Incubation Period in the 2,2,4,4-Tetramethyl-1- piperidinyloxy-mediated Thermal Autopolymerization of Styrene: Kinetics and Simulations

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1 Incubation Period in the 2,2,4,4-Tetramethyl-1- piperidinyloxy-mediated Thermal Autopolymerization of Styrene: Kinetics and Simulations ENRIQUE SALDÍVAR-GUERRA, 1 JOSÉ BONILLA, 1 GREGORIO ZACAHUA, 2 MARTHA ALBORES-VELASCO 3 1 Centro de Investigación en Química Aplicada, Boulevare Enrique Reyna 140, Saltillo, Coahuila, 25100, México 2 Escuela Superior de Ingeniería Química e Industrias Extractivas, Zacatenco, CP 07738, México DF 3 Facultad de Química, Universidad Nacional Autónoma de México, CU Coyoacán, México DF Received 28 June 2006; accepted 26 August 2006 DOI: /pola Published online in Wiley InterScience ( ABSTRACT: Mechanisms and simulations of the induction period and the initial polymerization stages in the nitroxide-mediated autopolymerization of styrene are discussed. At C and moderate 2,2,4,4-tetramethyl-1-piperidinyloxy (TEMPO) concentrations ( M), the main source of radicals is the hydrogen abstraction of the Mayo dimer by TEMPO [with the kinetic constant of hydrogen abstraction (k h )]. At higher TEMPO concentrations ([N] > 0.1 M), this reaction is still dominant, but radical generation by the direct attack against styrene by TEMPO, with kinetic constant of addition k ad, also becomes relevant. From previous experimental data and simulations, initial estimates of k h & 1 and k ad & L mol 1 s 1 are obtained at 125 8C. From the induction period to the polymerization regime, there is an abrupt change in the dominant mechanism generating radicals because of the sudden decrease in the nitroxide radicals. Under induction-period conditions, the simulations confirm the validity of the quasi-steady-state assumption (QSSA) for the Mayo dimer in this regime; however, after the induction period, the QSSA for the dimer is not valid, and this brings into question the scientific basis of the well-known expression k th [M] 3 (where [M] is the monomer concentration and k th is the kinetic constant of autoinitiation) for the autoinitiation rate in styrene polymerization. VC 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: , 2006 Keywords: kinetics (polym.); living polymerization; modeling; radical polymerization; simulations INTRODUCTION In the last few years, nitroxide-mediated living radical polymerization has been widely used in the synthesis of controlled macromolecular structures. 1 6 Also, previous limitations of this Correspondence to: E. Saldívar-Guerra ( esaldivar@ ciqa.mx), Vol. 44, (2006) VC 2006 Wiley Periodicals, Inc. technique, such as slow reactions or poor control, have been successfully addressed and overcome to various extents by different authors Because of its practical implications, considerable attention has been given to the mechanism and kinetics of the nitroxide-mediated polymerization (NMP) of styrene, and a fair grasp of its general features seems to be well established, despite a still incomplete understanding of the mechanism of thermal styrene autoinitiation. 6962

2 THERMAL AUTOPOLYMERIZATION OF STYRENE 6963 Styrene is the monomer for which 2,2,4,4-tetramethyl-1-piperidinyloxy (TEMPO)-controlled polymerization seems to work best, and this fact has been attributed in part to the ability of styrene to thermally self-generate free radicals that help to maintain a reasonable reaction rate. 22 Most of the previous studies have focused on the stage at which (quasi-) steady-state kinetics for the nitroxide species has been reached (the equilibrium period in the notation of Fischer 18 ), in part because most of these works have addressed model systems in which initiation is achieved via an alkoxyamine (monomolecular process), leading to a rapid equilibration of the nitroxide stable radical and alkoxyamine concentrations. However, the region of nonequilibrium kinetics is quite important and can last for significantly long reaction times for systems controlled by the addition of a nitroxide radical (bimolecular systems), in which the rate of initiation (R i ) is low, such as systems with pure thermal autoinitiation or with a high ratio of the nitroxide to the initiator (above ca. 1.3). These systems exhibit an incubation or induction period and have been experimentally studied by several authors in bulk, in heterogeneous polymerizations (miniemulsions), 28,29 or in both. 30 Fischer 31 seems to be the first to study this regime by modeling. The study of this regime is important for several reasons. Keoshkerian et al. 32 reported on the striking difference observed when a miniemulsion-controlled polymerization system is initiated with a polymeric alkoxyamine and when it is initiated with a nitroxide radical and a radical initiator. Also, Cunningham et al. 28 reported on the complex behavior of the induction periods observed for TEMPO-mediated miniemulsion styrene polymerization when the ratio of a water-soluble initiator [potassium persulfate (KPS)] to TEMPO is varied. The same group 33 mathematically modeled this system and explained their experimental data via model simulations; however, they completely neglected hydrogen abstraction from the Mayo dimer by TEMPO, a reaction that has a significant effect on radical generation. A better understanding of the induction-period kinetics, together with studies of nitroxide partitioning between aqueous and organic phases, 34 could help to explain the behavior of nitroxide systems in aqueous, heterogeneous systems because nitroxide and radical partitioning could lead to nonsteady kinetics inside the polymer particles, as mounting experimental evidence suggests From a practical point of view, although it has been experimentally found that a ratio of TEMPO to the initiator of about 1.3 generally works well in bimolecular bulk NMP, 12 the presence and length of an induction period will be rather sensitive to the ratio used, as any shortage of the initiator will generate and increase this period. Also, bimolecular systems may present additional complex behavior, especially in heterogeneous reactions, 35 which may be difficult to interpret. Clearly, a better understanding of the induction period is needed. It was originally reported by Moad et al. 36 and then confirmed by other groups 25,26 that styrene autopolymerization in the presence of TEMPO results in an induction period that is about 1 order of magnitude shorter than the theoretically expected one based only on reported values of the rate of radical generation (e.g., see Hui and Hamielec 37 ) in thermal styrene autopolymerization in the absence of TEMPO. More recently, Kothe and Fischer 38 established a quadratic dependence of the rate of nitroxide consumption (which is directly linked to the rate of radical generation) on the monomer concentration during the induction period; this provides a key element for a more quantitative description of the incubation period. Styrene polymerizations exhibiting an induction period due to the presence of nitroxide radicals have been previously discussed in qualitative 25,26,39 and quantitative ways; 31,33,40 however, the quantitative descriptions have omitted key reaction steps. Therefore, a detailed quantitative analysis that encloses the period of induction as well as the period after the onset of the polymerization is still missing. In this work, experimental observations from our group and others of the TEMPO-mediated thermal autopolymerization of styrene, encompassing both periods and up to moderate conversions, are explained with a simple kinetic model and simulations that use minimum assumptions. Also, simplified limit-case analytical expressions valid before and after the induction period under certain assumptions are derived. The simulations also lead to further insight into the change of the regime from induction to the onset of polymerization, as well as the kinetics of pure autothermal styrene polymerization, and they raise questions about the kinetic expressions that have been widely used to correlate experimental data. 37

