Accurate Bond Energies of Biodiesel Methyl Esters from

Article pubs.acs.org/JPCA

Accurate Bond Energies of Biodiesel Methyl Esters from Multireference Averaged Coupled-Pair Functional Calculations Victor B. Oyeyemi,† John A. Keith,‡ and Emily A. Carter*,‡,§,∥ †

Departments of Chemical and Biological Engineering, ‡Mechanical and Aerospace Engineering, §Program in Applied and Computational Mathematics, and ∥Andlinger Center for Energy and the Environment, Princeton University, Princeton, New Jersey, 08544-5263, United States S Supporting Information *

ABSTRACT: Accurate bond dissociation energies (BDEs) are important for characterizing combustion chemistry, particularly the initial stages of pyrolysis. Here we contribute to evaluating the thermochemistry of biodiesel methyl ester molecules using ab initio BDEs derived from a multireference averaged coupled-pair functional (MRACPF2)-based scheme. Having previously validated this approach for hydrocarbons and a variety of oxygenates, herein we provide further validation for bonds within carboxylic acids and methyl esters, finding our scheme predicts BDEs within chemical accuracy (i.e., within 1 kcal/mol) for these molecules. Insights into BDE trends with ester size are then analyzed for methyl formate through methyl crotonate. We find that the carbonyl group in the ester moiety has only a local effect on BDEs. CC double bonds in ester alkyl chains are found to increase the strengths of bonds adjacent to the double bond. An important exception are bonds beta to CC or CO bonds, which produce allylic-like radicals upon dissociation. The observed trends arise from different degrees of geometric relaxation and resonance stabilization in the radicals produced. We also compute BDEs in various small alkanes and alkenes as models for the long hydrocarbon chain of actual biodiesel methyl esters. We again show that allylic bonds in the alkenes are much weaker than those in the small methyl esters, indicating that hydrogen abstractions are more likely at the allylic site and even more likely at bis-allylic sites of alkyl chains due to more electrons involved in π-resonance in the latter. Lastly, we use the BDEs in small surrogates to estimate heretofore unknown BDEs in large methyl esters of biodiesel fuels.

1. INTRODUCTION Biodiesel is a clean-burning renewable fuel produced by the transesterification reaction of vegetable or algal oils with methanol.1−3 Biodiesel typically contains large methyl esters (and sometimes ethyl esters) with long hydrocarbon chains of 16 to 18 carbon atoms, most of which have one to three CC double bonds in the hydrocarbon chain. For example, soybeanderived biodiesel is composed of methyl-octadeca-9,12dienoate (53.1%), octadec-9-enoate (21.8%), hexadecanoate (11.8%), octadeca-9,12,15-trienoate (8.0%), and octadecanoate (4.8%).4 The presence of oxygen atoms in the biodiesel ester group reduces soot emissions,5−7 while the presence of CC double bonds seems to increase soot production.8 Moreover, biodiesel produces less particulate emissions9−11 and lower amounts of aldehyde species (relevant in ozone formation),12 but it creates a slightly elevated level of NOx emissions relative to conventional diesel.13 The interplay of the various chemical features of the methyl ester structures leading to these observed properties of biodiesel combustion is not fully understood. This knowledge is needed to optimally utilize the fuel. Detailed understanding of combustion chemistry is challenging because the processes often involve up to thousands of species and elementary reactions, depending on a fuel’s © 2014 American Chemical Society

molecular composition and size distribution. The most readily measurable combustion properties are global observables like emission profiles, flame structures, ignition delays, and extinction times that are determined by the cumulative effect of many interrelated individual elementary processes. Deriving an understanding of combustion ultimately entails the difficult task of isolating the dominant individual elementary processes or species. Some experimental techniques can isolate overall reactions between two reactants to determine reaction rates (e.g., ref 14). However, such studies have not yet isolated specific elementary reactions involving the methyl esters of biodiesel. As an alternative or complement to experiment, combustion can be studied theoretically by generating and analyzing accurate ab initio thermochemical and kinetic data. This theoretical approach makes it possible to study elementary reactions as well as model global properties and therefore is more advantageous compared to experiments in this respect. Special Issue: Kenneth D. Jordan Festschrift Received: December 28, 2013 Revised: February 13, 2014 Published: March 13, 2014 7392

dx.doi.org/10.1021/jp412727w | J. Phys. Chem. A 2014, 118, 7392−7403

The Journal of Physical Chemistry A

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(CC) theory (e.g., CCSD(T)34 in refs 33 and 29). Popular semiempirical schemes such as Gaussian-n (G3,35,36 G3B3,37 G3(MP2)B3,37 G4,38 and G4(MP2)39) and the complete basis set (CBS-n) techniques (e.g., CBS-Q40 and CBS-QB340) are composite approaches based on single-reference QCISD(T) or CCSD(T) calculations. They are affordable for molecules consisting of up to about 10 heavy atoms and provide BDEs accurate to about 1−2 kcal/mol.33 It has been argued that single-reference CC’s inclusion of disconnected quadruple and higher excitations makes CC methods appropriate for modeling multiconfigurational systems (e.g., refs 41 and 42). For instance, the Weizmann-n methods (e.g., W441), include higher excitations from coupled cluster expansions to effectively account for MR character. Unfortunately, these calculations are so expensive that applications have been limited to fewer than five heavy atoms.33 We recently validated29 a composite ab initio scheme based on MR averaged coupled-pair functional (MRACPF2) theory that achieves chemically accurate BDEs in oxygenated molecules, as an alternative to CCSD(T) theory, which can be extended to other parts of PESs. BDEs are the only welldefined thermochemical kinetics metric to test an ab initio scheme’s accuracy against experimental benchmarks. They also are critical quantities to furnish for biofuel combustion kinetics models. We demonstrated29 that BDEs of simple oxygenated species (e.g., methanol and ethanol) require a less demanding MR treatment than BDEs of more complex carbonyl-containing compounds like aldehydes and carboxylic acids (vide infra). Building on that benchmark study, herein we report MRACPF2 BDEs of small to midsize (C1−C4) methyl esters and small (mostly unsaturated) hydrocarbons related to sections of the actual large biodiesel molecules. We focus on the smaller esters to understand how ester groups affect biodiesel combustion properties. Section 2 describes our MR composite approach for computing BDEs. Section 3 presents method validation results. Sections 4 reports our predicted BDEs for small C1−C4 esters and relevant hydrocarbon surrogates and provides comparisons to BDEs from other methods. Section 5 culminates with best estimates of large biodiesel ester BDEs based on those we predicted for the small ester and hydrocarbon surrogates. Section 6 offers concluding remarks.

