Quantum Chemistry Investigation of Secondary Reaction Kinetics in

Article pubs.acs.org/JPCA

Quantum Chemistry Investigation of Secondary Reaction Kinetics in Acrylate-Based Copolymers Danilo Cuccato, Evangelos Mavroudakis, and Davide Moscatelli* Dipartimento di Chimica, Materiali ed Ingegneria Chimica, Politecnico di Milano, Via Mancinelli 7, 20131 Milano, Italy S Supporting Information *

ABSTRACT: Recently, a growing amount of attention has been focused on the influence of secondary reactions on the free radical polymerization features and the properties and microstructure of the final polymer, particularly in the context of acrylate copolymers. One of the most challenging aspects of this research is the accurate determination of the corresponding reaction kinetics. In this paper, this problem is addressed using quantum chemistry. The reaction rate coefficients of various backbiting, propagation, and β-scission steps are estimated considering different chain configurations of a terpolymer system composed of methyl acrylate, styrene, and methyl methacrylate. The replacement of methyl acrylate radical units with styrene and methyl methacrylate globally decreases the backbiting probability and shifts the equilibrium toward the reactants, while the effect of replacing adjacent units is weaker and more dependent upon the specific substituting monomer. Propagation kinetics is affected primarily by the replacement of the radical units, while this effect appears to be particularly effective on midchain radical reactivity. The overall results clarify the different physicochemical behavior of chain-end, midchain, and short-branch radicals as a function of copolymer composition, providing new insights into free radical polymerization kinetics.

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INTRODUCTION The industrial and commercial relevance of poly(acrylates) and their high level of production worldwide reflect the wide applicability of these materials. Acrylate-based polymers are primarily used for surface treatments (e.g., coatings, films, latex paints, and pigments), biomedical applications (e.g., drug delivery systems and biocompatible adhesives), and additives for processing (e.g., thickeners, flocculants, and binding and plasticizing agents). Moreover, acrylate-based polymers are employed in the manufacture of a large variety of materials that display remarkable performance characteristics (e.g., synthetic rubbers and fibers, glues, and acrylic glass) because of their breakage resistance and elasticity. Poly(acrylates) are primarily produced by free radical polymerization (FRP), and a large set of acrylic esters with different functions can be combined with themselves or other vinyl monomers [e.g., styrene (St), methacrylates, vinyl chloride, and fluorinated monomers] to obtain high-performance copolymer formulations. The copolymerization of acrylic monomers with St for the manufacture of automotive coatings belongs in this context.1 The polymerization behavior of acrylic compounds can differ significantly from the ideal because of the demonstrated effect of backbiting and other secondary reactions. In this regard, the polymerization process can be affected and the investigation of the reaction kinetics can be subject to the interference of the intramolecular chain transfer.2,3 Particularly critical cases include high-temperature processes, such as those performed to obtain low-molecular weight resins and oligomers.4,5 Moreover, close correlation has been observed between the peculiarity of the most relevant secondary steps and certain © XXXX American Chemical Society

features of the resulting polymer, such as the degree of branching and the formation of gel fractions.6−8 The importance of investigating the kinetics of secondary reactions and the determination of their rate coefficients has recently become evident.9 The development of pulsed-laser polymerization (PLP) analysis has resulted in numerous attempts to estimate accurate rate coefficients for reactions that involve or form midchain radicals (MCRs). Interesting results have been obtained, particularly for butyl acrylate (BA).6,10−12 Moreover, the computational study of polymerization kinetics using quantum chemistry has facilitated the indepth investigation of the kinetics of specific steps of the secondary reactions of acrylic polymers in addition to polymerization systems not previously examined using the experimental PLP analysis.13−18 Most previous studies of secondary reactions in acrylate polymerization have focused on only homopolymer systems. However, it is interesting to consider the secondary steps that are active in copolymer formulations. Several studies of acrylate copolymer systems at high temperatures have been performed, in particular concerning the polymerization of BA and St.1,5,19 Under these polymerization conditions, the activity of the MCRs of poly(butyl acrylate) (PBA) is enhanced, whereas their kinetics is likely to be affected by the presence of a St monomer and by St units in the chains. Moreover, β-scission reactions yield a significant portion of polymer chains with terminal Received: February 27, 2013 Revised: April 30, 2013

A

dx.doi.org/10.1021/jp402025p | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

Article

Figure 1. Reaction schemes of the investigated backbiting steps. Notation adopted for the replacement units: tr, chain-end radical unit; mr, MCR unit; pu, penultimate unit adjacent only to the MCR site; mu, intermediate unit between chain-end and midchain radical sites. For MA units, R is COOCH3 and X is H. For St units, R is C6H5 and X is H. For MMA units, R is COOCH3 and X is CH3.

