Article pubs.acs.org/JPCA
Role of Carbon−Carbon Phenyl Migration in the Pyrolysis Mechanism of β‑O‑4 Lignin Model Compounds: Phenethyl Phenyl Ether and α‑Hydroxy Phenethyl Phenyl Ether Ariana Beste*,† and A. C. Buchanan, III‡ †
Joint Institute for Computational Sciences, The University of Tennessee, Oak Ridge, Tennessee 37831, United States Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States
‡
S Supporting Information *
ABSTRACT: We investigate phenyl shift and subsequent β-scission reactions for PhCHXCH·OPh [X = H, OH], which are part of the pyrolysis mechanism of phenethyl phenyl ether (PPE) and α-hydroxy PPE. PPE and its derivatives are model compounds for the most common linkage in lignin, the β-O-4 linkage. We use density functional theory to locate transition states and equilibrium structures and kinetic Monte Carlo in combination with transitionstate theory for kinetic simulations. Oxygen−carbon and carbon−carbon phenyl shift reactions proceed through cyclic intermediates with similar barriers. However, while subsequent β-scission of the oxygen−carbon shift products proceeds with virtually no barrier, the activation energy for β-scission of the carbon−carbon shift products exceeds 15 kcal/mol. We found that about 15% of β-radical conversion can be attributed to carbon−carbon shift for PPE and α-hydroxy PPE at 618 K. Whereas the oxygen−carbon shift reaction has been established as an integral part of the pyrolysis mechanism of PPE and its derivatives, participation of the carbon−carbon shift reaction has not been shown previously.
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INTRODUCTION Providing sustainable energy sources is one of the most important tasks of our time. Biomass lends itself as a promising alternative precursor for fuels and chemicals; it is renewable and reduces our carbon footprint during its life cycle.1−3 Whereas the production of bioethanol and biodiesel from glucosecontaining polymers in plant biomass is an established industrial process, large-scale applications for lignin, which is the third major component in biomass, are lacking. Lignin is an amorphous, three-dimensional polyphenolic substance whose exact structure is very difficult to resolve and depends on the origin and isolation technique. Often, model compounds are utilized to obtain insight into the mechanistic details of lignin conversion.4−7 The use of model compounds has also opened the opportunity for computational methods to be successfully applied to the study of lignin chemistry. A variety of groups have researched bond breaking in dilignols,8−12 which is relevant to conversion processes, while others have investigated the coupling of monolignols13−17 to understand what governs lignin formation and linkage distribution in vivo. Furthermore, mechanical properties,18 NMR shifts,19 binding energies in ionic liquids,20,21 and reaction energies and barriers for addition reactions with transition metals22 have been determined for lignin model compounds using computational methods. Lignin models are characterized by the linkage that they represent. The β-O-4 ether linkage is the most common linkage © 2012 American Chemical Society
and is modeled by phenethyl phenyl ether (PhCH2CH2OPh, PPE). Several mechanisms were proposed for the thermolysis of PPE.23−27 In contrast to pyrolysis experiments where short residence time and low concentration lead to the preference of unimolecular reactions,11,27 at moderate temperatures (300− 450 °C) and concentrations, the pyrolysis of PPE proceeds via a free radical chain mechanism.28 For the latter experimental setup, product distributions and substrate conversion rates were recorded as a function of naturally occurring substituents on PPE.6 In particular, the α/β-selectivity was measured, which is the ratio of product formation through the α- and β-pathways of hydrogen abstraction to form the corresponding radical intermediates, PhCH·CH2OPh and PhCH2CH·OPh. Paralleling the experimental work, we used computational methods to study key mechanistic steps involved in the pyrolysis of PPE and oxygen-substituted PPE,29 including homolytic cleavage,30,31 hydrogen abstraction,32−34 and oxygen−carbon phenyl migration.35 In our previously reported work, substituents were located on aromatic rings. In yet unpublished work, A. C. Buchanan, III and co-workers have examined the thermal decomposition of α-hydroxy PPE, where a hydroxy substituent is situated on the aliphatic ether bridge. New products were identified indicating that a carbon− Received: October 22, 2012 Revised: November 28, 2012 Published: November 29, 2012 12242
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Figure 1. Shift reactions of β-radical intermediates in the pyrolysis of PPE and α-hydroxy PPE with subsequent β-scission reactions investigated in this work.