Global properties are obtained from numerical simulations of up to thousands of elementary species and their reactions. With a comprehensive combustion scheme in place, specific individual elementary reactions can be adjusted to test their individual effect on a global combustion property. The accuracy of these models naturally depends on the accuracy of the thermochemical kinetics parameters used, such as bond dissociation energies (BDEs), which are the focus of the present work. BDEs are used within or outside of chemical kinetics models to rationalize the relative rates of competing pyrolytic pathways. Experimental studies have determined BDEs for some small methyl esters (e.g., methyl formate and methyl acetate), but these carry high levels of uncertainty. Large biodiesel ester BDEs are typically not known because they have low vapor pressures and are difficult to volatilize for measurements.15,16 Theoretical studies have reported more BDEs for several small to midsize esters (methyl formate,17−19 methyl acetate,18 methyl butanoate,18,20 ethyl propanoate,20 and isopropyl butanoate21) using various methods from group additivity22 to ab initio calculations. However, the theoretical data for larger methyl esters are less common because reliable group additivity values are not generally available, and accurate ab initio calculations on the large molecules are prohibitively expensive. Nevertheless, some researchers23,24 have estimated enthalpies of formation for large esters and their radical components using schemes based on Kohn−Sham density functional theory (KSDFT).25,26 BDEs may be retrieved from these enthalpy data. We return to this subject, given the incomplete BDE data for methyl esters and the uncertainties in available data. The pathway for bond dissociation is inherently a multiconfigurational (“multireference” or MR) process best treated by methods based on multideterminant zeroth-order wave functions.27−29 Description of all energetically accessible points on the bond-breaking potential energy surface (PES) with a single consistent approach requires use of MR methods. Singlereference techniques (e.g., KS-DFT, CCSD(T), Gn-methods, etc.) successfully account for electron correlation in equilibrium species and thus are in fact sufficient for BDE calculations (which only require energy differences between equilibrium species), unless the equilibrium species themselves have significant MR character (vide infra).29 We prefer to test and use MR methods for BDEs because our ultimate goal is to be able to describe entire PESs, which requires a consistent level of theory across the PES that does not exhibit the divergences frequently exhibited by single-reference methods. Most of the computed BDEs available in the literature for methyl esters were derived from empirical group additivity relationships or calculated using single-reference techniques. Of the theoretical methods available, group additivity provides the simplest and least computationally demanding means for estimating molecular BDEs, but it also comes with substantial uncertainty since it is empirically derived and specific group values are limited to available data. While useful in general computational chemistry applications, single-reference KS-DFT calculations are usually not sufficiently accurate for BDE calculations (see refs 30−33). One KS-DFT study using a variety of exchange-correlation (XC) functionals (LDAs, GGAs, meta-GGAs, and hybrid meta-GGAs) yielded BDEs with mean signed errors greater than 4 kcal/mol, though double hybrid functionals were reported to be accurate to ∼2 kcal/mol.33 BDEs have also been computed with higher-level postHartree−Fock ab initio methods such as coupled cluster

2. MULTIREFERENCE SCHEME AND COMPUTATIONAL DETAILS We begin by outlining key elements of the computational scheme described in greater detail elsewhere.29 BDEs are defined as the energy of the supermolecule (i.e., the molecule with the breaking bond extended to the dissociation limit, defined here as 10 Å, with all other internal degrees of freedom relaxed in each fragment) minus the energy of the molecule with all bonds at their nondissociated equilibrium values. We define De (electronic energy change upon dissociation), D0 (enthalpy of dissociation at 0 K), and D298 (enthalpy of dissociation at 298 K) for a generic bond A−B as De = ΔEMRACPF2

(1)

D0 = De + ΔZPE

(2)

D298 = D0 + ΔH0 → 298

(3)

ΔZPE and ΔH0→298 are the zero-point energy and thermal correction differences between dissociated fragments (A• + 7393



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challenging to calculate since larger active spaces are needed to describe their MR character.29 We therefore extend our earlier validation tests to include small carboxylic acids and esters, namely, formic acid (FA), acetic acid (AA), methyl formate (MF), and methyl acetate (MA). The small sizes of these molecules allowed a detailed investigation of the active space requirements for each bond in the carboxylic acid/ester groups. Furthermore, direct comparison to experiment is possible for all bonds in the four molecules. The experimental data help guide a preliminary assessment of our scheme, but we also revisit these data since they often have large uncertainties. We group the small acid and ester single bonds into five classes: RC(O)O−R′, ROC(O)−R′, RC(O)−OR′, RC(O)OCH2−H, and ROC(O)CH2−H, where R and R′ are either −H or −CH3 (Figure 1). Experimental D298s were taken from the extensive

