Density Functional Theory Study of Addition Reactions of Carbon

(24) DFT is often a cost-effective theoretical procedure, widely employed in a ..... log A = 8.2 for MOH and MEst; and finally, log A = 7.5 for tertia...
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J. Phys. Chem. A 2011, 115, 52–62

Density Functional Theory Study of Addition Reactions of Carbon-Centered Radicals to Alkenes D. Moscatelli,* M. Dossi, C. Cavallotti, and G. Storti Department of Chimica, Materiali e Ingegneria Chimica “Giulio Natta”, Politecnico di Milano, 20131 Milano, Italy ReceiVed: August 12, 2010; ReVised Manuscript ReceiVed: NoVember 3, 2010

Addition reactions of carbon-centered radicals to unsaturated compounds have been studied using quantum chemistry. Following the review by Fischer and Radom (Angew. Chem., Int. Ed. 2001, 40, 1340.), the radicals were grouped in four different families, and the alkenes were selected from among those typical of polymer productions. All of the kinetic constants were calculated using density functional theory and classic transition state theory. Geometries of reactants, products, and transition states were determined at the B3LYP/6311+G(d,p) level of theory, whereas reaction enthalpies, activation energies, and kinetic constants were estimated using different basis sets. By comparative evaluation of the results obtained with different basis sets, the best computational approach for each kinetic step was identified. As a result of this study, a computational methodology suitable for investigating a large number of kinetic pathways typical of freeradical polymerization processes is proposed. 1. Introduction Reactions of addition of carbon-centered radicals to alkenes are of major importance in polymer chemistry. In fact, propagation reactions play a paramount role in determining the molecular weight distributions of the final polymer, as well as the product composition in copolymerization. Moreover, reactions between radicals and unsaturated compounds often lead to the production of branched polymer chains or, more generally, of defects such as internal unsaturations or small branches due to head-to-head propagation. In all cases, these reactions have to be properly characterized, and knowledge of the corresponding thermodynamic and kinetic parameters is essential to a reliable modeling of the polymer properties. With reference to free-radical polymerization, considerable progress has been made in obtaining reliable and accurate kinetic information by experimental methods.1-5 Using sophisticated experimental techniques such as pulsed laser polymerization followed by size-exclusion chromatography (PLP-SEC), absolute rate constants for radical reactions and their activation parameters can be directly evaluated.6-12 However, the experimental determination of rate coefficients is often time-consuming or even complicated to achieve because it is hard to study separately a specific kinetic step out of all the possible reactions. Moreover, some kinetic steps, such as secondary reactions (e.g., backbiting, β-scission, hydrogen abstraction, etc.), cannot be directly determined by these techniques.13,14 In such cases, a valuable alternative is represented by theoretical predictions of thermodynamic and kinetic parameters based on quantum mechanics (QM). In fact, the rapid and continuing increase in computer power has enabled the use of ab initio QM to determine with sufficient accuracy different properties such as molecular geometries (bond lengths, bond angles, and torsional angles), energetic reaction profiles, vibrational frequencies of molecular species, transition state structures, and reaction frequency factors. In particular, ab initio calculations permit these useful properties to be obtained * Corresponding author. E-mail: [email protected].

explicitly and directly.15-18 The quality of the corresponding results depends on the size of the selected basis set: Apart from the required computational effort, the larger the basis set, the more accurate the simulation results. On the other hand, there is clear evidence that the deviation from experimental data is sometimes enlarged by increasing the basis set. This is mainly due to a partial and fortuitous compensation of different aspects that contribute to the determination of the correlation energy. Therefore, validating simulations are always needed prior to the study of a reactive system to assess the accuracy and reproducibility of the employed computational approach. In this work, the reactivity between 10 different carboncentered radicals and a large number of alkenes, representative of monomers widely used in polymer synthesis, has been systematically studied. Simulations have been performed with three different basis sets, ranging from an inexpensive and less accurate level of theory to an expensive and more precise one. The first aim of this study was to identify the best computational approach for each specific kinetic step. Because the behavior of these reactive systems suggests that it might be possible to correlate rate coefficients to the structural characteristics of the radicals and monomers,15 the second goal of this work was to provide an accurate and cost-effective computational methodology for a huge number of reactions grouped by the chemical behavior of reactants (both radicals and alkenes). Once an efficient computational approach has been defined, its potential with respect to the understanding and evolution of many reaction steps of primary importance in polymer science could be fully exploited.19-23 The radicals investigated in this study are those reported by Fisher and Radom:15 methyl radical (CH3•); benzyl radical (Bn); three primary radicals, cyanomethyl (MCN) and tert-butoxycarbonylmethyl (MEst) with an electron-acceptor substituent and hydroxymethyl (MOH) with an electron-donor substituent; four tertiary radicals, tert-butyl (tBu) and 2-hydroxy-2-propyl (POH) with only donor substituents and 2-tert-butoxycarbonyl-2-propyl (PEst) and 2-cyano-2-propyl (PCN) with two donor and one acceptor substituents; and finally trifluoroacetonyl radical (FAc)