migration of the β-radical intermediate that is formed by hydrogen abstraction during PPE pyrolysis.35 Rate constants determined in that study are used as part of the input for the kinetic Monte Carlo simulations discussed herein. All phenyl shift reactions proceed through intermediates, for which multiple conformers are located. Prior to intermediate formation, prestructures are formed, which are distinct from the β-radicals by conformational change. We take into account reaction pathways leading to each distinct intermediate and pathways leading from each intermediate to the shift products, which we call reaction channels in the following. We assume that conformational changes do not occur before β-scission. βscission yields postcomplexes before dissociation to final products. Rate constants for elementary reaction steps are calculated with our own Python code, which is interfaced to the NWChem program package. We apply transition-state theory39 and include a Wigner tunneling correction40 to compute unimolecular reaction rate constants. Anharmonic effects are incorporated for low-frequency vibrations up to 110 cm−1, which corresponds to about five modes per transition state and equilibrium structure. Within the independent mode approximation, the anharmonic potentials are obtained by displacement along the normal modes. Nine energy points per mode are used for a fourth-order polynomial fit. The anharmonic vibrational partition functions are computed with the semiclassical Wigner−Kirkwood approximation, as outlined in ref 34. The commonly used hindered rotor approximation for anharmonic motion is only applicable if hindered rotation and low-frequency vibration are separable, which is typically not the case in medium-sized organic molecules such as the lignin model compounds considered here. Prestructures for intermediate formation during the shift reactions are within about 2 kcal/mol of the β-radicals. Assuming corresponding low barriers, we impose equilibrium conditions to prestructure generation. The forward rate constant for the shift reaction can then be expressed as k′f = kfKpre, where kf is the forward rate constant for the shift reaction with the prestructure as the reactant and Kpre is the equilibrium constant for the prestructure formation from the β-radical. We also assume thermodynamic equilibrium between the poststructures of the β-scission reactions and the final products because dissociation is barrierless. The back reaction of the βscission is expressed as kb′ = kb/Kpost, where kb is the rate
carbon shift reaction is also taking place, which has previously not been reported.28 For unsubstituted PPE, products of the new shift reaction are identical to products of the α-pathway. Because the α/β-selectivity is measured as the product ratio through the α- and β-pathways, the participation of the carbon−carbon shift reaction would erroneously increase the experimental result. The goal of the work presented here is to understand the role and significance of the carbon−carbon shift reaction in the pyrolysis mechanism of α-hydroxy PPE and PPE. For PPE, this information is unattainable by experiment, without perhaps the use of tedious isotopic labeling experiments, which highlights the significance of computational research in combination with experimental guidance. In this article, we use density functional theory (DFT) to study carbon−carbon and oxygen−carbon phenyl migration in the β-radical intermediates resulting from hydrogen abstraction on PPE and α-hydroxy PPE during pyrolysis. Because βscission reactions follow the rearrangements to yield final products and because they are potentially the rate-determining steps in the rearrangement process, we also investigate the βscission reactions. Rate constants calculated with transitionstate theory are input into kinetic Monte Carlo simulations.
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COMPUTATIONAL DETAILS
All electronic structure calculations are carried out with the NWChem program package.36 We employ unrestricted density functional theory, specifically, the M06-2X kinetic functional.37 We use the NWChem “xfine” integration grid to avoid inaccuracies due to the grid size.38 Geometry optimizations and frequency calculations are performed with a mixed doubleζ basis set.34 Frequency analysis shows one imaginary mode for all transition states except one, even though we set the convergence criteria for the saddle point search to tight (see NWChem36 manual). However, displacement along the imaginary mode with frequency 17.8i cm−1 reveals that the structure is at a minimum along that mode. The corresponding potential is shown in the Supporting Information. Final energies were obtained with the 6-311++G** basis set. Reaction energies and barriers given in the reaction profile are zero-point-corrected. Partial atomic charges were computed by a fitting procedure of the quantum mechanical electrostatic potential on selected grid points. Figure 1 shows the reactions included in this work. Previously, we investigated the oxygen−carbon phenyl 12243
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Figure 2. Transition states and equilibrium structures for the reaction channels of the carbon−carbon phenyl shift and subsequent β-scission reactions of the β-radical of α-hydroxy PPE. CCf: carbon−carbon bond that is formed during the shift reaction; CCb: carbon−carbon bond that is broken during the shift reaction; CCr: carbon−carbon bond that is not changed during shift reaction and is part of the epoxy ring; COb: carbon− oxygen bond that is broken during β-scission; bond distances in Å.