B•) and the undissociated molecule (AB), all at their equilibrium geometries. Geometries, vibrational frequencies, ΔZPE, and ΔH0→298 are computed with DFT-B3LYP43 and a triple-ζ basis set (6-311G(2d,p)44 or cc-pVTZ;45 the two basis sets give largely similar ΔZPE, ΔH0→298, and BDEs; see Table SI-1 of Supporting Information. Electronic energies are calculated from either multireference singles and doubles configuration interaction (MRSDCI)27,46 or its size-extensive variants.46−52 We previously compared29,53 electronic energies calculated from MRSDCI, multireference averaged coupled-pair functional (MRACPF and MRACPF2) methods,50,52 and MRSDCI with the a posteriori sizeextensivity Davidson−Silver correction (MRSDCI-DS).49 That work showed that our scheme functions most accurately and reliably with MRACPF2 electronic energies. Only MRACPF2 results are reported below. The complete active space self-consistent field (CASSCF)54 method is used as the reference multiconfigurational wave function for the MRACPF2 calculations. Sufficient CASSCF active space sizes vary depending on the nature of the breaking bond.29 The CAS(2e,2o) active space consisting of two electrons in two orbitals (σ and σ*) of the dissociating bond is appropriate for most bonds in acids and esters. Important πelectron resonance effects arise for breaking bonds beta (β) to the carbonyl group of acids and esters, and so a CAS(4e,4o) active space consisting of the π and π* orbitals of CO and the σ and σ* orbitals of the dissociating bond is needed. Lastly, a CAS(8e,6o)29 active space is needed for acyloxyl radicalproducing bonds. For the MRACPF2 calculations of any particular bond, we used a superset of references from important CASSCF configurations of the equilibrium geometry and supermolecule of the bond, where an important CASSCF configuration is defined as one with wave function coefficient larger than 0.05. We extrapolate the reference energies (Eref) and the correlation (Ecorr) energies to the CBS limit using cc-pVDZ (X = 2) and cc-pVTZ (X = 3) basis sets with the scheme due to Truhlar:53,55,56 EXref = EXref + Aref X −3.4

(4)

EXcorr = EXcorr + Acorr X −2.4

(5)

Figure 1. Experimental,64 CCSD(T), and MRACPF2 D298s for small acids (formic and acetic acids) and esters (methyl formate and acetate). R, R′ = H or Me. MRACPF2 D298s using references defined at the CAS(4e,4o) and CAS(8e,6o) levels are labeled by shaded boxes. Other D298s were obtained using references defined at the CAS(2e,2o) level.

compilation of Luo.64 Other reference BDEs were computed when possible from the ATcT enthalpies of formation database.65−67 The Luo reference reports multiple BDE measurements based on a variety of experimental techniques including spectroscopy and kinetics studies. For instance, five D298s ranging from 93.9−98.7 kcal/mol are listed for H− CH2C(O)OH.68−70 The Luo recommended value of 95.3 ± 2.9 kcal/mol from collision-induced dissociation69 was measured in 1994, while a value of 98.7 ± 0.8 kcal/mol from calorimetry was obtained later in 2001.70 It is not obvious which of the five entries is the most accurate, so we note the full range of experimental data when validating computed energies. In fact, computed BDEs such as those reported herein are important to help resolve experimental uncertainties. The experimental BDEs plotted in Figure 1 are those recommended in the Luo compilation. 3.1. Trends in MRACPF2 BDEs (Comparison of BDEs within Bond Groups). Figure 1 shows BDEs from experiment as well as MRACPF2 and CCSD(T) calculations. A T1 amplitude threshold of 0.02 is typically used to identify multireference character. RC(O)O radicals have CCSD(T) T1 amplitudes of 0.020−0.023, and RC(O) and ROC(O) of 0.022−0.025, meaning multireference effects may be important for bond breaking involving these radicals. Other radicals and all closed shell molecules have T1 values below 0.020 (see Table SI-2 of Supporting Information for individual T1 values). We found that radical wave functions arising from RC(O)O−R′ dissociation are highly multiconfigurational

CASSCF calculations were performed within MOLCAS 7.8.57 TigerCI58−62 was used for all MRACPF2 calculations. Geometry optimization and vibrational frequency calculations were performed with GAMESS-US44 using the program’s default convergence parameters. We also compare BDEs from the single-reference CCSD(T)34 theory using the same basis set extrapolation procedure above. The CCSD(T) calculations were performed within MOLPRO.63

3. VALIDATION AND BENCHMARKING OF MRACPF2 BDES FOR SMALL ACIDS AND ESTERS Biodiesel fuels are typically composed of methyl esters, CH3OC(O)R, where R is a hydrocarbon chain of 16 to 18 carbon atoms and contains between zero to three CC double bonds. Our previous investigation of small hydrocarbons indicates chemically accurate BDEs for bonds on the saturated hydrocarbon end of the methyl esters will be obtained with MRACPF2 and the minimal CAS(2e,2o) active space.53 However, BDEs for bonds beta to double bonds are more 7394