Specifically, terpolymer chains composed of units of methyl acrylate (MA), St, and methyl methacrylate (MMA) are considered. MA and MMA have been selected as representative monomers of the acrylate and methacrylate families, respectively, for the performance of simulations that adopt polymer chain models with a reasonable number of atoms. A comprehensive kinetic copolymerization scheme is presented, and the specific reaction rate coefficients are estimated using a consolidated computational procedure. This paper aims to evaluate physicochemical parameters that are not accessible using an experimental approach but are of substantial interest in the development of complex kinetic models of acrylate copolymerization.

double bonds, which act as macromonomers with respect to further initiation reactions. Other studies have been conducted on BA−St copolymerization at low temperatures and noted the effect of a decrease in the polymerization rate and the gel content as a result of the presence of St.20−22 Kinetic modeling of terpolymer resins that include methacrylates has been performed to determine the impact of the secondary reactions on the polymerization properties in addition to PLP studies in estimating the composition-averaged propagation rate coefficient.23,24 Furthermore, Monte Carlo simulations have recently been used to investigate the kinetic behavior of acrylic functional copolymers to correlate the secondary reaction kinetics with the polymer microstructure and composition.25 Despite these attempts to accurately model copolymerization features, the detailed evaluation of secondary reaction kinetics and MCR reactivity is complicated if the additional parameter of copolymer composition is considered. For instance, in the case of backbiting, the probability of a radical shift along the backbone is likely to be affected not only by the nature of the radicals involved (i.e., chain-end or midchain radicals) but also by the different monomer units that may be found in the proximity of these radicals. Nevertheless, at this stage, computational tools have proven to be a useful means of overcoming the limits of the experimental approach. Therefore, in this paper, a quantum chemistry approach is applied to investigate the physicochemical features of the secondary reaction kinetics in acrylic-based copolymers with a focus on the intramolecular chain-transfer, propagation, and β-scission reactions as well as the difference in reactivity between chainend radicals, MCRs, and short-branch radicals (SBRs).

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COMPUTATIONAL DETAILS

Density functional theory (DFT) is adopted to evaluate the thermodynamic and kinetic parameters of the investigated reactions.26 On the basis of previous studies,14,27 the B3LYP functional is primarily applied to perform geometry optimizations and frequency calculations and to detect transition state structures.28,29 Afterward, single-point calculations and energy estimations are performed at the MPWB1K level of theory for the reactants, products, and transition state structures that correspond to the reactions being studied.30 In the combined B3LYP/MPWB1K approach, the 6-31G basis set with added polarization functions is adopted.31 The rate coefficients are calculated using the conventional transition state theory, which implies the calculation of the reaction rate constant using the Arrhenius formula as defined by eq 1: B



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Figure 2. Reaction schemes of the investigated propagation steps. Notation adopted for the replacement units: tr, chain-end radical unit; mr, MCR unit; pu, penultimate unit adjacent to the radical site of the reactant; mo, attaching monomer. For MA units, R is COOCH3 and X is H. For St units, R is C6H5 and X is H. For MMA units, R is COOCH3 and X is CH3.

Figure 3. Reaction scheme of the investigated β-scission steps. Notation adopted for the replacement units: mr, MCR unit; pu, penultimate unit that is adjacent to the radical site of the reactant and that becomes a terminal radical unit in the product. For MA units, R is COOCH3 and X is H. For St units, R is C6H5 and X is H. For MMA units, R is COOCH3 and X is CH3.

k(T ) = A e−Ea /(kbT ) = κ(T ) TS TS TS TS Q trans Q rot Q vib Q el N ∏i r

i i i Q trans Q rot Q vib Q eli

■

k bT 0 1 − Nr (C ) h

RESULTS AND DISCUSSION The rate coefficients of the backbiting, propagation, and βscission reactions are estimated taking into account the polymer chains in a syndiotactic configuration according to our previous studies (i.e., the conformation of the backbone that exhibits an intermediate kinetic behavior among the atactic configurations that are commonly obtained through FRP)14,27 and composed of a number of monomer units sufficient to describe the reacting system (Figures 1−3). Specifically, 1:5 backbiting and MCR propagation are simulated considering linear tetramer radicals (Figures 1a and 2b), whereas the reactions of backbiting and the further propagation of SBRs are studied taking into account the pentamer branched radicals created after the propagation of the MCRs (Figures 1c and 2c). Internal j:j+4 backbiting and βscission are modeled by adopting larger suitably designed MCRs (Figures 1b and 3). Finally, chain-end propagation is studied considering linear trimer radicals (Figure 2a). As a general procedure applied to all of the kinetic steps investigated, the rate coefficients of reactions that involve poly(methyl acrylate) (PMA) homopolymer chains are adopted as benchmark values for investigation of the relative effect of