constant for the back reaction of the β-scission reactions with the poststructure as the reactant and Kpost is the equilibrium constant for poststructure dissociation. We use the kinetic Monte Carlo code SPPARKS,41 which was developed at the SANDIA National Laboratory, for the kinetic simulations. The simulation temperature of 618 K is chosen to allow for comparison to experimental data on the pyrolysis of α-hydroxy PPE. The volume of the simulation cell is assigned to 2.0 × 10−14 L. Initial molecule count of the PPE β-radical is set to 50 × 103, and initial molecule counts of the two possible conformers of the β-radical of α-hydroxy PPEs are
set to 25 × 103 each. This corresponds to a radical concentration of 4.2 × 10−6 M. The PPE simulation was run for 1.0 μs, and the α-hydroxy PPE simulation was run for 4.3 μs. Product ratios are not sensitive to the variation of the simulation cell volume or the initial molecule count. The relatively high initial radical count was chosen for convenience (resulting in a visually appealing smooth graph). In particular, increasing the simulation volume and, thereby, reducing the radical concentration leads to identical simulation results. All rate and equilibrium constants used in the simulation are given in the Supporting Information. Additionally, we provide a list of 12244
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Figure 3. Reaction profile of the oxygen−carbon and carbon−carbon shift reactions of the β-radical intermediates in the pyrolysis of PPE and αhydroxy PPE; energy differences are zero-point-corrected and given in kcal/mol. Reaction channels with highest contribution to product formation are shown: channel 2 for the carbon−carbon shift PPE; channel 4 for the carbon−carbon shift α-hydroxy PPE; channel 3 for the oxygen−carbon shift PPE; and channel 6 for the oxygen−carbon shift α-hydroxy PPE.
pyrolysis of α-hydroxy PPE. Participating species are depicted in the Supporting Information. Similar to the oxygen−carbon shift reaction, the carbon− carbon shift reaction of the PPE β-radical proceeds through an intermediate. In the intermediate, a three-membered ring of carbon atoms is formed. Opening of the ring leads to a shortening of the ether bridge by one carbon atom, which has become a side chain and is the new radical center. We located two distinct conformers for the intermediate resulting in two reaction channels, as shown in the Supporting Information. Again, due to the existence of two conformers for the β-radical formed from α-hydroxy PPE, the number of distinct intermediate conformers and the number of reaction channels double. Transition states, intermediates, and prestructures for the different channels of the carbon−carbon shift reaction of the β-radical of α-hydroxy PPE are displayed in Figure 2. Following oxygen−carbon and carbon−carbon shift reactions, β-scission takes place. In the oxygen−carbon shift products, the carbon−carbon bond next to the oxygen radical center cleaves, while in the carbon−carbon shift products, the oxygen−carbon bond breaks. Prior to dissociation into final products, postcomplexes are formed. Dissociation of the postcomplexes into benzaldehyde and benzyl radical (oxygen−carbon shift) and styrene and phenoxy (carbon−carbon shift) proceeds without a barrier. Transition states and postcomplexes for the β-scission reactions are included in the Supporting Information and in Figure 2. Figure 3 displays the reaction profile for the shift reactions with subsequent β-scission reactions. For each reaction, only the channel with the highest contribution to product formation
prefactors and activation energies in the Supporting Information, which were obtained from Arrhenius plots in the temperature range of 580−660 K for the β-scission and the phenyl migration reactions, where we apply steady-state conditions for the latter.