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O)CH3) = 111.4 kcal/mol). (2) Disagreement between MRACPF2 D298s and the experimental BDEs are highest for C−H bonds alpha (α) to the carbonyl group (middle of Figure 1). (3) Calculated methoxy C−H bonds disagree with experiment by ∼4 kcal/mol. We discuss these discrepancies next. 3.2.1. BDE Trends within Acyloxyl-Producing Dissociations. The experimental D298 BDEs of the O−H bonds in formic and acetic acids are 112 ± 3 kcal/mol,16 suggesting both are the same. Other experimental sources give formic acid BDEs as 111.7 ± 0.372 or 111.5 kcal/mol73 and acetic acid BDEs as 106.5,74 110,73,75 or 112 kcal/mol.76 Thus, the formic acid experimental BDEs are consistently ∼112 kcal/mol, while acetic acid experimental BDEs vary more, with values ranging from 106−112 kcal/mol. We conclude it is likely that the O−H BDE in acetic acid is indeed lower than that in formic acid. This would be consistent with the calculated results for which MRACPF2 and CCSD(T) D298s suggest that the O−H bond in acetic acid is weaker by as much as 3 kcal/mol. Upon H dissociation from formic acid and acetic acid, OC(O)H structurally relaxes by 3.4 kcal/mol, while OC(O)CH3 relaxes by 5.1 kcal/mol at the DFT-B3LYP/6-311G(2d,p) level. The different degree of relaxation explains the lower O− H bond energy of acetic acid. We also analyzed different rotational conformers of acetic acid and acetyloxyl radical (Figure 2) since methyl group rotation is the only difference in

and require CAS(8e,6o) active spaces to achieve sufficient accuracy in MRACPF2 BDEs (see Table SI-3 of Supporting Information for MRACPF2 electronic energies and BDEs of all CAS(8e,6o) calculations).29 C−H and C−C bonds β to the carbonyl group also require CAS(4e,4o) active spaces due to resonance stabilization of the resulting radical electron by the carbonyl π electrons.29 For bonds in the acids and esters considered here, CAS(2e,2o) D298s are larger than CAS(4e,4o) D298s by 1.6−2.7 kcal/mol (see Table SI-4 of Supporting Information). BDEs calculated at the CAS(4e,4o) and CAS(8e,6o) levels are highlighted in Figure 1; all other BDEs are computed at the CAS(2e,2o) level. CCSD(T) D298s and MRACPF2 D298s largely agree for bonds requiring only a CAS(2e,2o) active space. In these cases, BDEs differ by less than 1 kcal/mol (with the exception of a 2.5 kcal/mol difference for HOC(O)−H). CCSD(T) BDEs are larger than MRACPF2 BDEs by 2.2−4.1 kcal/mol for bonds requiring active spaces beyond CAS(2e,2o) for MRACPF2, and these may be cases in which CCSD(T) is inaccurate. Some MRACPF2 (and single-reference CCSD(T)) BDE trends disagree with experimental trends and exhibit deviations of up to ∼5 kcal/mol. We first analyze trends in MRACPF2 BDEs here and then compare computed and experimental BDEs below in section 3.2. The most dramatic difference is apparent on the left side of Figure 1, where we see that the O−H bonds of formic acid and acetic acid are stronger than O−C bonds in methyl formate and methyl acetate by over 20 kcal/mol. The observed BDE trends can be mostly understood by considering the intrinsic strength of O−H versus O−C bonds, which are largely determined by the relatively worse overlap of the O and C sp3-like orbitals versus better overlap of the O sp3 with the H 1s orbital. Differences in structural relaxation of radical fragments formed after bond dissociation can also partly explain the differences in BDEs. To get a feel for the magnitude of these effects, we compute qualitative geometry relaxation energies at the DFTB3LYP/6-311G(2d,p) level.71 The O−H bond is also stronger than the O−C bond in these molecules because the methyl radical in O−C dissociation relaxes from a pyramidal to planar structure upon bond breaking while of course the H in O−H dissociation has no internal degrees of freedom. This lowers the energy of the methyl radical by 6.2 kcal/mol, thereby decreasing the O−C bond energy. Likewise, O−C single bonds in formic acid and acetic acid are about 8 kcal/mol stronger than those in methyl formate and methyl acetate (leftmiddle of Figure 1), partly because of steric repulsion destabilizing the methoxy versus the hydroxyl in the ester versus the acid and also because OH radicals structurally relax less than the methoxy radical (OH does not relax at all and OCH3 relaxes by 2.1 kcal/mol). Lastly, in the R′OC(O)−R bond class (right-middle of Figure 1), C−H bonds are stronger than C−C bonds by 5.3−5.9 kcal/mol largely because here the methyl radical relaxes by 7.2 kcal/mol. 3.2. Discrepancies between MRACPF2, CCSD(T), and Experimental BDEs. Several discrepancies between computed and experimental BDEs are evident in Figure 1: (1) Experimental BDEs are constant for bonds within the two acyloxyl groups (left side of Figure 1), while computed MRACPF2 and CCSD(T) BDEs are not (exptl, D298(H− OC(O)H) = 112 ± 3 kcal/mol, D298(H−OC(O)CH3) = 112 ± 3 kcal/mol; MRACPF2, D298(H−OC(O)H) = 111.4 kcal/mol, D298(H−OC(O)CH3) = 108.9 kcal/mol; CCSD(T), D298(H−OC(O)H) = 114.3 kcal/mol, D298(H−OC(

Figure 2. BDEs from rotamers of acetic acid and acetyloxyl radical.