× e−Ea /(kbT ) (1)

where kb and h are the Boltzmann and Planck constants, respectively, T is the temperature, Nr is the number of reactants, and C0 is the standard state concentration. The activation energy (Ea) of the process is calculated as the difference between the electronic energy of the transition state and that of the reactants, including the zero-point energy. The translational, rotational, vibrational, and electronic partition functions (Q) of the transition state and of the ith reactant are identified with superscripts TS and i, respectively. The quantum tunneling effect is considered for backbiting reactions because of its relevance to hydrogen transfer phenomena. The tunneling factors, κ(T), are estimated on the basis of the Eckart model.32 All of the quantum chemistry calculations are performed using Gaussian 09.33 C



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Table 1. Calculated Values of the Activation Energy (Ea,fw), Rate Coefficient (kbb,fw), Radical Equilibrium Parameter (ΔE*), and Forward (fw)-to-Backward (bw) Backbiting Rate Constant Ratio (Keq) for the Investigated Backbiting Reactions Based on the Reaction Schemes and the Notation of the Replacement Units Reported in Figure 1a reaction A.0 A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 B.0 B.1 B.2 C.0 C.1 C.2 C.3 C.4

PMA tr-S tr-M mr-S tr-S, mr-S tr-M, mr-S pu-S pu-M mu-S mu-M PMA mr1-S mr1,2-S PMA tr-S tr-M mr-S tr-S, mr-S

Ea,fw (kJ mol−1)

kbb,fw (s−1)

ΔE*

50.7 68.4 81.2 58.5 84.9 85.9 55.1 47.9 49.2 50.4 69.8 91.6 101.0 59.1 64.9 81.3 74.7 99.8

× × × × × × × × × × × × × × × × × ×

7.1 −0.5 −4.2 10.1 1.2 −2.0 3.4 5.9 11.6 4.7 0.0 −3.8 0.0 2.3 −9.2 −5.0 6.9 −3.9

AAAȦ ↔ AȦ AA AAAṠ ↔ AȦ AS AAAṀ ↔ AȦ AM ASAȦ ↔ AṠAA ASAṠ ↔ AṠAS ASAṀ ↔ AṠAM SAAȦ ↔ SȦ AA MAAȦ ↔ MȦ AA AASȦ ↔ AȦ SA AAMȦ ↔ AȦ MA AAAȦ A ↔ AȦ AAA AAAṠA ↔ AȦ ASA ASAṠA ↔ AṠASA A(AȦ )AA ↔ A(AA)Ȧ A A(AṠ)AA ↔ A(AS)Ȧ A A(AṀ )AA ↔ A(AM)Ȧ A A(AȦ )SA ↔ A(AA)ṠA A(AṠ)SA ↔ A(AS)ṠA

4.12 4.53 2.19 1.60 4.85 9.62 1.81 2.36 2.59 3.18 1.09 5.43 1.38 7.44 7.40 4.53 1.11 7.19

3

10 10 10−1 102 10−2 10−3 103 104 103 103 10 10−3 10−3 103 10 10−1 102 10−3

Keq 1.29 1.21 1.78 6.05 5.25 1.22 2.88 3.29 1.67 1.04 1.00 6.32 1.00 1.05 1.13 5.70 1.06 2.18

× × × × × ×

102 10−2 10−3 102 10−2 10−3

× 10 × 102 × 10 × 10−2

× × × ×

10−5 10−4 104 10−2

a

In the reaction formulas, the characters A, S, and M correspond to the MA, St, and MMA units, respectively. The rate constants and parameters were evaluated at 323 K. Additional parameters calculated for backbiting reactions are reported in Table S1 of the Supporting Information.

