not the rate at the average activation energy. Instead, the apparent activation energy is − ∂ ln⟨k⟩G/∂ β = ∆E0 − βσ 2.
(8)
Equation (8) highlights the importance of at least two temperatures: the temperature during catalyst synthesis, which influences ρ(∆E), ˜ and the temperature during catalyst operation which determines the activation barriers that can be surmounted. To illustrate realistic sizes of the ∆E0 and σ
parameters, we extract ρ(∆E) ˜ from the site energy vs. activation energy profile for the Mo-catalyzed metallacycle rotation examined by Goldsmith et al.53 If the synthesis temperature is 700 K, then a Boltzmann distribution applied to the previous results gives an activation energy distribution with mean ∆E µ = 53.1 kJ/mol and standard deviation σ = 11.5 kJ/mol. We stress that this is only an illustrative calculation. The synthesized catalyst need not correspond to an equilibrium structure at any temperature. Now, suppose the catalyst is operated near 450 K. Equation (8) predicts that the siteaveraged activation energy is only 20 kJ/mol. This example illustrates the degree to which sites on the low-activation energy tail of the distribution dominate the reactivity: in fact, it suggests that sites three standard deviations below the mean activation energy dominate the observed kinetics. More broadly, it suggests that the effects of quenched disorder might be even larger than the more widely appreciated effects of different density functionals and basis sets. The Gaussian example also illustrates the need to thoroughly characterize the low energy tail of the activation energy distribution. Goldsmith et al. did not actually identify any site with activation energies below 30 kJ/mol, yet the Gaussian approximation says they should be important. Do sites with activation energies three standard deviations below the average even exist? In general, the rate-weighted distribution ⟨k⟩−1 k(∆E) ρ(∆E) ˜ emphasizes low activation energy tails of the distribution, making this tail critical for accurate predictions. For example, consider a uniform block (B) distribution with the same mean and standard deviation as the Gaussian (G) distribution. √ block distribution√ is √ The corresponding ρ˜ B(∆E) = (2σ 3)−1 for ∆E µ − σ 3 < ∆E < ∆E µ + σ 3, and ρ˜ B(∆E) = 0 otherwise. The√site-averaged rate constant is √ ⟨k⟩ B = A exp[− β∆E µ ]sinh[ βσ 3]( βσ 3)−1, which depends very differently on the operating temperature compared to the rate that emerged from the Gaussian distribution. Figure 2 shows that the difference in shape causes a large difference in apparent activation energies for the Gaussian- and blockshaped models. The statistical framework can also be used to predict the results of active site counting experiments. The fraction of sites that are active can be expressed as the fraction of sites f X % which are responsible for X% of the activity. This quantity can be computed in two steps,
FIG. 2. (a) Showing a block-distribution (B) and a Gaussian distribution (G), each with the same mean activation energy ∆E µ and standard deviation σ. The corresponding k-weighted distributions, ⟨k⟩−1k ρ B and ⟨k⟩−1k ρ G , are also shown for the case βσ = 2. (b) The apparent activation energy E a for a block-distribution (B) exceeds that from a Gaussian (G) even though both distributions have the same first moments. The apparent activation energies are expressed in k BT units and thus represent the activation energies that would be extracted from an Arrhenius plot covering a narrow temperature interval.
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computationally. The statistical framework shows that the low activation energy tail in the activation energy distribution is particularly important for predicting kinetic properties. This letter should both motivate and guide the development of new computational algorithms and the modeling of catalysts with isolated sites on amorphous supports. FIG. 3. Illustrating how an activation energy distribution obtained from the distribution of active site structures identifies the fraction of sites responsible for 90% of the reaction.
X% = ⟨k⟩−1
∆E X %
d∆E ρ(∆E)k(∆E), ˜
(9)
−∞
which identifies the appropriate region ∆E < ∆E X % of the activation energy tail as shown in Figure 3. A second integration gives the corresponding fraction of active sites in the tail of the distribution ρ˜ B(∆E), ∆E X % f X% = d∆E ρ(∆E). ˜ (10) −∞
Figures 2 and 3 demonstrate the need for more powerful methods to probe the lowest reaches of the activation energy distribution computationally. Unless we can accurately predict how rapidly ρ˜ B(∆E) decays at low activation energies, we cannot predict overall kinetics, activation energies, or the results of active-site counting experiments. The framework developed here can potentially be generalized to predict averaged kinetics from rate expressions like those that emerge from Langmuir-Hinshelwood-Hougen-Watson analyses.66 Finally, the statistical framework in this letter can help reveal which aspects of a computational model are most critical for obtaining accurate predictions. For example, relative errors in the averaged rate constant, δ ln ⟨k⟩ = ⟨k⟩−1 δ ⟨k⟩, can be expanded (to first order) in the errors δ ρ(x), δ A(x), and δ∆E(x). As might be expected, ⟨k⟩ is most sensitive to errors in the mapping from x to ∆E(x): δ ln ⟨k⟩ = ⟨δ ln ρ(x)⟩k + ⟨δ ln A(x)⟩k − ⟨δ β∆E(x)⟩k . Perhaps less expected, the formalism also shows that all factors are primarily important near the peak in the k-weighted distribution, with errors in other locations being inconsequential. For appropriate systems, it may eventually be possible to relax the assumption that the distribution is quenched-in by the synthesis protocol. The actual distribution of site structures might evolve over time, likely affecting a type of catalyst deactivation. One could, in principle, augment the present framework with time-dependent population balance equations71 for time-dependent distributions ρ(x,t) or for ρ(∆E,t). However, the more immediate challenge, as emphasized in this work, is to understand the initial distributions ρ(x, 0) and ρ(∆E, 0). CONCLUSIONS
Atomistically detailed models of amorphous materials and surfaces are becoming available, and quantitative ab initio studies of catalysis on the quenched-distribution of surface sites will follow. We have outlined (1) how the distribution of activation energies can be constructed from a distribution of active site structures, (2) how apparent activation energies can be extracted from the activation energy distribution, and (3) how active site counting results can be predicted
ACKNOWLEDGMENTS
The authors thank the Department of Energy, Basic Energy Sciences for support under Grant No. DE-FG0203ER15467. 1J. C. Mol, “Olefin metathesis over supported rhenium oxide catalysts,” Catal.
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