**5. Quantum-chemical calculations**

426 Nitroxides – Theory, Experiment and Applications

MNP. This means that RO

shown in Table 2.

because the reaction with monomer is dominant.

radical by a unimer one can be expressed as

Let us return to the MNPCHPCPSBstyrene system. When the concentration of MNP is extremely high (over 0.3 mol L-1), the propagating styryl radical Pn is one unit long, so we can designate it as P1. The simplified reaction scheme may be written as below (Scheme 3). Now, a question arises as to whether this scheme is complete. In particular, if any RO generated after the CHPC decay does not react with MNP, is it safe to say that this radical does not interact with polymeric RAFT agent? Indeed, when the reaction is carried out in an inert solvent like benzene, the system CHPCMNPPSB generates polystyrene adducts with

experiment is repeated but with PBN in place of MNP, the radical accumulation is doubled, since PBN can capture the radicals bearing an unpaired electron on oxygen. So, in the absence of monomer, an appreciable portion of the formed RO can attack the polymeric RAFT agent. However, when this process is run in bulk styrene, we can neglect this reaction,

For the sake of simplicity we assume the rate coefficients of spin trapping of unimer and oligomer propagating radicals to be equal. Then the rate constant of substitution of oligomer

*kT a <sup>k</sup>*

*PT*

*sub*

**Scheme 3.** Simplified reaction scheme for MNPCHPCPSBstyrene system.

[MNP], mol L-1 [PSB], mol L-1 [CHPC], mol L-1 (a1) ksub, Lmol-1 s-1

**Table 2.** The experimental data obtained for systems containing MNP, PSB and CHPC in styrene

0.370 0.010 0.01 0.94 1105 0.316 0.029 0.005 0.71 2105

 <sup>1</sup> 1

where (a1) is the molar fraction of unimer radical adduct and kPT=4.6104 L mol-1 s-1.We have carried out experiments using PSB of Mn=2900 D; the conditions and the results are

*n*

[ ](1 ( )) ( )[ ]

is able to react with polymeric RAFT agent. When the

*a PZ* (29)

To help validate the kinetic results from the spin-trapping technique, we now compare the experimental results with the corresponding predictions from high-level ab initio molecular orbital theory calculations. Unlike experimental approaches, quantum chemistry offers the opportunity to calculate the rate and equilibrium coefficients for any (chemically-controlled) individual reaction within a complicated multistep processes directly, assuming only the Schrödinger equation and the values of a few fundamental constants (such as the masses and charges of the electron, proton and neutron). In this way, quantum chemical predictions are independent of any kinetic model-based assumptions and may thus be used to test their validity. Admittedly, because the many-electron Schrödinger equation does not have a simple analytical solution, numerical approximations must be made, introducing a potential source of error instead. Whilst very accurate methods exist, these usually require large amounts of computer power, with the computational cost scaling with the size of the system. Nonetheless, in recent years, rapid and continuing advances in computational resources and the development of cost-effective accurate quantum-chemical methods have made possible the chemically accurate prediction of rate coefficients for oligomeric reactions relevant to conventional and controlled radical polymerization processes (Coote et al., 2009, Lin et al., 2010).

The calculations described in this section all use the same high-level ab initio molecular orbital theory methods, which are described in full in the original studies (Coote et al., 2006, Izgorodina et al., 2006, Lin et al., 2011). Owing to their computational cost, we have focused only on the calculation of the equilibrium constant for the addition-fragmentation reaction K=kad/kfr. All geometries and frequencies are calculated using relatively low-cost DFT procedures which have been shown to be suitable for this purpose due to systematic error cancellation (Coote et al., 2002). We always conduct a thorough search of conformational space and have developed an efficient algorithm for this purpose called Energy-Directed Tree Search (Izgorodina et al., 2007). Having located our geometries, we then calculate very accurate energies using an ONIOM approximation (Izgorodina et al., 2006, 2007, Lin et al., 2009) to the W1 method, which in turn approximates CCSD(T) calculations at the infinite basis set limit via extrapolation procedures and has kJ accuracy provided single reference methods are appropriate (Martin et al., 1999). To this end, we divide the chemical reaction into a series of layers: an inner core that captures the reaction center; a core that includes the reaction center and all alpha substituent effects; the full system. In forming the core and inner core deleted substituents are replaced with hydrogens to maintain chemical valency and all geometries are relaxed. The inner core is then studied with W1 and a slightly lowercost composite method, G3(MP2)-RAD (Henry et al., 2003), the core is studied with G3(MP2)-RAD and RMP2/6-311+G(3df,2p) and the full system is studied with RMP2/6- 311+G(3df,2p). The W1 energy on the full system is then approximated as the sum of the RMP2/6-311+G(3df,2p) energy, plus a correction for RMP2 versus G3(MP2)-RAD error (as measured for the core) and a further correction for the G3(MP2)-RAD versus W1 error (as measured for the inner core). In this way RMP2 is used only to measure the remote substituent effects (primarily dispersion effects) for which it is well suited, and the more difficult task of modeling the reaction itself is performed using higher-level methods. Having obtained geometries, frequencies and energies for reactants and products, equilibrium constants are then calculated using the standard textbook formulae for the statistical thermodynamics of an ideal gas under the harmonic oscillator and rigid rotor approximation; further corrections are then made by treating low frequency torsional modes as hindered internal rotations. Finally, gas-phase results are corrected to the solution phase by means of a thermodynamic cycle in which the free energies of solvation are calculated using continuum models such as PCM (Miertus et al., 1981).