10.1021/jp107619y  2011 American Chemical Society Published on Web 12/09/2010

DFT Study of Additions of C-Centered Radicals to Alkenes with a stronger electron-acceptor substituent. These radicals can be divided into four different groups: nonpolar nucleophilic radicals (Me and Bn), nucleophilic radicals (tBu, POH, and MOH) ambiphilic-electrophilic radicals (PEst, PCN, MEst, and MCN), and electrophilic radicals (FAc). The reactivity of these radicals was investigated toward 13 different alkenes: two olefins (propylene and isobutylene), three chlorinated compounds (vinyl chloride, the monomer of the second most produced polymer product in the world after polyolefins; isopropenyl chloride; and vinylidene chloride), styrene and R-methylstyrene (materials for a large number of copolymers used as insulations, fibres, coatings, etc.), methyl acrylate and methyl methacrylate (highly reactive monomers, typically used for acrylic glass productions, coatings, etc.), acrylonitrile and methacrylonitrile (primarily used to synthesize fibers), acrylaldehyde (used in the synthesis of polyester resins and polyurethane), and 2-methoxy propylene (used in elastomer production). The optimized molecular structures of all radicals are shown in Figure 1a, and all alkene molecules are sketched in Figure 1b. This work is organized as follows: The computational details are first reported in section 2. In section 3, the capabilities of the selected basis sets are tested by comparing calculated reaction enthalpies with experimental data. Then, the activation energy for each reactive step is determined with the two most promising basis sets. At the end of the section, some considerations on the radical-alkene reactivity as a function of the molecular structures are presented. Finally, implications of this work on future investigations (section 3.4) along with the Conclusions are reported. 2. Computational Details All thermodynamic and kinetic parameters were computationally determined with the density functional theory (DFT).24 DFT is often a cost-effective theoretical procedure, widely employed in a huge number of computational studies involving reactive systems similar to those examined in this study.1,2,19,21,22,25 On the other hand, recent computational results have shown how DFT methods sometimes lead to nonsystematic errors.26,27 In all DFT calculations, the exchange and correlation energy wereevaluatedwiththeBecke3parametersandtheLee-Yang-Parr functionals (B3LYP).28,29 These functionals can be less accurate than others proposed in the literature even if they are well assessed and largely used.21,30 Simulations were performed using three different basis sets: the all-electron 6-31 basis set with added polarization functions [6-31G(d,p)] as a medium-size basis set, the 6-31G(d) as a small basis set, and the all-electron 6-311 basis set with added polarization and diffuse functions [6-311+G(d,p)] as a large basis set.28 To obtain the most accurate description of the molecular structure, the geometries of all investigated compounds (reactants, products, and transition states) reported in this work were determined at the highest level of theory. On the other hand, all kinetic parameters (reaction enthalpies, activation energies, and kinetic constants) were calculated using different basis sets to compare their abilities to reproduce experimental data. All quantum chemical calculations of radicals were performed using a spin multiplicity of 2 and an unrestricted wave function (UB3LYP).31,32 To reduce the computational time for the optimization of the molecular structures and to minimize possible errors in the molecular energetic determination, geometries were first determined using the LigPrep program distributed by Schrodinger.33 Through this code, a wide conformational search was performed using a molecular mechanics force field refined to optimize the structures of ligands involved

J. Phys. Chem. A, Vol. 115, No. 1, 2011 53 in intramolecular complexes. This step is of primary importance to exclude optimized geometries found during simulations representative of a local minimum in the potential energy surface, leading to an error in the following evaluation of the energetic parameters. Moreover, it is worth noticing that both trans and gauche radical attacks can occur in the transition state and that the transition state geometry representative of the most stable configuration was always used. Obtained geometries are considered as first guess for the next geometry optimizations performed at the higher level of theory. As a general rule, all geometries were fully optimized with the Berny algorithm and were considered stable only after vibrational frequencies had been calculated and the absence of imaginary vibrational frequencies had been verified.29 All calculations were performed with the Gaussian 03 suite of programs, and all pictures were drawn using PyMol 1.3.34,35 Activation energies were calculated as the difference between the electronic energy of the transition state and that of the reactant, including zero-point energies (ZPE). Enthalpy changes were determined at 298 K including the thermal correction terms

Ea ) (EE + ZPE)TST -

∑

(EE + ZPE)i

(1)

i)reactants

∆H )

∑

(EE + ZPE + TC) -

j)products

∑

(EE +

i)reactants

ZPE + TC)i

(2)

where EE represents the electron energy, ZPE is the zero-point energy, and TC is the thermal energy correction. For reactions involving carbon-centered radicals and unsaturated monomers, the rate constant can be expressed through the Arrhenius equation. Two parameters are then required to determine the kinetic constant: the frequency factor (Ap) and the activation energy (Ea). These parameters are evaluated through the classic transition state theory (TST) as

-Ea/kbT

k(T) ) ApTe

kb ) T h

rot vib el qTS qTS qTS

∏

qrotqvibqel

e-Ea/kbT

(3)

reactants

where kb represents the Boltzmann constant; h is Plank’s constant; T is the temperature; qTS represents the molecular partition function of the transition state (rotational, vibrational, and electronic, as specified by the superscript); and Ea is the activation energy, calculated as shown in eq 1. The transition state structures were identified employing the synchronous transit guided method and were characterized by a single imaginary vibrational frequency.29 3. Results and Discussion The investigated radicals, divided into the four groups previously reported, exhibit a peculiar reactivity with respect to alkenes that is strongly affected by their own electronic structure. Because the studied alkenes can be represented as a mono- or disubstituted ethylene molecule (mono- or disubstitution occurs on the same carbon atom), this reactivity is also affected by the alkene substituent. Experimental evidence on this reactivity is reported in detail in ref 15 and briefly summarized here for the sake of clarity. Methyl (Me) and benzyl (Bn), which do not exhibit polar effects, were chosen as

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Figure 1. (a) Optimized molecular structures of the radicals considered in the simulations. Geometries were determined at the B3LYP/6-311+G(d,p) level of theory. (b) Investigated monomers.

representative radicals because they lead to the same chemical behavior (although Bn is less reactive than Me by more than 2 orders of magnitude, because of the benzyl radical resonance). Nucleophilic radicals react more slowly than Me with electronrich alkenes (propylene, isobutylene, 2-methoxy propylene, etc.) but faster with electron-deficient ones (methylacrylate, acry-

lonitrile, acrylaldehyde, etc.). Because of the strong cyano or carboxylic acceptor substituent, primary ambiphilic-electrophilic radicals (MEst, MCN) are generally more reactive than Me, whereas their tertiary counterpart reacts more slowly (about 2 orders of magnitude in terms of kinetic constant). In general, carboxylic substituents make the relative addition faster than

DFT Study of Additions of C-Centered Radicals to Alkenes

J. Phys. Chem. A, Vol. 115, No. 1, 2011 55

TABLE 1: Atomic Charges for Radicals Employed in the Simulationsa radical Me Bn

TBu

PCN

a

atom

charge

radical

atom

charge

radical

atom

charge

C1 C1 C2 C3 C4 C5 C6 C7 C1 C2 C3 C4 C1 C2 C3 C4 N1

-0.50 (0.17) -0.39 (0.15) 0.07 -0.07 (0.11) -0.19 (0.13) -0.07 (0.12) -0.19 (0.13) -0.07 (0.11) -0.04 -0.12 (0.04) -0.12 (0.04) -0.12 (0.04) 0.03 0.30 -0.16 (0.08) -0.16 (0.08) -0.50