complexity between formic and acetic acids. MRACPF2 energies of equilibrium geometry conformers (A, B, and C) of acetic acid differ by only 0.6 kcal/mol, while supermolecule conformers D and E are equal in energy. Thus, −CH3 rotation does not produce significantly different BDEs in acetic acid. Again, the difference between formic acid and acetic acid BDEs seems to be purely a result of greater geometric relaxation after bond scission in acetic acid. The forgoing discussion applies to the C−O bonds in methyl formate and methyl acetate as well, for which experimental and computed BDEs all show a nonconstant trend. However, our computed CAS(8e,6o)MRACPF2 energies are lower than the CCSD(T) and experimental energies (exptl, D298(CH3OC(O)H) = 91.7 ± 3 kcal/mol, D298(CH3OC(O)CH3) = 90.9 ± 3 kcal/ mol; MRACPF2, D298(CH3OC(O)H) = 86.4 kcal/mol, D298(CH3OC(O)CH3) = 85.0 kcal/mol; CCSD(T), D298(CH3OC(O)H) = 90.6 kcal/mol, D298(CH3 OC(O)CH3) = 88.1 kcal/mol). The reason for the large discrepancy is not clear, given that we had shown earlier29 that the MRACPF2 BDE for the analogous H−O bond in formic acid exhibits much smaller errors (1−3 kcal/mol, depending on 7395



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Table 1. MRACPF2 D298s (kcal/mol) in C1−C4 Methyl Esters: Methyl Formate (MF; R = H), Methyl Acetate (MA; R = Me, n = 0), Methyl Propanoate (MP; R = Et, n = 1), and Methyl Butanoate (MB; R = Pr, n = 2); the Breaking Bond Is Indicated by (−)

which reference data one compares to). However, since CCSD(T) also shows deviations of up to ∼3 kcal/mol, more refined measurements to reduce the uncertainty in these values may be warranted. We simply note here that our BDEs appear systematically low by 1−3 or 5−6 kcal/mol when forming the formyloxyl or acyloxy radicals, respectively, by far the largest errors (if they are errors) that we have encountered. Clearly more work is needed to establish these BDEs more precisely. 3.2.2. R′OC(O)−H BDEs. CCSD(T) and MRACPF2 give similar D298s of R′OC(O)−H bonds (right-middle of Figure 1), but the experimental D298s for these bonds are much lower than the computed values. We investigated the R′OC(O)−H group further using HOC(O)−H as a prototype. The HOC(O) radical has two conformers: trans-HOC(O) and cis-HOC(O). The trans-HOC(O) conformer is more stable by 1.3 kcal/mol at the DFT-B3LYP/6-311G(2d,p) level, in agreement with DFT-M06-2X (1.4 kcal/mol) and CCSD(T) (1.7 kcal/mol) predictions. DFT-B3LYP vibrational frequencies are in good agreement with the two available experimental frequencies (maximum differences: 141 cm−1 for the H−O stretch and 29 cm−1 for the CO stretch.)77,78 and with high level ab initio frequencies from CCSD(T) quadratic force field calculations of Fortenberry et al.29,79,80 (maximum difference of 50 cm−1). Thus the discrepancy between theory and experiment is likely not due to the geometries or vibrational frequencies. The reference D298 for this bond is 96.0 ± 1.0 kcal/ mol from experiment and 98.5 ± 0.2 kcal/mol from ATcT (calculated from ATcT D0 + DFT-B3LYP ΔH0→298 = (97.0 ± 0.14) + 1.5 kcal/mol). The 4−5 kcal/mol discrepancy between reference (experiment and ATcT) and computed BDEs remain with a full valence active space.29 Another possible source of error is that basis set extrapolation based on cc-pVDZ energies fails in this case. MRACPF2 D298s for trans-HOC(O)-H are 93.8, 97.7, and 101.9 kcal/mol within the cc-pVDZ, cc-pVTZ, and cc-pV∞Z basis sets, respectively. The large difference between cc-pVDZ and cc-pVTZ energies may have led to an overestimated cc-pV∞Z BDE. 3.2.3. Methoxy C−H BDEs (RC(O)OCH2−H). Lastly, we discuss BDE for the C−H bond in the methoxy group of methyl acetate. MRACPF2 and CCSD(T) are in near perfect agreement in predicting a D298 of ∼100.5 kcal/mol for HC( O)OCH2−H and CH3C(O)OCH2−H. The lone experimental value available is for CH3C(O)OCH2−H with a D298 of 96.7 kcal/mol. We believe this BDE should be re-examined experimentally.

breaking bond

MF

MA

MP

MB

RC(O)OCH2−H CH3OC(O)(CH2)nCH2−H C−H beta to CO C−H gamma to CO CH3OC(O)−R C−C beta to CO C−C gamma to CO RC(O)−OCH3 RC(O)O−CH3