characterized by a strongly decreased backbiting probability, which is comparable with that of hydrogen abstraction in polystyrene (PS), and exhibits an activation energy value that agrees with previous quantum chemistry calculations for PS 1:5 backbiting.34 Finally, the combined effects of a chain-end unit of MMA and a midchain unit of St, which result in the greatest decrease in the 1:5 backbiting probability, are clearly demonstrated by reaction A.5. The presence of St and MMA in the penultimate (pu) and intermediate (mu) unit positions, with reference to Figure 1a, slightly affects the backbiting rate coefficient. More specifically, a penultimate unit of St appears to inhibit the abstraction of the tertiary hydrogen on the nearby MA unit (reaction A.6), whereas an MMA penultimate unit (reaction A.7) enhances the rate coefficient with respect to that of benchmark reaction A.0. Previous studies noted the importance of considering the internal backbiting reaction in addition to the conventional 1:5 shift among hydrogen transfer steps that involve a six-member transition state ring.14,27 Reversible j:j+4 backbiting reactions (Figure 1b) have been investigated using the rate coefficient determined for PMA (reaction B.0 in Table 1) as a benchmark rate coefficient. This reaction exhibits a rate coefficient value that is substantially smaller than that of reaction A.0 because the reactant is a less reactive tertiary radical and 1 order of magnitude smaller than the corresponding value previously calculated for PBA.14 In reaction B.1, the reactant radical is on a St unit, which implies a decrease in the backbiting probability toward a tertiary MA radical with respect to reaction B.0. The same behavior with respect to the internal backbiting that involves PMA is observed for reaction B.2 of the hydrogen transfer between two MCRs of St. These results are consistent with the observations reported for backbiting on terminal radicals (reactions A.1 and A.3). Moreover, the activation energy value of the backbiting between two MA units requires approximately 20 kJ mol−1 less than the backbiting between MA and St units. This difference in activation energy is

the replacement units in the resulting copolymer chains. This choice was made in an attempt to provide a comprehensive estimation of the kinetic parameters for a copolymer system in which the acrylate plays a dominant role. Therefore, the effect of St and MMA comonomers is studied with the aim of emphasizing their behavior as modifiers of the polymerization process. Backbiting Reaction Kinetics. The computational results for the investigated hydrogen shift reactions and the corresponding nomenclature (A.0−A.9 for 1:5 backbiting of the chain-end radical, B.0−B.2 for internal j:j+4 backbiting, and C.0−C.4 for 1:5 backbiting of SBR) are reported in Table 1. For the benchmark 1:5 backbiting on the PMA homopolymer (reaction A.0), the obtained rate coefficient is comparable with that of PBA, according to our previous work.14 On the basis of the literature, such agreement between PMA and PBA backbiting kinetics is expected.9,13 However, the lack of accurate experimental data, specifically for the backbiting of PMA, prevents a better validation of the computational result. The first analysis of the copolymer composition effect on 1:5 backbiting kinetics is performed by changing the terminal and midchain MA units. If the terminal unit (tr, as reported in Figure 1a) is replaced with St (reaction A.1), the 1:5 shift becomes 2 orders of magnitude slower than for the PMA homopolymer, which agrees with the greater stability of the St radical because of its resonance in the aromatic ring. A similar but stronger effect is observed for an MMA replacement unit (reaction A.2), where the terminal radical is now located on a less reactive tertiary carbon with respect to the secondary carbon that characterizes both MA and St terminal radical units. When the midchain unit (mr, as reported in Figure 1a) is replaced by St (reaction A.3), the rate coefficient is again reduced compared with that of benchmark reaction A.0 (although to a shorter extent) as a result of the weakened tendency of the benzyl hydrogen to be abstracted. Both effects are combined in reaction A.4, in which both terminal and midchain units are replaced with St. This reaction is D



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increased homogeneously to approximately 30−35 kJ mol−1 in the case of backbiting between two St units. The last series of backbiting that was investigated concerns the hydrogen transfer reaction that occurs on SBR (Figure 1c). The 1:5 shift on the branched chain-end radical of PMA (reaction C.0 in Table 1) is faster than the conventional 1:5 backbiting of a linear chain (reaction A.0) by a factor of nearly 2. Although the activation energy of the former reaction is higher, in that case, the backbiting probability is enhanced by the closer configuration of the reactant and transition state, which determines an increase in the pre-exponential factor, as discussed in our previous study of BA.14 With regard to the other investigated reactions, all of the corresponding rate coefficients are smaller than that of reaction C.0, which was selected as a benchmark value for the backbiting of SBR, as for 1:5 backbiting. A comparison of the rate coefficients of the backbiting of SBR with the parameters of the corresponding 1:5 shift reactions indicates that if only the terminal unit is replaced (reactions C.1 and C.2) negligible changes in the activation energy are observed, whereas the rate coefficients are slightly enhanced. In contrast, for reactions C.3 and C.4, in which the midchain unit is also modified, the energy barrier increases to a degree similar to that observed for benchmark reaction C.0 compared with reaction A.0. Radical Stability and Backbiting Equilibrium. To generalize the results and classify the investigated reactions according to the relative order of reactivity, we studied the radical stability between the reactants and the products of the investigated backbiting steps. For this purpose, a radical equilibrium parameter (ΔE*) is defined (eq 2), which considers the energy difference between the reactant and the product with reference to a forward R → P backbiting reaction. Notably, ΔE* is unaffected by the sources of inaccuracy that typically characterize the evaluation of the pre-exponential factors using quantum chemistry calculations. Thus, this parameter is well suited to determining the radical stability with good accuracy. ΔE* =