PEst

C1 C2 C3 C4 C5 C6 C7 C8 O1 O2 C1 C2 C3 F4 F5 F6 O7

0.10 -0.08 (0.03) -0.08 (0.03) 0.33 0.19 -0.09 (0.04) -0.09 (0.04) -0.09 (0.04) -0.45 -0.29 -0.16 (0.14) 0.44 0.14 -0.09 -0.09 -0.09 -0.42

POH

C1 C2 C3 O1 C1 O1 C1 C2 N1 C1 C2 C3 C4 C5 C6 O1 O2

-0.06 -0.06 (0.04) -0.06 (0.04) -0.48 (0.42) -0.14 (0.11) -0.49 (0.43) -0.35 (0.21) 0.43 -0.50 -0.20 (0.12) 0.50 0.25 -0.13 (0.04) -0.13 (0.04) -0.13 (0.04) -0.48 -0.35

FAc

MOH MCN MEst

Data in parentheses are representative of the charge of hydrogen atoms bonded to the relative heavy atom, numbered as reported in Figure

1.

cyano groups (about 10 times). Finally, electrophilic radicals react faster than Me with electron-rich alkenes but more slowly with electron-deficient ones (the opposite of nucleophilic radicals). The structures of all of the considered molecules, optimized at the B3LYP/6-311+G(d,p) level of theory, are shown in Figure 1a, and atomic partial charges are collected in Table 1. Electrostatic potentials were calculated at the B3LYP/6-31G(d) level, and charges were fit using the RESP formalism as reported in ref 36. In the first part of this section, enthalpy changes calculated using all of the selected basis sets are compared with experimental data. The results show how the use of the smallest basis set leads to the worst prediction. Thus, to save computational time, all successive simulations were performed with the medium and large basis sets. In subsection 3.2, a cost-effective procedure to determine kinetic constants is developed, and activation energies for all possible reactions between the radicals and alkenes are presented and compared with experimental data. In subsection 3.3, the reactivity of radical-alkene pairs is investigated as a function of their molecular structure. 3.1. Reaction Enthalpies. Enthalpy changes for the different reactions involving Me radical and monomers shown in Figure 1b (with the addition of ethylene, ethyl vinyl ether, and vinyl silane) were determined with all basis sets. Calculated reaction enthalpies are compared with experimental data in Table 2. A close inspection of these values clearly shows how the smallest basis set is the least adequate for estimating this thermodynamic parameter, thus indicating that the addition of p diffuse function to H atoms is fundamental to correctly describe the energetics of the molecules under investigation. Although the 6-31G(d) basis set is cost-effective, able to estimate enthalpies with an error in the (5 kcal/mol range (for reported reactions), and has been reported in previous works37,38 as being suitable for studying radical reactions, its prediction capabilities are often improved by larger basis sets. On the other hand, higher basis sets do not always imply higher accuracy. This evidence was also reported in the pioneering paper of Fischer and Radom, where it was found that a basis set larger than 6-311+G(d,p), namely, 6-311+G(3df,2p), led to the poorest enthalpy determination among 35 different level of theory.15 In fact, enthalpy changes calculated for the reactions between 2-methoxy propylene, methylacrylate, acrylonitrile, styrene, and Me radical

TABLE 2: Comparison between Theoretical (UB3LYP) and Experimental Reaction Enthalpies for the Addition of a Me Radical to Different Unsaturated Compounds (CH2dCXY)a alkene X

Y

6-31G(d)

6-31G(d,p)

6-311+G(d,p)

exp15

H H H Me Me H H H Me Cl H H H Me H

H Me Et Me OMe OEt Cl SiH3 Cl Cl CO2Me CN CHO CN Ph

-24.8 -25.1 -24.6 -24.6 -22.3 -24.9 -27.9 -27.1 -27.5 -31.7 -23.2 -33.7 -33.2 -34.9 -33.8

-24.4 -24.7 -24.8 -22.3 -21.6 -24.5 -27.4 -26.6 -27.0 -31.2 -24.6 -33.3 -33.9 -34.5 -33.0

-21.1 -21.6 -20.3 -20.7 -18.0 -21.2 -23.6 -23.0 -23.2 -27.4 -18.9 -29.5 -28.7 -29.6 -29.8

-22.3 -22.0 -22.5 -22.4 -21.3 -22.7 -24.1 -22.9 -22.4 -28.6 -26.6 -31.8 -27.7 -29.9 -34.4

a

Energies reported in kcal/mol.

are better described through the medium-size basis set. In general, the evaluated thermodynamic properties are slightly underestimated by both computational methods. Nevertheless, this trend cannot be taken as a rule because, for some reactions, (i.e., vinyl silane, acrylaldehyde) the value is overestimated. With the exception of acrylaldehyde (which exhibits discrepancies from the experimental value of up to 5 kcal/mol), the accuracy of the predictions achieved using the two large-sized basis sets is satisfactory, being within the range of (2.5 kcal/ mol in all cases. Often reaction enthalpies are exploited to estimate activation energies through the empirical Evans-Polanyi-Semenov (EPS) equation.15,39-41 Accordingly, a linear relationship between activation energy and enthalpy change for a given reaction is presumed

Ea ) C + R∆Hr

(4)

where Ea and ∆Hr are the activation energy and reaction enthalpy, respectively, whereas C and R are two parameters. The accuracy of this equation is often questionable; however,

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TABLE 3: Reaction Enthalpies for the Addition of Carbon-Centered Nonpolar Nucleophilic, Nucleophilic, Ambiphilic-Electrophilic, and Electrophilic Radicals to Different Unsaturated Compounds (CH2dCXY)a alkene X