100.6

100.4 97.3

100.9 103.1 92.4

100.3

93.8

93.7 83.2

100.3 86.4

100.0 85.1

100.5 86.0

100.4 102.7 92.1 99.9 93.6 81.6 88.6 99.8 86.0

tables would be the same across this row, and our calculations show this is a valid assumption. However, trends in other rows sometimes have notable outliers, and the nature of the breaking chemical bond explains this. For example, in the second row of the table we see that the primary C−H D298s for the alkyl chains in MP and MB are very similar (MRACPF2 D298s of 103.1 and 102.7 kcal/mol, respectively), but the corresponding C−H D298 for MA is noticeably weaker (97.3 kcal/mol). The existence of π-resonance stabilization due to the carbonyl group in the radical fragment in MA explains this difference. Similar stabilization arises for any breaking bond that is β to a carbonyl group. Now note that the β C−H bond in MA (CH3OC( O)CH2−H), a terminal bond (found in the second row of Table 1), is about 5 kcal/mol stronger than the internal β C−H bonds of MP and MB (third row of Table 1). Larger geometric relaxation of the long chain radicals explains these trends. The gamma (γ) C−H bond in MB (fourth row) is much stronger than the β C−H bonds of MP and MB because of the lack of resonance stabilization in the resulting radical. For C−C D298s in CH3OC(O)−R, the C−C bonds in MA, MP, and MB are very similar (93.8, 93.7, and 93.6 kcal/mol, respectively), but all are considerably weaker than the C−H bond in MF (100.3 kcal/mol). This trend is because breaking these C−C bonds again produces an alkyl fragment that undergoes geometric relaxation resulting in a lower BDE. Breaking the C−H bond in MF yields an H atom and an sp2-hybridized fragment that undergoes only small structural relaxation, making C−H BDEs on average larger than C−C BDEs. Of the two C−O bonds, the one producing acyloxyl radicals have much lower bond strengths (last row, ∼86 kcal/mol versus second to last row, ∼100 kcal/mol) due to resonance stabilization coupled with the relative extent of geometric relaxation of the methyl versus methoxy radicals. However, the β C−C bonds (D298(MP β C− C) = 83.2 kcal/mol and D298(MB β C−C) = 81.6 kcal/mol) are the weakest of all due to both alkyl radical relaxation and the very stable, allylic-like, π-resonance-stabilized radicals formed. The BDE of the MP β C−C bond is larger because of the smaller relaxation energy of the resulting methyl radical compared to the ethyl radical formed in MB C−C scission. The C−C bond γ to the carbonyl is much stronger due to the lack of resonance stabilization in the resulting radical. Beyond methyl esters, there is interest in ethyl esters3,20,81,82 because substituting renewable ethanol for petroleum-derived methanol in the transesterification process makes biodiesel a more sustainable fuel. We therefore also computed MRACPF2 D298s in a representative ethyl ester, namely ethyl propanoate

4. BDES IN C1−C4 ESTER AND HYDROCARBON SURROGATES OF BIODIESEL METHYL ESTERS 4.1. Trends in BDEs from Methyl Formate to Methyl Butanoate. We begin by reporting MRACPF2 D298s for the C1−C4 methyl esters. Table 1 organizes groups of chemical bonds in the five esters to readily distinguish qualitative trends (and variations in trends) in BDEs from different chemical bond types. Based on the results of the previous section, we expect certain BDEs to be more accurate than others: methyl C−H, C−C, and non-acyloxyl/formyloxyl-producing C−O bonds are expected to be reliable while the acyloxyl/ formyloxyl-producing C−O bonds may be too low by a few kcal/mol. Trends are expected to be reliable, and therefore, the discussion uses the raw MRACPF2 predictions to discuss trends. Table 1 illustrates several points. Consider the first row of D298s in RC(O)OCH2−H bonds. D298s from group additivity 7396



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not necessarily unique (i.e., a choice of different isodesmic reactions may give different BDEs) and also not the same as the supermolecule approach used here. Thus, the comparisons here are not directly relatable to standard CCSD(T) or CBS-QB3 calculations. Qualitative BDE trends are the same for all methods/schemes, however. BDEs from MRACPF2 and CBSQB3-Iso, the only other ab initio-based scheme, agree within 0.6 kcal/mol for four bonds (CH3OC(O)−CH2CH2CH3, CH 3 OC(O)CH(CH 2 CH 3 )−H, CH 3 OC(O)CH 2 − CH2CH3, and CH3OC(O)CH2CH2−CH3). The group additivity estimates for these bonds differ more (by 1.2−3.7 kcal/mol) with the exception of Dooley et al.’s group additivity value for CH3OC(O)CH2CH2−CH3, which agrees with MRACPF2 within 0.3 kcal/mol. MRACPF2 D298s for the acyloxyl C−O bonds in methyl esters are all ∼86 kcal/mol (Table 1). These are consistent with CBS-QB3-Iso (87.0 kcal/ mol) and group additivity (86.7 kcal/mol) D298s for methyl butanoate (Figure 4). There are important quantitative differences between the methods for other bonds. First, both the group additivity methods and MRACPF2 give a D298 of around 100.4 kcal/mol for the primary C−H ester bond, while CBS-QB3-Iso gives a lower energy of 98.9 kcal/mol. Second, the CBS-QB3-Iso BDE is larger than the MRACPF2 BDE by 1.5 kcal/mol for CH3O−C(O)CH2CH2CH3, but lower than the MRACPF2 BDE by 1.2 kcal/mol for CH3OC( O)CH2CH(CH3)−H. These differences in bond energies are significant and may have important effects on predictions by kinetics models. 4.2. Methyl Crotonate vs Methyl Butanoate. The majority of the esters in commercial biodiesel are unsaturated. Thus, the unsaturated surrogate methyl crotonate (MC) may better model biodiesel fuel chemistry than methyl butanoate (MB).8 We discuss the role of C−C bond unsaturation in methyl esters by comparing BDEs for MB and MC (Figure 5).