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RESULTS AND DISCUSSION The reactions that are discussed in the following are given in Figure 1. This choice is based on experimental evidence. In particular, the β-radical of PPE (and aryl-substituted PPEs) does not have a competitive β-scission path because this would require the cleavage of an sp2−sp3 bond generating a phenyl radical. In previous work,35 we investigated the oxygen−carbon phenyl shift reaction of the β-radical intermediate resulting from hydrogen abstraction on PPE. We located three distinct conformers for the shift intermediates with corresponding transition states. For completeness, these structures are given in the Supporting Information. Previously calculated reaction barriers for the oxygen−carbon shift reaction of the PPE βradical were in good agreement to published data on the rearrangement of the 1,1-diphenylethoxyl radical.42 The reported first barrier, intermediate energy, and second barrier are 9.9, 7.3, and 10.7 kcal/mol, respectively, with corresponding values in ref 35 of 8.1, 5.2, and 11.1 kcal/mol. Hydrogen abstraction on the β-carbon of α-hydroxy PPE can occur at the cis or trans position to the hydroxy substituent, resulting in two conformers for the corresponding β-radical intermediate. This yields six phenyl shift intermediate conformers and, therefore, six reaction channels for the oxygen− carbon phenyl migration of the β-radical intermediate in the 12245
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is shown. We observe that the oxygen−carbon and the carbon− carbon shift reactions have comparable reaction profiles. The carbon−carbon shift reaction has a lower first barrier by 2.8 kcal/mol for the β-radical of PPE and by 2.6 kcal/mol for the βradical of α-hydroxy PPE than the oxygen−carbon shift reaction. Because the rate constants to cross the second barrier are much larger than the rate constants for intermediate formation, we would expect similar participation of the oxygen−carbon and carbon−carbon shift reactions in the pyrolysis mechanism judging from the shift reactions alone. For instance, the steady-state rate constant at 618 K for the lowest channel of the carbon−carbon shift reaction of the β-radical of PPE is 2.2 × 106s−1, and the corresponding rate constant for the oxygen−carbon shift reaction is 5.4 × 106 s−1. However, the continuation of the reaction profile in Figure 3, which depicts the β-scission reactions, shows a big difference between the reaction paths following the shift reactions. Whereas the oxygen−carbon reaction path has virtually no barrier for the βscission reaction, the barrier for β-scission of the alternative carbon−carbon reaction path has a barrier of 17.5 kcal/mol for the β-radical of PPE and 15.4 kcal/mol for the β-radical of αhydroxy PPE for the reaction channels shown in Figure 3. (The negative barrier for the β-scission of the oxygen−carbon shift product of the β-radical of PPE is an artifact of the zero-point correction, for which a harmonic correction is used. Rate constants are calculated using anharmonic corrections.) On the basis of the results for the β-scission reactions, we expect that the major contribution to the pyrolysis mechanism stems from the originally proposed28 oxygen−carbon shift reaction. Along the reaction profile, we notice that the α-hydroxy substituent affects reaction barriers and energy differences between equilibrium structures. The largest increase in energy by the substituent is observed for the intermediate of the carbon−carbon shift reaction. Shift products and third transition states are stabilized by the α-hydroxy substituent to a similar degree, resulting in comparable third barriers for the βscission reactions of PPE and α-hydroxy PPE shift products. Final products are also stabilized. Qualitatively, the same conclusions can be drawn for the reaction paths of the β-radical of PPE and α-hydroxy PPE. Further, we investigate why there is such a big difference between the β-scission transition states of the carbon−carbon and oxygen−carbon shift products. Because the α-hydroxy substituent induces a similar energetic shift in the migration products and in the β-scission transition states, we analyze the corresponding β-scission transition states for PPE only. The spin densities of TS3 for the reaction channels with the highest contribution to product formation for the β-scission reactions of the carbon−carbon shift and the oxygen−carbon shift product are given in the Supporting Information and do not reveal a significant difference. In both transition states, the single electron is located at the radical center and delocalizes into the opposite phenyl ring through the breaking bond. We then calculated partial charges for the transition states and the shift products. The difference between the partial charges in the transition state and the shift product indicates charge polarization during β-scission. We added the charge differences within each fragment of the transition states, which are divided according to the separation into final products. The summed partial charge differences of the molecular fragments are shown in Figure 4. In the transition states, polarization occurs through partial electron donation from the fragment that carries the single electron in the final products to the fragment that
Figure 4. Partial charge differences in electrons between transition states and shift products within molecular fragments. Shaded areas indicate the molecular fragments within which the partial charge differences have been added: blue: electron gain; red: electron loss; left: TS3, channel 2, oxygen−carbon shift; right: TS3, channel 3, carbon−carbon shift for the PPE β-radical.