nonpolar nucleophilic Y

Me

Bn

nucleophilic tBu

POH

ambiphilic-electrophilic MOH

PEst

PCN

MEst

electrophilic MCN

FAc

H Me -24.7 -8.7 -13.8 -14.1 -19.3 -5.6 -5.8 -18.1 -13.8 -14.6 Me Me -22.3 -5.3 -10.8 -11.2 -16.6 -2.3 -2.3 -15.6 -12.3 -13.0 Me OMe -21.6 -6.3 -10.6 -11.2 -16.7 -1.6 -3.0 -14.9 -12.5 -17.8 H Cl -27.4 -10.7 -16.7 -16.8 -22.3 -8.1 -6.9 -20.5 -15.9 -17.2 Me Cl -27.0 -10.3 -15.6 -15.8 -22.0 -7.1 -6.1 -20.3 -15.8 -17.2 Cl Cl -31.2 -14.2 -19.1 -18.8 -25.1 -10.6 -8.7 -24.1 -18.7 -20.9 H CO2Me -24.6 -15.4 -19.8 -20.6 -25.5 -12.1 -10.8 -23.3 -18.8 -19.3 Me CO2Me -23.3 -18.8 -19.8 -21.0 -26.9 -11.6 -10.7 -24.9 -20.6 -21.9 H CN -33.3 -17.5 -23.0 -23.0 -27.9 -14.3 -13.3 -26.2 -20.3 -20.3 H CHO -33.9 -16.5 -24.2 -24.6 -29.4 -15.0 -12.9 -27.5 -22.1 -24.3 Me CN -34.5 -18.8 -23.5 -23.7 -29.3 -15.0 -13.0 -27.4 -22.8 -23.7 H Ph -33.0 -16.6 -22.4 -22.8 -28.2 -14.1 -13.5 -26.3 -22.4 -23.4 Me Ph -32.8 -15.8 -20.7 -23.9 -28.0 -12.3 -11.8 -26.1 -22.3 -23.2 b Reaction enthalpies for the addition of carbon-centered nonpolar nucleophilic, nucleophilic, ambiphilic-electrophilic and electrophilic radicals to different unsaturated compounds (CH2dCXY). Energies determined at the B3LYP/6-311+G(d,p) level of theory are reported in kcal/mol. Alkene Nonpolar nucleophilic Nucleophilic Ambiphilic-electrophilic Electro-philic X Y Me Bn tBu POH MOH PEst PCN MEst MCN FAc H Me -20.2 -5.3 -10.5 -10.5 -16.2 -2.2 -2.7 -14.6 -11.7 -11.4 Me Me -21.2 -4.7 -10.9 -11.1 -16.3 -2.3 -2.9 -15.0 -12.5 -12.7 Me OMe -22.6 -8.0 -11.7 -12.1 -18.3 -4.0 -4.0 -15.9 -14.7 -15.2 H Cl -23.6 -6.8 -13.7 -13.5 -18.7 -5.1 -4.4 -17.2 -13.3 -14.0 Me Cl -23.1 -6.4 -12.6 -12.4 -18.3 -4.1 -3.6 -17.0 -13.1 -14.0 Cl Cl -27.3 -10.5 -16.5 -21.9 -22.0 -8.0 -6.6 -20.9 -16.2 -17.2 H CO2Me -26.8 -12.4 -16.1 -17.1 -22.1 -8.3 -7.4 -20.0 -16.0 -16.0 Me CO2Me -28.2 -13.2 -17.0 -16.9 -23.0 -8.6 -8.2 -22.0 -18.3 -19.1 H CN -29.5 -14.0 -19.8 -19.3 -24.1 -11.1 -9.2 -22.7 -17.4 -18.6 H CHO -30.6 -13.7 -21.1 -21.0 -25.3 -12.3 -10.6 -24.3 -19.5 -20.5 Me CN -30.7 -15.5 -20.3 -20.1 -25.7 -11.7 -10.2 -24.1 -20.0 -20.3 H Ph -29.7 -13.0 -19.5 -19.5 -24.9 -10.9 -10.9 -23.2 -20.0 -20.5 Me Ph -29.1 -12.6 -17.5 -17.6 -24.6 -9.0 -9.1 -22.9 -19.8 -20.2 a

Energies determined at the B3LYP/6-31G(d,p) level of theory and reported in kcal/mol.

the predicted trend between Ea and ∆Hr can be reasonably precise for classes of reactions showing the same reactive behavior. According to this equation, a linear correlation is operative between reaction enthalpy and activation energy; therefore, a poor estimation of the first is expected to directly affect the accuracy of the estimated activation energy. As a consequence, to save computational time, all simulations were performed without the smallest basis set, using two basis sets only. The results of these simulations are summarized in Table 3. Even though a direct assessment of the accuracy of all of these data is difficult because of the lack of experimental values, some comparisons among the estimated values can be done profitably. Generally, the results obtained with the two basis sets are in good agreement, meaning that the calculated enthalpy changes differ by less than 3 kcal/mol. Sometimes, the discrepancy between the two used basis sets is negligible, as for the case of isobutylene, for which the reaction enthalpies calculated for all selected radicals differ by less than 1 kcal/mol. Estimation of exothermicity leads to systematically smaller values using the largest basis set. The only exception was found for the 2-methoxy propylene, for which data using the smallest basis set were smaller by about 1.5 kcal/mol. Comparing data reported in Table 2 for 2-methoxy propylene and Me, it appears that the results obtained with the 6-31G(d,p) basis set are more reliable. Even though a direct comparison with experimental data is difficult to perform, a partial validation of the obtained results can be carried out based on the EPS linear relationship. In fact, previously reported experimental trends on the reactivity between radicals and monomers are qualitatively in agreement with those predicted by computation. Higher exothermicity

(about 15 kcal/mol) is found for Me than Bn. For nucleophilic radicals, enthalpies are in agreement with the high reactivity against electron-deficient monomers. For the ambiphilic-electrophilic radical family, the enthalpy data reproduce the expected trends in terms of reactivity of carboxylic- and cyano-substituted and primary radicals versus tertiary ambiphilic-electrophilic ones. In fact, the values for the electrophilic radical are also aligned with those of MCN radical. Finally, it is worth noticing that DFT ab initio simulations with the selected basis sets provide useful results with accuracies that are sometimes below the experimental error. Furthermore, many values determined in this work are not available in the literature and can be considered reasonable first guesses for modeling applications. 3.2. Activation Energies. As already mentioned, the determination of kinetic constants for the addition of radicals to alkenes requires evaluating pre-exponential factors (A) and activation energies (Ea). Estimation of energetic parameters (such as enthalpy changes or activation energies) can lead to results with accuracies that depend on the applied computational method. On the contrary, the determination of A (and of the corresponding entropy change) is usually less precise. Even though many efforts are currently focused on improving such predictions (i.e., hindered-rotor corrections for the lower vibrational frequencies), a fully satisfactory method to be used systematically to evaluate A or entropy change for propagation reactions is currently not available.1,19,42-45 To overcome this difficulty, a hybrid computational-experimental procedure was applied in this work. As reported by Fisher and Radom, for polyatomic radicals reacting with alkenes, log A ranges within a narrow interval, from 6.5 to 8.5 L/(mol/s).15 First, this means that the rate constant is mainly affected by Ea. Second, starting