Table 3 summarizes the calculated and experimental values of the equilibrium constant K(=*k*ad/*k*fr) for the addition of the tert-butyl (*t*Bu) radical to the various low molecular weight RAFT agents. The calculations in this table were performed for the same conditions (20°C, benzene solution) as the experiments; the energies in solution in this case were calculated using our W1-ONIOM procedure described above with PCM-UAHF. Further computational details are available in the original reference (Chernikova et al., 2011); data for BB and BC differ slightly from this previous work as the conversion from the gas-phase to solutionphase standard state was accidentally applied twice in this earlier study. In cases where the leaving group(s) on the RAFT agent is not *t*Bu, there are multiple addition/fragmentation reactions possible and all pathways were considered computationally. As noted above, the fragmentation of the *t*Bu radical is observed exclusively in the experiments when the RAFT agent is BB and fragmentation of the Bz radical is observed exclusively when the RAFT agent is BC. In this latter case, the experiments were unable to measure the equilibrium constant for the addition/fragmentation reactions because the intermediate radical was too short-lived and only the substitution rate was reported.

Examination of Table 3 shows that, in general, there is good accord between the computed and measured equilibrium constants, though with one important exception. For TB the agreement is excellent (within a factor of 1.5), while for TC the theoretical calculations underestimate experiment by approximately 2 orders of magnitude, an error still within the reasonable limits of uncertainty of both techniques. However, for BB the calculated and experimental equilibrium constants show very good agreement (within a factor of 2), but theory predicts that fragmentation of the benzyl radical is preferred whilst experiment detects *t*Bu radical exclusively. Although the experimental observation is difficult to argue with, the conclusion that *t*Bu undergoes fragmentation more readily than benzyl radical is


using continuum models such as PCM (Miertus et al., 1981).

short-lived and only the substitution rate was reported.

and all geometries are relaxed. The inner core is then studied with W1 and a slightly lowercost composite method, G3(MP2)-RAD (Henry et al., 2003), the core is studied with G3(MP2)-RAD and RMP2/6-311+G(3df,2p) and the full system is studied with RMP2/6- 311+G(3df,2p). The W1 energy on the full system is then approximated as the sum of the RMP2/6-311+G(3df,2p) energy, plus a correction for RMP2 versus G3(MP2)-RAD error (as measured for the core) and a further correction for the G3(MP2)-RAD versus W1 error (as measured for the inner core). In this way RMP2 is used only to measure the remote substituent effects (primarily dispersion effects) for which it is well suited, and the more difficult task of modeling the reaction itself is performed using higher-level methods. Having obtained geometries, frequencies and energies for reactants and products, equilibrium constants are then calculated using the standard textbook formulae for the statistical thermodynamics of an ideal gas under the harmonic oscillator and rigid rotor approximation; further corrections are then made by treating low frequency torsional modes as hindered internal rotations. Finally, gas-phase results are corrected to the solution phase by means of a thermodynamic cycle in which the free energies of solvation are calculated

Table 3 summarizes the calculated and experimental values of the equilibrium constant K(=*k*ad/*k*fr) for the addition of the tert-butyl (*t*Bu) radical to the various low molecular weight RAFT agents. The calculations in this table were performed for the same conditions (20°C, benzene solution) as the experiments; the energies in solution in this case were calculated using our W1-ONIOM procedure described above with PCM-UAHF. Further computational details are available in the original reference (Chernikova et al., 2011); data for BB and BC differ slightly from this previous work as the conversion from the gas-phase to solutionphase standard state was accidentally applied twice in this earlier study. In cases where the leaving group(s) on the RAFT agent is not *t*Bu, there are multiple addition/fragmentation reactions possible and all pathways were considered computationally. As noted above, the fragmentation of the *t*Bu radical is observed exclusively in the experiments when the RAFT agent is BB and fragmentation of the Bz radical is observed exclusively when the RAFT agent is BC. In this latter case, the experiments were unable to measure the equilibrium constant for the addition/fragmentation reactions because the intermediate radical was too