contains a double bond in the final product. We rationalize that the higher electronegativity of the oxygen atom compared to that of the carbon atom in the electron-receiving fragment in the transition state of the oxygen−carbon shift product facilitates the polarization. In contrast, the oxygen atom in the transition state of the carbon−carbon shift product is located in the electron-donating fragment. All reaction channels shown in the Supporting Information and in Figure 2 for the shift and subsequent β-scission reactions have been used in the kinetic Monte Carlo simulations at 618 K. The depletion of the β-radicals of PPE and α-hydroxy PPE and the formation of benzaldehyde and styrene as a function of time are displayed in Figures 5 and 6. Benzyl and phenoxy
Figure 5. Mole fraction as a function of time of the reactant β-radical of PPE (bPPE), of the product of the oxygen−carbon shift pathway (benzaldehyde), of the product of the carbon−carbon shift pathway (styrene), and of the carbon−carbon shift products (shift products 1 and 2) at 618 K.
radicals are formed at the same rate as benzaldehyde and styrene, respectively, and are not included in the graphs. The simulations have been started with equivalent molecule counts of β-radicals of PPE and α-hydroxy PPE (the β-radical of αhydroxy PPE is represented by 50% of each conformer). The molecule count of the shift intermediates, the oxygen−carbon shift products, and the pre- and poststructures stays zero during the simulation. We observe a small molecule count of the carbon−carbon shift product for the β-radical of PPE at the beginning of the simulation (see Figure 5), which did not 12246
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Table 1. Zero-Point-Corrected, 0 K Energy Barriers (Forward) in kcal/mol for All Channels Included in the Simulation of the Oxygen−Carbon and Carbon−Carbon Shift Reactions with Subsequent β-Scission for the β-Radical of PPE and α-Hydroxy PPE, Overall Contribution to Product Formation at 618 K channel
TS1
TS2
TS3a
contribution [%]
Oxygen−Carbon Shift and β-Scission Reactions: PPE 1 19.9 2.4 4.5 9.3 2 19.5 2.5 5.7 29.8 3 17.3 2.9 −0.6 60.9 Oxygen−Carbon Shift and β-Scission Reactions: α-Hydroxy PPE 1 20.9 2.3 0.7 2.8 2 22.0 1.9 −0.2 2.8 3 22.0 1.1 0.6 1.2 4 17.1 2.4 0.4 43.4 5 20.9 1.1 −0.3 4.0 6 18.7 2.0 0.1 45.8 Carbon−Carbon Shift and β-Scission Reactions: PPE 1 20.1 4.0 17.8 27.8 2 15.3 5.4 17.5 72.2 Carbon−Carbon Shift and β-Scission Reactions: α-Hydroxy PPE 1 21.2 0.2 14.7 37.8 2 21.2 1.5 14.5 13.0 3 15.7 5.4 15.4 12.9 4 15.6 3.5 15.4 40.0
Figure 6. Mole fraction as a function of time of the two reactant conformers of the β-radical of α-hydroxy PPE (bPPE(OH)1 and 2), of the product of the oxygen−carbon shift pathway (benzaldehyde), of the product of the carbon−carbon shift pathway (styrene(OH)), and of the carbon−carbon shift products (shift products 1, 2, 3, and 4) at 618 K.
appear for the β-radical of α-hydroxy PPE (see Figure 6). Overall reaction time is shorter for the β-radical of PPE than that for α-hydroxy PPE. This is due to the fact that intermediate formation during the carbon−carbon as well as the oxygen−carbon shift reaction is faster for the β-radical of PPE. Participation of the oxygen−carbon shift reaction versus the carbon−carbon shift reaction in the pyrolysis mechanism of PPE is, with 86 versus 14%, predicted to be similar to the participation in the pyrolysis mechanism of α-hydroxy PPE, with 85 versus 15%. The selectivity of the oxygen−carbon versus carbon−carbon shift reaction for α-hydroxy PPE (85:15) agrees quite well with preliminary pyrolysis results of Buchanan and co-workers, which gives a selectivity of ∼80:20. Table 1 provides the reaction barriers for the shift and βscission reactions for all conformers included in the simulations. Also given are the contributions of each conformer to product formation. There is a strong correlation between the barrier for intermediate formation and conformer participation. An exception is channel 3 of the carbon−carbon shift reaction of the β-radical of α-hydroxy PPE, which has a low contribution to product formation and is superseded by channel 1. This is due to the relatively high barrier of shift product formation (TS2) for channel 3 compared to that of channel 1. Finally, we would like to point out that the final products of the simulations include benzyl and phenoxy radicals. In the complete pyrolysis mechanism, these radicals abstract hydrogen atoms on PPE and α-hydroxy PPE at different rates. This potentially affects the distribution between products formed through the carbon−carbon pathway, where phenoxy is produced, and products obtained through the oxygen−carbon pathway, which yields benzyl. However, the rate constant of hydrogen abstraction on PPE is 5 orders of magnitude slower and the rate constant for hydrogen addition to the β-radical (backward reaction) is 3 orders of magnitude slower32 than βradical conversion through the shift reactions. As a consequence, hydrogen abstraction is largely decoupled from subsequent shift and β-scission reactions, and conclusions from this work will hold when hydrogen abstraction is taken into account in the simulation of PPE pyrolysis.