DFT Study of Additions of C-Centered Radicals to Alkenes TABLE 4: Activation Energies for the Addition of Carbon-Centered Nonpolar Nucleophilic Radical to Different Unsaturated Compounds (CH2dCXY)a alkene X H Me Me H Me Cl H Me H H Me H Me

Y Me Me OMe Cl Cl Cl CO2Me CO2Me CN CHO CN Ph Ph

6-31G(d,p)

6-311+G(d,p)

Me

Me

4.8 6.7 5.9 4.0 4.1 2.8 4.9 2.3 1.8 0.3 1.0 2.7 3.0

Bn 10.4 11.7 10.9 8.1 8.2 6.4 5.2 6.0 5.4 4.1 4.8 6.5 7.0

6.9 5.3 5.8 5.7 5.6 4.2 3.6 3.5 3.3 1.3 2.5 4.2 4.4

Bn 12.3 10.7 10.7 10.4 10.5 8.5 6.8 7.2 7.2 5.2 6.5 8.4 8.7

exp15 Me 6.6 6.2 6.0 5.7 5.4 4.1 4.0 3.8 3.7 3.6 3.6 4.2 4.1

J. Phys. Chem. A, Vol. 115, No. 1, 2011 57 TABLE 5: Calculated Activation Energies for the Addition of Carbon-Centered Nucleophilic Radical to Different Unsaturated Compounds (CH2dCXY) Compared with Experimental Values (in Parentheses)15 a alkene

Bn 9.7 9.3 7.9 7.9 7.0 7.0 6.9 6.3 7.4 7.5

a Energies determined at both the B3LYP/6-31G(d,p) and B3LYP/6-311G+(d,p) levels of theory and compared with experimental data; reported in kcal/mol.

from a large number of experimental data available in the literature, it is possible to identify an average value of A selected within the reported range, characteristic of an entire family of reacting radicals. As a result, log A ) 8.5 can be set for primary radicals such as Me, Bn, MCN, and FAc; log A ) 8.2 for MOH and MEst; and finally, log A ) 7.5 for tertiary radicals (tBu, PCN, POH, PEst). Note that this simplification is the same as that applied in ref 15 to estimate activation energies. The resulting hybrid computational-experimental procedure is as follows: For a given reaction, the pre-exponential factor is set equal to one of the reported average values depending on the radical involved. In other words, A is assumed here to be fully independent of the alkene molecule. Then, the activation energy is determined as reported in section 2. The kinetic constants estimated in this way are more accurate than those estimated in a fully predictive way. Because this hybrid computational-experimental procedure was aimed to assess a suitable computational approach, only activation energies are explicitly reported in this work; the values of A, computationally determined for a large number of reactive steps, are reported in the Supporting Information. Interestingly, these values are in qualitative agreement with the selected average A values (higher reactivity for primary radicals and lower for more hindered tertiary ones), and they confirm that A is more affected by the radical than the alkene. A comparison between the experimental and calculated activation energies, for both the medium and large basis sets, is reported in Table 4 for reactions of Me and Bn. All of the computational results are in good agreement with the experimental ones. More in detail, the differences between experimental and calculated activation energies at the highest level of theory for Me radical are below 1 kcal/mol in all cases, with the exception of acrylaldehyde (∆Ea ) 2.3 kcal/mol). The same basis set gives comparable results also for the Bn reactivity (the largest discrepancy was found for acrylaldehyde with a ∆Ea ) 1.7 kcal/mol). With the exception of the aromatic monomers (styrene and R-methylstyrene), the alkenes are listed in all tables in order of increasing rate constant for methyl radical additions. This behavior is well predicted using the large basis set, whereas both the accuracy and agreement with this trend are less when the medium basis set is used. The findings for the methyl radical are in agreement with those reported by Fischer and Radom.15 They found that the UB3-LYP procedure gives satisfactory

nucleophilic radical

X

Y

tBu

H Me Me H Me Cl H Me H H Me H Me

Me Me OMe Cl Cl Cl CO2Me CO2Me CN CHO CN Ph Ph

7.5 (6.2) 9.8 (6.4) 11.5 (7.1) 5.4 (4.5) 6.3 (4.7) 4.2 (2.7) 2.5 (2.0) 3.0 (2.3) 1.5 (1.1) 0.3 (1.4) 1.4 (1.7) 3.9 (3.3) 4.9 (3.8)

POH 3.4 5.8 7.5 0.8 1.7 5.8 0.0 0.0 0.0 0.0 0.0 2.3 2.0

(-) (-) (6.0) (-) (4.3) (2.2) (0.0) (0.6) (0.0) (0.0) (0.0) (2.4) (3.2)

MOH 2.5 3.9 4.0 3.7 0.7 6.6 1.7 1.6 0.0 0.0 2.8 2.4 3.1

(8.3) (8.3) (8.4) (6.5) (7.0) (5.1) (3.6) (3.7) (3.3) (2.9) (3.6) (5.6) (5.5)

a Energies determined at the B3LYP/6-31G(d,p) level of theory and reported in kcal/mol.

TABLE 6: Activation Energies for the Addition of Carbon-Centered Ambiphilic-Electrophilic and Electrophilic Radicals to Different Unsaturated Compounds (Ch2dCXY) Compared with Experimental Values (in Parentheses)15 a alkene X H Me Me H Me Cl H Me H H Me H Me

Y

ambiphilic-electrophilic PEst

Me 16.4 (-) Me 12.1 (-) OMe 12.5 (7.9) Cl 8.9 (-) Cl 9.1 (7.3) Cl 7.8 (6.0) CO2Me 6.4 (6.0) CO2Me 6.7 (5.4) CN 5.9 (5.6) CHO 4.4 (5.8) CN 5.2 (5.3) Ph 6.2 (5.1) Ph 7.0 (5.1)

electrophilic

PCN

MEst

MCN

FAc

10.8 (-) 13.2 (-) 12.0 (7.9) 10.4 (-) 10.6 (7.6) 9.5 (6.7) 8.1 (7.0) 8.2 (6.1) 8.1 (5.9) 6.5 (6.2) 7.2 (6.3) 7.4 (5.8) 8.2 (5.8)

4.7 (4.8) 5.6 (4.3) 2.7 (4.5) 4.3 (4.8) 4.1 (4.4) 3.3 (4.1) 3.1 (3.7) 2.7 (3.2) 2.9 (3.7) 1.9 (3.9) 1.6 (3.4) 1.8 (2.0) 1.8 (2.5)