Examination of Table 3 shows that, in general, there is good accord between the computed and measured equilibrium constants, though with one important exception. For TB the agreement is excellent (within a factor of 1.5), while for TC the theoretical calculations underestimate experiment by approximately 2 orders of magnitude, an error still within the reasonable limits of uncertainty of both techniques. However, for BB the calculated and experimental equilibrium constants show very good agreement (within a factor of 2), but theory predicts that fragmentation of the benzyl radical is preferred whilst experiment detects *t*Bu radical exclusively. Although the experimental observation is difficult to argue with, the conclusion that *t*Bu undergoes fragmentation more readily than benzyl radical is

**Table 3.** Comparison of theoretical and experimental equilibrium constants (K, L mol-1 20°C, benzene solution)

not easy to accept at face value. In particular, as is well known, the benzyl radical (whose radical stabilization energy is 61.0 kJ mol–1) is significantly more stable than *t*Bu (23.0 kJ mol–1) and should thus fragment preferentially (Coote et al., 2010). Indeed for BC, experiment does detect the preferential fragmentation of benzyl (instead of *t*Bu), a result in accord with theory. At present we are unable to explain the exclusive detection of *t*Bu radical in the experimental study of BB, but one tantalizing possibility is that there may be another (as yet undetected) reaction channel available to the benzyl radical (but not the *t*Bu radical) in the presence of the dithiobenzoate RAFT agent BB that is not significant in the presence of the trithiocarbonate RAFT agent BC. Such a process, if it exists, may help to resolve the apparent inconsistencies in the experimental and theoretical data for dithiobenzoate mediated polymerizations of styrene, and further work on this problem is currently underway.

To study the effects of chain length on the RAFT equilibrium, we considered the reactions of *t*Bu radical with dithiobenzoates bearing various oligomers of polystyrene and poly(methyl methacrylate) as leaving groups. We also considered the addition of these same oligomers to the TB RAFT agent. The substitution (or chain transfer) reactions studied experimentally are the overall reaction that results when pairs of corresponding polyRAFT and TB addition reactions are subtracted. The calculated equilibrium constants (calculated new as part of the present work but with some species taken from our earlier studies) are shown in Figure 14. As in the data above, calculations were performed using our W1-ONIOM method for energies, the PCM-UAHF solvation model but, due to the larger size of the species, the harmonic oscillator approximation was used for the gas-phase partition functions. This latter approximation adds an additional uncertainty of approximately one order of magnitude to our results but should not affect the qualitative trends.

From Fig. 14 we see that chain length effects on the equilibrium constant for the RAFT agent, in reactions with a constant attacking radical, are relatively small. However, not surprisingly, the equilibrium constant is much more sensitive to the nature of the attacking radical, especially with respect to its terminal and penultimate units where K values can stretch over as much as 10 orders of magnitude with trends that are very specific to the reaction under study. However, beyond the penultimate position convergence is quite rapid. With respect to reactions involving the *t*Bu radical with polymeric RAFT agents, it is clear that the former is a much poorer leaving group than either the polystyryl or polymethyl methacrylate attacking radicals and this is consistent with the experimental observation that they undergo rapid substitution.

**Figure 14.** Chain length dependence of the equilibrium constant (K, L mol–1, 20°C, benzene solution) in RAFT polymerization of methyl methacrylate and styrene

### **6. Conclusion**

The most convincing argument for the currently accepted mechanism of RAFT polymerization is the direct monitoring of the radical intermediates formation by means of ESR (Chernikova et al., 2004, Hawthorne et al., 1999, Golubev et al., 2005, Kwak et al, 2002). The concentrations of these intermediates and their changes during the process provide valuable data for the kinetic modeling of RAFT polymerization. If the radical intermediates are relatively stable, they are accumulated in concentrations sufficient to be observed by ESR. However, if the intermediates are active, one cannot detect them in a reaction mixture. The use of spin traps can help to overcome this disadvantage and thereby extend the possibilities for the kinetic investigations of this process. We have found that the MNP and C-phenyl-N-butyl nitrone spin traps and their adducts are stable in the presence of RAFT agents, and this allows us to determine the nature of the trapped radicals from their ESR spectra. Through careful design of the reaction conditions (especially the use of low concentrations of active radicals) spin traps allow us to simplify the kinetic analysis by outcompeting certain side reactions, in particular termination and re-addition processes.

430 Nitroxides – Theory, Experiment and Applications

observation that they undergo rapid substitution.