Ea,bw − Ea,fw − (E P − E R ) = RT RT

In summary, a terminal unit of any monomer type is more inclined to shift the radical to an MCR of St than to an MCR of MA, whereas SBRs tend to become MCRs to a lesser extent than the corresponding chain-end radicals. Moreover, the ΔE* values are only positive (i.e., the equilibrium is in favor of the MCR) for 1:5 backbiting from terminal units of MA and for reaction A.4’s reproduction of the hydrogen transfer in PS. With regard to the backbiting from an MCR of MA, the radical is more likely to be shifted toward MMA than St and, again, more likely toward St than MA. Additionally, the MCR is more likely to be transferred toward another MCR than to an SBR and, again, more likely toward an SBR than to a terminal unit. The only exception is reaction C.1, whose equilibrium appears to be more shifted to the terminal St unit than both the corresponding backbiting to an MCR (reaction B.1) and the corresponding backbiting to an SBR of MMA (reaction C.2). Finally, the same observations made concerning an MCR of MA can be extended to the backbiting that occurs from an MCR of St, with reference to the ΔE* values listed in Table 2. An exception in this case is represented by reaction C.4. Thus, with respect to SBRs, the terminal unit of St is far more effective in stabilizing the radical and enhancing the backward backbiting reaction (reactions C.1 and C.4). Because the prediction of radical reactivity with respect to the backbiting equilibrium has been demonstrated to agree with the physicochemical behavior of the radical species, in this paper, the kinetic study focuses on the estimation of the backbiting equilibrium constant (Keq) for the investigated reactions. This parameter, which is a function of ΔE*, is defined as the ratio of the forward to the backward backbiting rate coefficients (eq 3). The value of Keq quantifies the backbiting equilibrium and accounts for the effect of preexponential factors, unlike ΔE*. Therefore, the calculated Keq values can be used as additional kinetic parameters to characterize the backbiting reactions. Because these parameters are evaluated as the ratios between absolute rate coefficients, they can be considered free from certain systematic errors introduced in the pre-exponential factor that are often responsible for decreased accuracy in quantum chemistry calculations.

(2)

Keq =

Radical equilibrium parameters have been calculated for all of the investigated backbiting reactions (Table 1). A comparison of the calculated ΔE* parameters referenced to the most relevant backbiting reactions (i.e., leaving aside penultimate and intermediate unit effects) is given in Table 2.

=

chain-end radical of MA chain-end radical of St chain-end radical of MMA MCR of MA MCR of St

=

k bb,bw

P P Q vib Q rot R R Q vib Q rot

A fw × e(Ea,bw − Ea,fw )/(RT ) Abw

× eΔE

*

(3)

Specifically, Keq values can be used to directly calculate the backward backbiting rate coefficients (kbb,bw) reported in Table S1 of the Supporting Information. The equilibrium constants of 1:5 backbiting reported in Table 1 clarify that when the terminal radical unit of MA is replaced (reactions A.1, A.2, A.4, and A.5) the forward backbiting kinetics is slowed and the equilibrium is strongly shifted toward reactants, with Keq values on the order of 10−2 to 10−3. With regard to the replacement of penultimate and intermediate units, the effect is milder (reactions A.3 and A.6−A.9), and although the corresponding equilibrium constant values are decreased compared with that of reaction A.0, they are still >1 (i.e., the backbiting equilibrium continues to be shifted toward the MCR). With regard to internal backbiting, reaction B.1 follows the behavior of the corresponding 1:5 shift (reaction A.1) even if the forward and backward rate coefficients are considerably slowed as a result of

Table 2. Sequences of the Calculated Radical Equilibrium Parameters (ΔE*) for the Most Relevant Backbiting Reactions Investigateda ΔE*

reference radical unit

k bb,fw

A.3 > A.0 > C.3 > C.0 A.4 > A.1 > C.4 > C.1 A.5 > A.2 > C.2 C.1bw > C.2bw > A.2bw > B.1bw > A.1bw > B.0 > C.0bw > A.0bw C.4bw > A.5bw > B.2 > A.4bw > B.1 > C.3bw > A.3bw

a

The subscript bw denotes the backward backbiting step and indicates that the corresponding ΔE* value from Table 1 must be considered with the opposite sign. E