6.5 (5.6) 7.3 (5.7) 8.1 (5.0) 6.3 (5.6) 5.8 (5.5) 5.3 (5.1) 4.8 (4.4) 4.2 (4.0) 5.4 (4.4) 3.8 (5.2) 4.1 (4.2) 3.3 (3.7) 3.4 (3.4)

3.1 (4.4) 3.6 (2.6) - (-) 3.6 (4.8) 2.7 (-) 2.2 (4.1) 3.7 (4.7) 1.9 (3.3) 3.5 (5.4) 1.5 (5.0) 1.7 (4.4) 0.6 (2.3) 0.7 (2.2)

a Energies determined at the B3LYP/6-31G(d,p) level of theory and reported in kcal/mol.

results employing both small [6-31G(d)] and large [6-311+G(3df,2p)] basis sets, and they determined R2 correlations with experimental barriers for the 6-31G(d) and the 6-311+G(d,p) basis set of 0.77 and 0.81, respectively. The correlations obtained from data reported in Table 4 led to R2 values equal to 0.67 and 0.81 for the methyl radical, whereas values of 0.77 and 0.74 were found for the benzyl radical. Reported R2 values confirm again that the accuracy is not strictly related to the basis set size. All of the activation energies calculated at the medium level of theory are reported in Tables 5 and 6 for nucleophilic, ambiphilic-electrophilic, and electrophilic radicals. For ambiphilic radicals, the best results were obtained for tBu (R2 ) 0.95) and POH (R2 ) 0.64) radicals, whereas predictions for MOH (R2 ) 0.19) reactivity were affected by discrepancies with experimental data as large as 6 kcal/mol. In particular, this is true for reactions involving alkenes with electron-donor substituents (propylene, isobutylene, and 2-methoxy propylene). For ambiphilic and electrophilic radicals, the results are in agreement with the experimental data, and often the errors are below 1 kcal/mol. The only exception is for 2-methoxy propylene, which supports the conclusion that the 6-31G(d,p) approach gives less accurate results when an oxygen atom is directly bonded to the

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TABLE 7: Activation Energies for the Addition of Carbon-Centered Radicals to Different Unsaturated Compounds (CH2dCXY)a alkene X

Y

H Me Me H Me Cl H Me H H Me H Me

Me Me OMe Cl Cl Cl CO2Me CO2Me CN CHO CN Ph Ph

ambiphilicelectrophilic

nucleophilic

electrophilic

tBu POH MOH PEst PCN MEst MCN 8.5 8.2 8.8 6.8 7.7 5.4 3.8 4.2 2.7 1.5 2.9 5.4 6.6

8.3 7.4 7.8 5.5 4.9 1.1 0.0 2.3 0.0 0.0 0.0 4.1 1.6

7.9 6.5 7.3 5.7 6.1 7.9 5.1 3.2 2.0 1.1 2.2 4.1 5.0

18.4 10.9 10.3 10.7 11.0 9.3 8.5 8.6 7.6 5.8 7.2 8.3 8.6

12.6 11.4 10.7 11.7 11.5 10.6 9.7 9.7 9.4 7.5 7.7 8.9 9.7

6.5 4.4 4.1 5.8 5.5 4.6 4.7 3.9 4.6 3.6 3.2 3.2 3.7

7.3 5.7 4.9 7.5 7.0 6.3 6.3 5.4 6.9 5.4 5.4 4.5 4.6

FAc 4.6 2.0 5.1 4.0 4.0 5.7 3.3

-

-

4.2 3.6 1.2 1.3

a Energies determined at the B3LYP/6-311+G(d,p) level of theory and reported in kcal/mol.

reactive center (POH, MOH, 2-methoxy propylene). After the activation energy determination at the medium level of theory, the energetic parameters were calculated at the highest level of theory [6-311+G(d,p)], and the results are summarized in Table 7. The calculated values for nucleophilic radicals provide inconsistent indications. In fact, for the MOH radical, the estimation of the activation energy is systematically improved with the largest basis set (R2 ) 0.66), leading to a mean error of about 1.5 kcal/mol. On the other hand, the tBu reactivity is better described by the 6-31G(d,p) basis set, and apart from the 2-methoxy propylene case, the deviation is below 1 kcal/mol (R2 determined with the higher basis set equal to 0.92). Finally, for the POH radical, the two methods lead to similar results (R2 ) 0.77). The comparison in the determination of kinetic parameters for ambiphilic-electrophilic radicals shows interesting results: Activation energies between radicals and alkene with electron-donor substituents (propylene, isobutylene, and 2-methoxy propylene) are determined with higher accuracy employing the 6-311+G(d,p) basis set, whereas parameters for alkenes with electrophilic substituent are better described with the smallest basis set. For the reactivity of FAc (electrophilic radical), the best results are always obtained with the largest basis set. The correlations with experimental activation energies for the abovereported reactions are provided in the Supporting Information. At this point, some general comments on the reported results can be made. Independently of the selected basis set, the DFT approach is suitable for the determination of the activation energy of the addition reactions of carbon-centered radicals to alkenes with good accuracy. With a few exceptions, all of the results are in agreement with the experimental data, and the discrepancies are often below the experimental error. About the prediction capabilities of different basis sets in terms of kinetic parameters, it has to be pointed out that an optimal choice is not easy to make a priori. In fact, in some cases, the largest basis set provides better results (i.e., electrophilic radicals); in others cases, the best choice is the smallest basis set (i.e., ambiphilic-electrophilic radicals with alkene substituted with an electron-acceptor substituent); and in rare cases, the option is not clear (i.e., for the POH system). Finally, it is worth mentioning that the differences between the calculated results using one or the other of the two basis sets range from 0.1 to 2.0; that is, the estimations are practically equivalent. In fact, apart from the values in Table 4, all other experimental activation energies were estimated starting from the value of