a
Negative barriers are an artifact of the zero-point correction for which a harmonic approximation is used. Rate constants are calculated using anharmonic correction.
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CONCLUSIONS Using kinetic Monte Carlo based on transition-state rate constants, we simulated the oxygen−carbon and carbon− carbon shift reactions with subsequent β-scission of the βradical of PPE and α-hydroxy PPE, which are model compounds of the β-O-4 linkage in lignin. While the oxygen−carbon shift reaction has been identified as being part of the pyrolysis mechanism of PPE,28 the carbon−carbon shift reaction yields products identical to products of the αpathway, where hydrogen is abstracted on the α-carbon of PPE, and could, therefore, not be recognized by experiment. We found that about 15% of β-radical conversion can be attributed to carbon−carbon shift for PPE and α-hydroxy PPE at 618 K. This value is in good agreement with preliminary experimental data from the pyrolysis of α-hydroxy PPE. In addition to benzaldehyde and styrene, benzyl and phenoxy radicals are βscission products. Those react further, abstracting hydrogen on PPE and α-hydroxy PPE. Because hydrogen abstraction is a few orders of magnitude slower than the shift reactions with subsequent β-scission, the reactions investigated in this work are largely decoupled from reactions initiated by simulation products. Therefore, our results are representative for the entire pyrolysis mechanism, where hydrogen abstraction is considered. Reaction profiles of the oxygen−carbon and carbon−carbon shift reactions are similar; both proceed through a cyclic intermediate with comparable energy barriers. The difference between oxygen−carbon and carbon−carbon shift pathways can only be recognized by analyzing subsequent β-scission. Whereas the oxygen−carbon shift products cleave with virtually no barrier, the carbon−carbon shift products possess a barrier of above 15 kcal/mol for β-scission. We believe that this is due 12247
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to the charge separation in the transition state, which is facilitated when the electron-receiving fragment contains the ether oxygen. Although the α-hydroxy substituent affects reaction barriers and energy differences, results are not qualitatively changed by the substituent. The detailed understanding of the gas-phase kinetics and mechanisms of the pyrolysis of PPE and its derivatives is an important foundation for future research aimed at selective catalytic transformations of these lignin model compounds.
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ASSOCIATED CONTENT
S Supporting Information *
Potential along a spurious imaginary mode; list of rate and equilibrium constants used in the simulation at 618 K; list of prefactors and activation energies obtained from Arrhenius plots in the temperature range of 580−660 K for the β-scission reactions and the phenyl migration reactions, applying steadystate conditions for the latter; transition states and equilibrium structures for the reaction channels of the oxygen−carbon phenyl shift and subsequent β-scission reactions in the β-radical of PPE and α-hydroxy PPE; transition states and equilibrium structures for the reaction channels of the carbon−carbon phenyl shift and subsequent β-scission reactions in the β-radical of PPE; spin densities; and complete refs 36 and 41. 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.
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ACKNOWLEDGMENTS This research was sponsored by the Division of Chemical Sciences, Geosciences, Office of Basic Energy Sciences, U.S. Department of Energy and was performed in part using the resources of the National Center for Computational Sciences at Oak Ridge National Laboratory under Contract DE-AC0500OR22725. It was also supported by an allocation of advanced computing resources provided by the National Science Foundation; computations were performed on Kraken at the National Institute for Computational Sciences. A portion of this research was conducted at the Center for Nanophase Materials Sciences, which is sponsored at Oak Ridge National Laboratory by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy.
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REFERENCES
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