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the involvement of the MCR. Finally, for SBRs, the stabilizing effect on the chain-end radical that results from the nearby branch and the corresponding enhancement of the backward reaction rate are reflected in a global shift of the equilibrium toward the reactant compared with the corresponding kinetics of 1:5 backbiting. The only exception is reaction C.3, in which the presence of an MCR on an St unit strongly decreases the backward shift. Propagation Reaction Kinetics. Kinetic parameters and rate constants for all of the investigated propagation reactions along with the corresponding nomenclature (D.0−D.6 for the propagation from a chain-end radical, E.0−E.4 for the propagation from an MCR, and F.0−F.5 for the propagation from an SBR) are listed in Table 3. The effect of the polymer composition on the propagation probability was studied by replacing chain-end (tr) and midchain (mr) radical units, the attaching monomer (mo), and the penultimate unit (pu) (Figure 2).

First, the chain-end propagation reaction of the PMA homopolymer is examined with reference to Figure 2a. The obtained rate coefficient (reaction D.0) is found to be 1 order of magnitude smaller than the experimental data.35 The underestimation of the propagation rate coefficient is primarily the result of uncertainty in the determination of the preexponential factor, according to what has been observed for BA.14 A radical unit of St in place of MA decreases the rate constant value by approximately 2 orders of magnitude (reaction D.1), whereas the effect on the rate coefficient of the replacement of the attaching MA monomer with St (reactions D.2 and D.3) is nearly negligible. The ratio between the experimental values of the propagation rate constants of MA and St is ∼100,35,36 which is a factor of only 2 smaller than the ratio between the computational values calculated in this paper (kp,D.0/kp,D.3). This result confirms that a systematic error is largely responsible for the underestimation of the singlepropagation rate coefficients, whereas the estimation error is decreased for the rate constant ratios.14,32,37−39 By replacing the attaching monomer unit with MMA (D.4) and the penultimate unit with St or MMA (reactions D.5 and D.6), we did not observe significant effects on the propagation rate coefficient with respect to the benchmark MA propagation. Among all of the propagation reactions that involve an MA radical, those steps characterized by a replacement monomer unit (reactions D.2 and D.4) exhibit decreased activation energy values because of the larger electron density in the double bond of St and MMA compared with that of MA. Moreover, it is worth noting that the ratios of rate coefficients of reactions D.0 and D.4 are in agreement with literature experimental parameters.40 With regard to penultimate units (reactions D.5 and D.6), while an St replacement unit determines a decrease in the propagation rate coefficient, an MMA unit has the opposite effect. This different behavior of the two monomers with respect to the reactivity of the adjacent reacting site agrees with the conclusion regarding the penultimate unit’s effect on 1:5 backbiting, in which the rate coefficient with respect to a PMA chain was increased by an MMA penultimate unit but decreased by an St penultimate unit. The propagation of an MCR of MA is 2 orders of magnitude slower than that of the corresponding chain-end radical, and the rate coefficient of reaction E.0 is close to that obtained for BA.14 On the basis of Figure 2b, St and MMA replacement units change the MCR propagation rate coefficient homogeneously in the same way that they affect the chain-end radical propagation, although the effects are more strongly focused on the reduction of the rate constant value compared with the benchmark MA propagation (reaction E.0). The ratios of kp,E.i to kp,E.0 (where E.i refers to an MCR propagation reaction with generic replacement units) are globally smaller than those of the corresponding chain-end propagation reactions (i.e., the ratios of kp,D.i to kp,D.0, where D.i refers to a generic chain-end propagation among the reactions investigated), indicating that the stabilizing effect of replacement units on the MCR reactivity is more effective than on chain-end radicals. Moreover, as long as MCRs are involved, a replacement monomer unit does not affect the activation energy of the propagation of the MCRs (reactions E.2 and E.4). In the last set of propagation reactions studied (F.0−F.5 in Table 3), the reactivity of the SBRs is investigated to understand how the replacement units affect the tendency of the propagation rate constant to approach the chain-end propagation value as the propagating MCR is converted into a

Table 3. Calculated Values of the Activation Energy (Ea), Pre-Exponential Factor (A), and Rate Coefficient (kp) for the Investigated Propagation Reactions Based on the Reaction Schemes and the Notation of the Replacement Units Reported in Figure 2a reaction D.0