the rate constant and then applying a constant pre-exponential factor to the whole class of reactions. This procedure, even though its validity was confirmed by comparison with different approaches reported in ref 15, is affected by an error that can be estimated as being on the order of 1 kcal/mol, that is, comparable to the range of discrepancies mentioned above. For the EPS equation, the values of C and R estimated from the DFT-calculated enthalpy changes (Table 3) and activation energies are reported in the Supporting Information. As a result, it was found that the EPS relation leads to accurate predictions only in a few cases (i.e., for tBu and PCN radicals using the medium basis set, where the linear correlation gives an R2 value larger than 0.9). No improvement with the dimension of the basis set was found, and no relation with the chemical structure of the reacting radical was recognized. Therefore, these results seem fortuitous, and in general, it can be concluded that the EPS equation cannot be used to accurately determine kinetic constants starting from thermodynamic data estimated by computational methods. Similar results were reported by Fischer and Radom.15 To maintain the correlation between the activation energy and the reaction enthalpy, the authors introduced corrective factors to the EPS treatment to account for polar interactions. 3.3. Molecular Structures. As previously reported, quantum chemistry can be profitably applied to estimate geometries (including bond distances, bond angles, and dihedral angles) of reactants, products, and transition states. The latter, in particular, are extremely difficult to determine experimentally because of the evanescent nature of the transition state. Geometries were determined with reasonable accuracy using both the medium and large basis sets.38 In Figure 2, the geometry of a typical transition state for the addition of a carbon-centered radical (indicated by R) to an alkene is shown. The alkene is characterized by two substituents, X and Y, whereas the transition state geometry is characterized by three parameters: the bond distance between the carbon radical and the alkene carbon in the R position (d), the bond angle between the carbon radical and the alkene double bond (R), and the torsional angle (δ) relative to the distortion of the alkene planarity. Data reported in Figure 2 are those characterizing the transition state of the reaction between alkenes and methyl radical. The bond distance d increases from 2.39 Å for propylene to 2.49 Å for styrene, acrylonitrile, and acrylaldehyde. This trend follows the increase of reactivity of alkenes and can be explained by the higher activation energy required to bring the reactants close to each other. Also, the torsional angle of the transition state decreases from olefins to more reactive monomers; this behavior can be ascribed to the higher energy required to break the planarity of the sp2 carbon atoms. The bond angle R is less related to the reactivity of the system, but it is more affected by the steric effect of the alkene substituents. The same considerations as reported for the Me radical can be made regarding the geometries of all of the determined transition states. (The calculated distances are shown in Figure 3, and the corresponding numerical values are given in the Supporting Information.) A comparison between the reactivities of Bn and Me can be simply made in terms of d values. In fact, the distance d constantly decreases by about 0.15 Å from Me to Bn, because of the relative lower reactivity of the benzyl radical (see Figure 2). Similar qualitative considerations apply to the ambiphilicelectrophilic radicals: Larger values of d can be found for MEst and MCN primary radicals, which are more reactive than PEst and PCN tertiary radicals. More in detail, smaller values of d both from MEst to MCN and from PEst to PCN are due to the

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J. Phys. Chem. A, Vol. 115, No. 1, 2011 59

Figure 2. Transition state structure of radical addition to alkene. Data reported are relative to the addition of methyl (Me) radical to different alkenes. In the last column, a comparison with the Bn reactivity is reported. Distances (d) are reported in angstroms, and angles (R and δ) are in degrees.

larger reactivity of radicals bonded to a tert-butoxycarbonyl group than to a cyano group (which is a more effective electronacceptor substituent). Finally, comparing nucleophilic and electrophilic radicals, the latter exhibit smaller values of d in agreement with their lower reactivities, whereas, as expected, d is smaller for POH than MOH. It is worth mentioning that the proposed correlation between the transition state geometry and the activation energy differs from that proposed by Fischer and Radom.15 In fact, in that work, the authors correlate the distance d to the reaction enthalpy and the activation energies to the charge transfer in the transition structure configuration. In contrast, by a close inspection of the geometries determined for the methyl radical addition employing the larger basis set, it is possible to observe that d correlates better with the activation energy (R2 ) 0.94) than with the enthalpy change (R2 ) 0.79). Moreover, apart from data for the POH and FAc radicals, the distance d correlates the experimental activation energy with accuracy comparable to the determined energetic barriers. In addition, for several radicals (i.e., MOH, Pest, PCN, MEst, and MCN), this accuracy is even higher than that determined by comparing the activation energy with both basis sets. (The data are given in the Supporting Information.) 3.4. Perspectives. This computational approach can be profitably used to investigate reactions involving carbon-centered radicals and alkenes, leading to reliable results after a few preliminary simulations aimed at identifying the best choice in terms of basis set. As an example, if the determination of the homopolymerization kinetic parameters for a popular system such as poly(methylacrylate) (PMA) is needed, the generic homopolymerization step can be simulated as the reaction between methylacrylate itself and MEst radical using the different basis sets. As reported in section 2, the reactivity of acrylate and this primary radical is better predicted when using

the 6-31G(d,p) basis set, which gives an activation energy of 3.1 kcal/mol, close to the value of 3.7 reported in ref 15. Then, using this basis set, it becomes easy to simulate the reaction between methylacrylate and a secondary carbon radical MA• such as that sketched in Figure 4: This radical has a structure much closer to that of the radical involved in the actual homopolymerization process. Conceptually, this procedure is the same as reported experimentally in ref 46, in which, to approximate the acrylate-radical reactivity, the 1-[(tert-butoxy)carbonyl]ethyl radical (EEst) was synthesized. The activation energy evaluated for this kinetic step is equal to 5.9 kcal/mol: This value is close to the 5.1 kcal/mol reported for the EEst radical, and it approaches that for the propagation of methylacrylate, measured by pulsed laser polymerization (PLP), which is 7.04 kcal/mol.47 The improvement in modeling the reactive system using a radical structure closer to that of the actual growing chain is then clear. Moreover, it is possible to further improve the radical simulation by increasing the complexity of the system, that is, considering larger radical species, for example, the dimer MA-MA• in Figure 4. If the reactivity of this dimer radical is studied, an activation energy of 7.4 kcal/ mol is calculated. This value is in very good agreement with that determined by PLP, which corresponds to the activation energy of propagation of a long growing-chain radical with methylacrylate. This value is already a meaningful approximation of the value of long chain radicals. In fact, previous computational results show how the propagation constant is a function of the degree of polymerization:48,49 This dependence is restricted to the first two propagation steps, after which the activation energy reaches a plateau value independent of the radical chain length. Finally, the proposed computational methodology can be applied to study the reactivity of free-radical copolymerization systems. The experimental study of such systems is always quite

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Figure 3. Distance d between carbon atoms in the transition state structure of the addition reactions between different unsaturated compounds: (a) ambiphilic-electrophilic radicals and (b) tBu, MOH, and FAc radicals (POH data, reported in the Supporting Information, are almost superimposed on those of tBu). kp12