(Figure 3). Most BDEs in the alkyl chain of ethyl propanoate (i.e., bonds on the right side of CO in Figure 3) are similar to

Figure 3. MRACPF2 D298s BDEs of methyl propanoate (MP) and ethyl propanoate (EP). Weakest C−H, C−C, and C−O bonds are marked with dashed ellipses around the bonds.

those for comparable bonds of methyl esters. The β C−H and C−C D298s are larger by 0.8−0.9 kcal/mol in EP compared to MP. However, these differences are within the uncertainty margins of the calculations. On the other hand, the terminal ester C−H bond (i.e., bonds on the left side of CO in Figure 3) is 3 kcal/mol stronger than those in the methyl esters. The secondary C−H bond in the ethyl group of ethyl propanoate is similar to the terminal C−H bond in the methyl esters (99.6 vs 100.9 kcal/mol). This suggests the noncarbonyl oxygen plays a role in lowering bond strengths γ to the carbonyl, perhaps via a hyperconjugative effect from lone pair electrons on the oxygen atom. The two C−O bonds in ethyl propanoate are stronger than the corresponding bonds in methyl propanoate. The acyloxyl C−O bond is stronger in ethyl propanoate by 2.7 kcal/ mol, and the second C−O bond involving the carbonyl C is stronger in ethyl propanoate by 0.6 kcal/mol. We next compare methyl butanoate D298s from MRACPF2 and other computational approaches in Figure 4 (see also Table

Figure 5. MRACPF2 BDEs of methyl butanoate (MB, top) and methyl crotonate (MC, bottom). The weakest C−H, C−C, and C−O bonds are marked with dashed ellipses around the bonds.

BDEs for interior (i.e., nonprimary) C−H bonds are considerably larger in MC than in MB (by 21.6 and 11.4 kcal/mol for C−H bonds β and γ to the carbonyl, respectively). Those C−H bonds in MC involve sp2-hybridized orbitals while those placed similarly in MB utilize sp3-hybridized orbitals. As discussed above, the sp2-hybridized centers do not structurally relax as much as sp3-hybridized centers do. This results in stronger vinylic C−H bonds in MC, a well-known trend for alkenes. On the other hand, the primary C−H BDE at the end of the alkenyl chain of MC is much lower (by 15.0 kcal/mol) than the C−H BDE in the analogous carbon position in MB due to incipient π-resonance stabilization in MC (by forming an allyl radical). Unlike C−H bonds in the alkyl chain, the C−

Figure 4. D298s from MRACPF2 compared with those from CBSQB3-Iso of El Nahas et al.20 and group additivity data of Dievert et al.84 and Glaude et al.85 A table of values can be found in the Supporting Information Table SI-5.

SI-5 of Supporting Information for precise numbers). Specifically, we compare our calculations to those from group additivity and isodesmic schemes83 with CBS-QB3 energies, which are extrapolated energies from CCSD(T) calculations (henceforth labeled: “CBS-QB3-Iso”). El Nahas et al.20 generated the CBS-QB3-Iso data using reaction sets that are 7397



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resonance stabilization. Vinylic single bonds (those involving sp2 carbon atoms) are stronger than others due to lack of structural relaxation. BDEs in 1-propene, 1-butene, and 1-pentene (Figure 7) illustrate convergence with distance from unsaturated CC

H bond on the ester methyl group is not affected by C−C saturation. Although CC double bonds have a strong local effect, as expected their presence is not felt past the OC bonds in biodiesel surrogates. As was the case for interior C−H bonds, C−C bonds in MC (sp2-hybridized bonds) are much stronger than C−C bonds in MB (sp3-hybridized bonds). In MB, the C−C bond γ to carbonyl is weaker than the C−C bond α to carbonyl (by 5.0 kcal/mol). The β C−C bond in MB is weaker still. This trend originates from production of two alkyl radicals (in β and γ C−C bond scission) compared to only one alkyl radical and one carbonyl radical (for α bond scission). The radicals that relax the most are associated with the weakest bonds, and the relaxation is greatest for the bulkiest sp3 alkyls (relieving steric repulsion). The weakest bonds by far are those that form an allylic-like radical upon bond dissociation (as in the β C−C bond in MB). The methoxy and ester C−O BDEs show little variation, again indicating that the carbonyl group decouples the methoxy group from the rest of the alkyl chain. 4.3. BDEs in Hydrocarbon Models for the Methyl Ester Alkyl Chain. The double bonds in unsaturated methyl esters of biodiesel are usually many positions removed from the ester group (first one is at the ninth carbon or farther from the C O). It is therefore reasonable to expect that the unsaturated portions of the hydrocarbon chains are unaffected by the ester moiety. We model bonds around these unsaturated portions of the chain with small unsaturated hydrocarbons: propene, 2butene, 1-butene, 3-hexene, and 1,4-pentadiene. In addition, we study butane for the saturated portions of the chain. This set of molecules were selected so computed BDEs in these hydrocarbons could be used in combination with those of the small esters above to predict BDEs in much larger methyl esters of biodiesel (vide infra). MRACPF2 and experimental BDEs for hydrocarbon bonds agree well in nearly all cases (see Supporting Information Table SI-6 for comparison with experiment as well as CBS-Q and G386). D298s in 2-butene and 3-hexene are reported in Figure 6, along with butane for comparison.87 CAS(2e,2o) and CAS(4e.4o)-MRACPF2 energies are shown for bonds β to the C C bond. The CAS(4e,4o) C−H BDEs are about 1.5 kcal/mol larger than those computed at the CAS(2e,2o) level, consistent with the results for β bonds in methyl esters. Again, bonds β to CC π bonds are weaker than others due to incipient π-

Figure 7. MRACPF2 D298s for 1-alkenes.

sites. The terminal primary methyl β C−H bond in propene is ∼3 kcal/mol stronger than the β C−H bonds in 1-butene and 1-pentene, due to less relaxation by the smaller alkyl radical. More importantly, bond strengths converge quickly as we move from 1-butene to 1-pentene (Figure 7). We also compare the effect of multiple double bonds by computing BDEs in 1,4-pentadiene (Figure 8). The D298 at bis-