PMA

D.1

tr-S

D.2

mo-S

D.3 D.4

tr-S, mo-S mo-M

D.5

pu-S

D.6

pu-M

E.0

PMA

E.1

mr-S

E.2

mo-S

E.3 E.4

mr-S, mo-S mo-M

F.0

PMA

F.1

tr-S

F.2

mo-S

F.3 F.4

tr-S, mo-S mo-M

F.5

tr-M

AAȦ + moA → AAAȦ AAṠ + moA → AASȦ AAȦ + moS → AAAṠ AAṠ + moS → AASṠ AAȦ + moM → AAAṀ ASȦ + moA → ASAȦ AMȦ + moA → AMAȦ AȦ AA + moA → A(AȦ )AA AṠAA + moA → A(SȦ )AA AȦ AA + moS → A(AṠ)AA AṠAA + moS → A(SṠ)AA AȦ AA + moM → A(AṀ )AA A(AȦ )AA + moA → A(AAȦ )AA A(AṠ)AA + moA → A(ASȦ )AA A(AȦ )AA + moS → A(AAṠ)AA A(AṠ)AA + moS → A(ASṠ)AA A(AȦ )AA + moM → A(AAṀ )AA A(AṀ )AA + moA → A(AMȦ )AA

Ea (kJ mol−1)

A (L mol−1 s−1)

kp (L mol−1 s−1)

14.9

3.42 × 105

1.35 × 103

28.8

3.49 × 105

7.61

7.5

1.94 × 104

1.19 × 103

28.2

1.89 × 105

5.30

9.7

1.62 × 105

4.39 × 103

16.0

4.40 × 105

1.14 × 103

10.0

6.43 × 104

1.53 × 103

17.3

1.30 × 104

2.08 × 10

36.8

6.40 × 104

7.27 × 10−2

16.2

5.25 × 103

1.25 × 10

39.0

3.61 × 104

1.78 × 10−2

19.9

9.14 × 103

5.63

16.0

5.63 × 105

1.46 × 103

30.8

1.03 × 105

1.07

16.0

3.19 × 105

8.27 × 102

38.4

8.49 × 104

5.29 × 10−2

11.6

2.02 × 105

2.70 × 103

23.9

7.96 × 104

1.09 × 10

a

In the reaction formulas, the characters A, S, and M correspond to the MA, St, and MMA units, respectively. The rate constants and parameters were evaluated at 323 K. F



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MA units are replaced with St or MMA in the mr and pu positions, the pre-exponential factor is decreased by at least 1 order of magnitude. Via replacement of one of these units with St (reactions G.1 and G.2), the activation energy value is not affected compared with that of reaction G.0. In contrast, via replacement of both units of St (reaction G.3), the activation energy value increases, which results in a rate coefficient that is smaller than that of PMA by 3 orders of magnitude. Finally, if the penultimate unit is MMA (reactions G.4 and G.5), the activation energy is decreased. Thus, the corresponding rate coefficient value is increased compared with that of benchmark reaction G.0. In the latter cases, the presence of a penultimate unit of MMA in addition to the MCR appears to be responsible for the enhanced β-scission probability, which agrees with the penultimate unit effect observed for the backbiting process and the propagation kinetics.

chain-end radical. The SBR propagation of the MA homopolymer [reaction F.0 (Figure 2c)] immediately reaches the chain-end propagation value. This process can be generalized to all of the propagation steps that involve an MA propagating unit (reactions F.2 and F.4). However, if the chain-end radical is an St or MMA unit (reactions F.1, F.3, and F.5), the propagation rate constant is still smaller than that of a chain-end radical by at least 1 order of magnitude. Nevertheless, in these cases, it is expected that the propagation rate coefficient approaches the value of the corresponding chain-end radicals starting from the third monomer addition step from an MCR.27 β-Scission Reaction Kinetics. The reaction kinetics of the β-scission steps that occur from MCRs is investigated. The obtained rate constants, parameters and the corresponding nomenclature (G.0−G.5) are listed in Table 4. The replacement units of St and MMA are selected to investigate different compositions of the MCR and the resulting products (Figure 3).