MA• + ST 98 MA-ST•

kp22

ST• + ST 98 ST-ST•

Figure 4. Computer-designed reactive radicals.

onerous, because of the large number of species and, therefore, of reaction steps involved. Usually, low-conversion experiments with different monomer mixture compositions are carried out to evaluate the ratios of the propagation rate coefficients of each monomer with each type of radical, so-called reactivity ratios. These same parameters can be estimated through the simplified approach reported below with reference to the industrially relevant system methylacrylate-styrene (MA-ST). Four different propagation constants (kpij) can be identified kp11

MA• + MA 98 MA-MA•

(5)

kp21

ST• + MA 98 ST-MA•

(6)

(7)

(8)

where kpij represents the propagation rate coefficient for the addition of monomer j to radical i, that is, an active chain with a radical on a monomer unit of type i. In the framework of the so-called terminal model,50 the reactivity of a polymer chain is determined by the last monomer unit only, and the four different rate coefficients in eqs 5-8 are needed to fully characterize the system evolution. When focusing on the polymer composition, such rate coefficients are lumped into two quantities only, namely, the previously mentioned reactivity ratios: r1 ) kp11/

DFT Study of Additions of C-Centered Radicals to Alkenes

J. Phys. Chem. A, Vol. 115, No. 1, 2011 61 TABLE 8: Comparison between Kinetic Parameters for the MA-ST Copolymerization Evaluated at the 6-31G(d,p) Level of Theory and the Experimental Valuesa calculated

experimental

radical i monomer j Ea log(A) kp at 25 °C r MA• MA• ST• ST•

MA ST ST MA

7.41 5.53 8.28 7.53

8.55 8.24 8.14 8.24

1304 11556 117 523

Ea

kp

0.11 7.04 15400 (20 °C)51 0.22 7.77 86 (25 °C)52 -

a Activation energies are kcal/mol; pre-exponential factors and kinetic constants in L/(mol s).

Figure 5. MA molar fraction in copolymer vs MA molar fraction in the corresponding monomer mixture. Symbols: Experimental data.53 Curves: dotted, experimental data from ref 15; dash-dotted, computational data using Bn and MEst radicals; continuous, computational data using MA• and ST• radicals. The gray continuous line indicates the diagonal.

kp12 and r2 ) kp22/kp21. The evaluation of these four kinetic constants requires the determination of the corresponding activation energies and pre-exponential factors. Using the experimental data from ref 15 for methylacrylate, styrene, Bn, and MEst radicals at 298 K, values of r1 ) 0.25 and r2 ) 2.56 were estimated. These values are significantly different from those reported in literature PLP studies of the copolymerization of MA and ST: r1 ) 0.19 and r2 ) 0.73.51 Whereas the r1 value is slightly overestimated, the prediction of styrene reactivity is completely wrong (larger instead of smaller than 1). These values were predicted by QM, using the 6-31G(d,p) basis set (cf. Tables 4 and 6): The simulations performed at the same temperature led to reactivity ratios equal to 0.12 and 0.11, respectively. Finally, the reactive radical was modeled as the secondary radical ST• shown in Figure 4, following the same approach as described above in the case of methylacrylate homopolymerization. Considering this radical structure, new values of the reactivity ratios of 0.11 and 0.22 were predicted. Therefore, the values of both reactivity ratios are closer to the experimental values, even though slightly underestimated. Regarding the overestimation of r2, it is worth noticing that QM calculations describe the reactivity of MA• better than that of ST• because of the difficulty in predicting the resonance of the radical over the benzene ring, as reported in a previous work.19 A final comment on the copolymerization of ST and MA can be made by considering the results reported in ref 15. In fact, the reported experimental data show a good correlation between propagation rates of copolymerization reactions and reaction rates calculated using selected small radicals. These findings validate the terminal unit model, demonstrating that the penultimate unit contributions to the MA-ST copolymer composition are not important. To evaluate the impact of such different values on the prediction of the copolymer composition, the classic Mayo and Lewis plot (instantaneous copolymer composition vs monomer mixture composition) was examined.50 In Figure 5, experimental copolymer compositions are compared with the different MA-ST composition curves calculated using the values of reactivity ratios from ref 15 or predicted by QM considering MEst and Bn or MA• and ST• as radicals. As expected, the worst description of the system was obtained using the Fisher and

Radom data15 directly, whereas the agreement with the experimental data was definitely better when the reactivity ratios predicted through QM simulations were used. In particular, the best agreement was found when MA• and ST• radicals were employed, even though improvements are indeed possible. All of the values of kinetic parameters predicted by QM using the 6-31G(d,p) basis set are reported in Table 8, along with the experimental values measured by PLP analysis.15,51 Once more, the proposed methodology provides reasonable predictions of the system reactivity, thus representing a powerful tool for investigating free-radical copolymerization systems. 4. Conclusions The reactions between different carbon-centered radicals and alkenes have been studied through quantum chemistry. Among all possible basis sets, two were selected and used to perform all simulations. It was found that both methodologies led to the determination of activation energies in agreement with the experimental data. (Often, differences between calculated and experimental data were below the experimental error.) It was also found that there was no clear evidence of improvements using the largest basis set; rather, the best method depends on the specific system under examination. In addition to the determination of kinetic parameters, the proposed computational approach can be used to determine the geometries of the involved transition states with reasonable accuracy; by analyzing such data, preliminary information about the reactivity of the different systems is readily accessed. Finally, the computational approach proposed here represents a powerful tool for investigating the reactivity of homo- and copolymer free-radical systems. Starting from the investigation of simple and elementary reactive steps, reasonable first-guess values of the kinetic parameters are produced that are useful for meaningful evaluations of reaction kinetics, molecular weight increases, and copolymer compositions. Acknowledgment. The authors acknowledge Prof. Robin A. Hutchinson of Queen’s University for fruitful discussions. Supporting Information Available: Calculated pre-exponential factors for a number of reactions involving carboncentered radicals and alkenes. Regression factors (R2) for correlations of calculated activation energies and values of d against experimental barriers. Parameters of the EPS equation for all radicals studied. Structures of reactants and transition states. This information is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Liang, K.; Dossi, M.; Moscatelli, D.; Hutchinson, R. A. Macromolecules 2009, 42, 7736.

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