RAFT polymerization of methyl methacrylate and styrene

**6. Conclusion** 

surprisingly, the equilibrium constant is much more sensitive to the nature of the attacking radical, especially with respect to its terminal and penultimate units where K values can stretch over as much as 10 orders of magnitude with trends that are very specific to the reaction under study. However, beyond the penultimate position convergence is quite rapid. With respect to reactions involving the *t*Bu radical with polymeric RAFT agents, it is clear that the former is a much poorer leaving group than either the polystyryl or polymethyl methacrylate attacking radicals and this is consistent with the experimental

**Figure 14.** Chain length dependence of the equilibrium constant (K, L mol–1, 20°C, benzene solution) in

The most convincing argument for the currently accepted mechanism of RAFT polymerization is the direct monitoring of the radical intermediates formation by means of ESR (Chernikova et al., 2004, Hawthorne et al., 1999, Golubev et al., 2005, Kwak et al, 2002). The concentrations of these intermediates and their changes during the process provide valuable data for the kinetic modeling of RAFT polymerization. If the radical intermediates are relatively stable, they are accumulated in concentrations sufficient to be observed by ESR. However, if the intermediates are active, one cannot detect them in a reaction mixture. The use of spin traps can help to overcome this disadvantage and thereby extend the possibilities for the kinetic investigations of this process. We have found that the MNP and C-phenyl-N-butyl nitrone spin traps and their adducts are stable in the presence of RAFT Reactions of addition and fragmentation are the principal stages in RAFT process. Two extreme cases can occur in different systems: radical addition to the RAFT agent resulting in formation of a stable intermediate, which one can observe via ESR (the model for this case is the TBMNP system); or formation of an unstable intermediate that does not accumulate in the system and immediately decomposes (BCMNP system). In the first case one can say that two different reactions of addition and fragmentation occur sequentially; in the second case – only the overall radical substitution is observed. In terms of the application of spin traps it is important to note that there are at least two different radicals in these reactions: one that is reacting, and another that is forming. In some cases, the forming radical may be an intermediate that is not caught by the spin trap.

For the successful application of spin-trap technique, particularly, for the estimation of kinetic constants of these reactions, the two following requirements must be met. The first requirement is that the ESR-spectra of spin trap adducts with reacting and leaving radicals should be different enough to determine quantitatively the contribution of each adduct. This requirement is easily met in the model studies presented here, and more generally in the first stage of RAFT polymerization where the propagating radical interacts with typical initial RAFT agents. We also showed that to some extent short oligomers of different chain lengths could also be discriminated. However, for study of the main equilibrium in RAFT, where the attacking and leaving radicals are essentially identical, this is more problematic and other strategies are required. The second requirement for successful application of the spin-trap technique is knowledge of the rate coefficients for the various spin trapping reactions. Unfortunately, there are not always reliable data available for these constants. At present we use the average value presented by Golubev et al. (2001). The development of accurate methods for the determination of trapping constants remains an ongoing challenge.

We have shown that the spin trapping technique can readily be used to measure substitution (and hence chain transfer) constants with a high level of accuracy. We have also used it to measure rate coefficients for the individual addition and fragmentation rate reactions that are in very good agreement with theoretical calculations for most systems studied to date. However, determination of these rate coefficients is still problematic for some systems, particularly if the intermediate radical is short lived. To measure these rate coefficients we use the method of competitive reactions (MCR), where the comparative reaction is the spin trapping. At the moment there is a lack of reliable information about the kinetics of these reactions. As a result, the determination of kinetic parameters is limited by the accuracy of the reference reaction, which is often unknown. In the literature the reproducibility of experimental data is usually mentioned, but the absolute accuracy is not. In the present work we have assumed that the absolute error accounts for half an order of

magnitude at least; in particular cases the error could be appreciably higher. This is one of the reasons why we use extensively the photolysis of MNP: for this reaction there is a reliable value of rate coefficient of tert-butyl radical spin trapping. The other challenge is that, in estimating kinetic constants, one also has to assume that other reaction channels are not significant. The generally good agreement with theoretical calculations suggests that these assumptions are likely to be reasonable most of the time, though the unusual results for BB are also a warning that these assumptions need to be continually re-evaluated for each new system studied. Nonetheless, it has to be stressed that the presence of the spin trap, by outcompeting most side reactions, does greatly minimize the impact of kinetic assumptions on the accuracy of the kinetic results, particularly when compared with standard polymerization conditions.

The spin trap technique therefore has much to offer the radical polymer field. For instance, the application of the spin traps has helped to solve a number of problems in complexradical polymerization and copolymerization (Golubev et al., 1978). They have also helped to address controversies related to spontaneous initiation and unusual inhibition of some polymerization systems (Zaremski et al., 1999). We hope that the results presented in this work will encourage the further development and application of these techniques for RAFT polymerization and other controlled radical polymerization processes.