Figure 8. MRACPF2 D298s illustrating convergence with active space at a bis-allylic bond site and the anomalously weak C−H bond at that site.

allylic site is converged at CAS(6e,6o) active space containing both sets of π/π* bonding/antibonding orbitals in addition to the σ/σ* orbitals. β C−H bonds at bis-allylic sites (placed between two double bonds) are weaker than those at allylic sites (with only one double bond) by ∼10 kcal/mol due to the extended π electron delocalization in the former. We also note that while α C−H bonds have similar bond strengths in alkenes with one and two CC bonds, the α C−C bonds in the molecules containing one CC bond are stronger than those with two CC bonds by about 13 kcal/mol. The α C−C bond in 1,4-pentadiene is much weaker again due to the formation of the resonance-stabilized allyl radical that does not form in the C−C cleavage of alkenes.

5. BDES IN LARGE METHYL ESTERS ESTIMATED FROM SMALL SURROGATES We now combine BDEs in small ester and hydrocarbon surrogates to estimate BDEs at similar bond sites in large biodiesel methyl esters (methyl stearate, methyl oleate, methyl linoleate, and methyl linolenate; Figure 9). The images distinguish which surrogate molecules are used for each BDE in the large ester. These models suggest that BDEs β to CC bonds are lower than those close to the ester group. Thus Habstraction and unimolecular decomposition via C−C cleavage are likely to occur first within the hydrocarbon chain in large methyl esters. Note that we tested the sensitivity of BDEs to conformational changes in methyl butanoate and found BDEs

Figure 6. MRACPF2 D298s in butane, 2-butene, and 3-hexene. Note that the MRACPF2 D298 for the β C−C bond of 3-hexene (71.6 kcal/ mol) is too low because the MRACPF2 calculations converged to an unphysical electronic structure, indicating the CAS(2e,2o) active space is insufficient. 7398



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Figure 9. continued

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Figure 9. (a) MRACPF2 BDEs in methyl stearate (methyl octadecanoate) obtained from calculations on methyl butanoate and butane surrogates. (b) MRACPF2 BDEs in methyl oleate (methyl octadec-9-enoate) obtained from calculations on methyl butanoate, butane, 3-hexene, and 1-pentene surrogates. (c) MRACPF2 BDEs in methyl linoleate (methyl octadeca-9,12-dienoate) obtained from calculations on methyl butanoate, butane, 1pentene, 3-hexene, and 1,4-pentadiene surrogates. For the marked (*) beta C−C bonds (see arrow from 1-pentene and the beta C−C bond in 3hexene group), we used the BDE from 1-pentene (75.0 kcal/mol) while other BDEs in the same group are from 3-hexene. (d) MRACPF2 BDEs in methyl linolenate (methyl octadeca-9,12,15-trienoate) obtained from calculations on methyl butanoate, butane, 1-pentene, 3-hexene, and 1,4pentadiene surrogates.

temperatures, readily breaking down the large biodiesel molecules into smaller fragments.

from different conformers are not significantly different. Therefore, although larger esters will adopt many conformational isomers, we expect BDEs of these larger methyl esters will not change significantly when different conformers are considered.

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ASSOCIATED CONTENT

* Supporting Information S

Table SI-1 reports comparison of BDEs and contributing terms when geometries and vibrational frequencies are obtained with B3LYP/cc-pVTZ and B3LYP/6-311G(2d,p). Table SI-2 lists T1 diagnostic values for small species in this study. Table SI-3 list energies for acyloxyl-radical-producing bonds. Table SI-4 compares CAS(2e,2o) and CAS(4e,4o) for bonds beta to C=O or C=C. Table SI-5 lists BDEs of methyl butanoate from MRACPF2, CBS-QB3, and group additivity. Table SI-6 compares MRACPF2 BDEs of hydrocarbons to those from experiments and those available from CBS-Q and G3. This material is available free of charge via the Internet at http:// pubs.acs.org.

6. CONCLUSIONS We have computed BDEs in small saturated and unsaturated methyl esters and hydrocarbons to model those moieties present in long chain biodiesel methyl esters. Benchmarking against the limited experimental data available indicates that certain bonds, particularly those that lead to formation of acyloxyl or formyloxyl radicals, still have a fairly high degree of uncertainty in their BDEs. Nevertheless, we are able to give serious estimates of the BDEs in large biodiesel molecules by superposing the BDEs obtained in surrogates. Qualitatively, we found that the ester group critically influences the BDEs of nearby bonds but its effect is minimal three or more bonds away from the ester. The presence of double bonds in the alkyl chain also affects BDEs and hence where hydrogen abstraction and bond cleavage will likely initiate. BDEs at the ester moiety (CH3OC(O)) are not affected by the length or level of saturation in the alkyl chain because the carbonyl group effectively separates the two ends of the esters. Double bonds in the alkyl chain strengthen bonds α but weaken bonds β to the double bond. Overall, C−C bonds β to CC are predicted to be the weakest in the alkyl esters, and these are likely the sites at which the fuels will likely pyrolize at high combustion

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AUTHOR INFORMATION

Corresponding Author

*(E.A.C.) E-mail: [email protected]. Fax: +1 609 258 5877. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported as part of the Combustion Energy Frontier Research Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award Number DE-SC0001198. 7400



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The work reported was performed on the TIGRESS high performance computer center at Princeton University, which is jointly supported by the Princeton Institute for Computational Science and Engineering and the Princeton University Office of Information Technology.

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