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CONCLUSION With reference to an acrylate-based terpolymer system, various secondary reactions were investigated using a computational method based on quantum chemistry. Backbiting, propagation, and β-scission reactions were examined with respect to differences in the nature of the radicals. With PMA as a starting point, the physicochemical behavior of active terpolymer chains in which relevant MA units (i.e., the chainend and midchain radical sites, the penultimate unit, the intermediate unit, and the attaching monomer) are replaced with St and MMA was investigated. Both St and MMA radicals were found to decrease the backbiting probability with respect to MA as a result of an increase in the radical stability and a strengthening of the C−H bond on the tertiary carbon. The replacement of units adjacent to the radical sites caused slight deviations from the benchmark backbiting rate coefficient of PMA, although St and MMA have opposite effects if the penultimate unit effect is considered. For internal backbiting and backbiting on SBR, the effect of composition on the reaction kinetics is comparable with that observed for 1:5 backbiting. Copolymer composition was also shown to affect the competition between forward and backward backbiting, in which radicals on the St and MMA units result in a shift of the equilibrium toward reactants. The propagation reactions appeared to be primarily affected by changing the chain-end radical unit, whereas other composition effects were nearly negligible. The reaction rate coefficients of the MCR propagation were homogeneously decreased with respect to chain-end radicals. However, the effects of the replacement units were demonstrated to be more effective on MCR reactivity. With regard to SBRs, the propagation rates attained values determined for chain-end radicals if the terminal unit was MA. In other cases, a stronger chain-length effect was found. Finally, the rate coefficients of β-scission reactions were homogeneously decreased by replacing MA units with St and MMA as a result of a decrease in the corresponding preexponential factors. In particular, the β-scission kinetics appeared to be affected primarily by the presence of penultimate MMA units, which determined and enhanced the rate coefficient. All of the reported results could be used to describe the physicochemical behavior of the secondary reaction active in acrylate-based copolymerization, providing that reasonable kinetic predictions could be made a priori. Calculated rate coefficients could be adopted as a reliable first source of

Table 4. Calculated Values of the Activation Energy (Ea), Pre-Exponential Factor (A), and Rate Coefficient (kβS) for the Investigated β-Scission Reactions Based on the Reaction Schemes and the Notation of the Replacement Units Reported in Figure 3a reaction G.0

PMA

G.1

mr-S

G.2

pu-S

G.3

mr-S, pu-S pu-M

G.4 G.5

mr-S, pu-M

AȦ AA → AADB + AȦ AṠAA → ASDB + AȦ AȦ SA → AADB + AṠ AṠSA → ASDB + AṠ AȦ MA → AADB + AṀ AṠMA → ASDB + AṀ

Ea (kJ mol−1)

A (s−1)

kβS (s−1)

115.6

1.02 × 10

14

2.09 × 10−5

115.9

1.37 × 1013

2.51 × 10−6

119.0

1.75 × 1013

9.80 × 10−7

128.4

1.28 × 1013

2.24 × 10−8

104.4

4.43 × 1012

5.74 × 10−5

109.8

2.26 × 1013

3.91 × 10−5

a

In the reaction formulas, the characters A, S, and M correspond to the MA, St, and MMA units, respectively, whereas the superscript DB identifies a terminal double bond. Rate constants and parameters were evaluated at 323 K.

Because right and left β-scission reactions exhibit nearly the same activation energy if the MCR is sufficiently far from the chain end, the composition of the reactants was changed to cover all of the desired chain configurations, whereas in agreement with results reported in the literature, only right βscission reactions are investigated.14,34 It should be noted that the kinetics of β-scission involving MCRs close to the chain end (i.e., MCRs that are likely to be formed by backbiting) is expected to be different from that of the representative reaction modeled in this work. Benchmark reaction G.0 exhibits an activation energy value that is comparable with that of BA, whereas a difference in the pre-exponential factor results in a βscission rate coefficient that is larger than that of BA by 2 orders of magnitude.14 Experimental studies of β-scission in hightemperature polymerization of acrylates report a rate coefficient that is larger than that calculated for BA in our previous work,14 and closer to that of MA estimated in this work.4,9 However, there is no experimental information that allows us to claim that the two acrylates are subject to β-scission with significantly different kinetics. With regard to other reactions in which the G



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parameters for a detailed description of the investigated terpolymer system to characterize the kinetics of various secondary reaction pathways in the context of acrylate-based copolymerization modeling.

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ASSOCIATED CONTENT

S Supporting Information *

Additional data. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: +39 02 2399 3135. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Prof. Robin A. Hutchinson of Queen’s University and Prof. José R. Leiza of the University of the Basque Country for their fruitful discussions. The research presented in this paper was supported by the European Union Seventh Framework Program (FP7/2007-2013) via Grant 238013.

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ABBREVIATIONS FRP, free radical polymerization; St, styrene; PLP, pulsed-laser polymerization; MCR, midchain radical; BA, butyl acrylate; PBA, poly(butyl acrylate); SBR, short-branch radical; MA, methyl acrylate; MMA, methyl methacrylate; DFT, density functional theory; PMA, poly(methyl acrylate); PS, polystyrene

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REFERENCES

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Quantum Chemistry Investigation of Secondary Reaction Kinetics in

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