3 6964 SALDÍVAR-GUERRA ET AL. EXPERIMENTAL Although several authors have reported data for the induction period of the thermal autopolymerization of styrene in the presence of TEMPO, we decided to collect additional data for two purposes: (1) to increase the number of experimental points from different laboratories with the aim of defining statistical trends in the behavior of this kind of system and (2) to explore firsthand the difficulties in measuring with precision the induction period. The induction period for systems at 120 8C was measured in our laboratory. Styrene from Aldrich was used (washed with 10 wt % solution of NaOH and dried with sodium sulfate). TEMPO from Aldrich was used as received. Styrene and TEMPO were weighed and placed in glass vials for a total solution volume of about 4.5 ml. Ultra-high-purity nitrogen was sparged in the solution for several minutes to purge the oxygen. The vials were then submerged in a preheated oil bath at the desired reaction temperature. Samples were taken out of the bath at preset times and quenched in iced water. The conversion was measured by the solid content after the samples were evaporated to dryness. A typical plot of the conversion versus the time for some of these runs is shown in Figure 1. The induction period can be estimated by linear extrapolation of the non-zero conversion time data until they cross the time axis. However, in some cases, there were some data points with very small conversions just before a well-defined change of slope marking the end of the induction period. In these cases, the induction period was estimated at the crossing point of the two lines with different slopes (see Fig. 1). Despite the presence of small polymer conversions in what is predominantly an induction period, we prefer to call this an induction period from here on. We speculate that those data points with very small conversions in the induction period represent the competition of propagation of a few monomer units with trapping of the oligomeric radical by TEMPO. Density data by Boundy and Boyer 41 were used for the calculation of the concentrations: g/ml at 120 8C (7.85 g mol/l) and g/ml at 125 8C (7.80 g mol/l). NUMERICAL SOLUTIONS Figure 1. Time conversion plots for the thermal polymerization of styrene in bulk at 120 8C in the presence of various TEMPO concentrations. [Color figure can be viewed in the online issue, which is available at The solutions of the systems of differential equations representing the evolution of the species concentrations were implemented in Fortran programs with an efficient algorithm (DDASL 42 ) for the solution of stiff systems of equations, and the calculations were checked independently with Matlab. RESULTS AND DISCUSSION Kinetic Scheme Scheme 1 includes the main reactions in a plausible mechanism of nitroxide-mediated radical autothermal polymerization of styrene during and after the induction period. The thermal autoinitiation of styrene has been studied in great detail, and up to date, the most viable dominant mechanism supported by spectroscopic evidence involves the Diels Alder reaction between two styrene molecules, as proposed originally by Mayo, 46,47 to form the so-called Mayo dimer (1). In the classical Mayo mechanism, 1 further reacts with a styrene molecule to yield two initiating radicals (2 and 3). Moad et al., 36 on the basis of quantitative spectroscopic evidence at 100 8C, proposed that when styrene is heated in the presence of TEMPO, this stable radical abstracts a hydrogen from 1 to form hydroxylamine (4) and an initiating radical (2), which is further trapped by TEMPO to yield 5. Species 5 was also spectroscopically detected by Devonport et al. 25 in similar experiments but at 130 8C, which gave further evi-

4 THERMAL AUTOPOLYMERIZATION OF STYRENE 6965 Scheme 1 dence of hydrogen abstraction from 1 by TEMPO. The rest of the reactions propagation, reversible capping of growing radicals by TEMPO, and irreversible termination are well established for this system. Kothe and Fischer 38 recognized that there is another chemical route to 4 and 5, which has the same overall stoichiometry as that shown in Scheme 1, involving the addition of TEMPO to dimer 1 followed by hydrogen abstraction from the cyclohexadienyl radical by a TEMPO molecule; however, this path seems less likely to occur than the route given in Scheme 1. Another reaction that can produce radicals and consume TEMPO is the direct addition of TEMPO to styrene to give the radical product in Scheme 2, which can further react with another TEMPO radical to yield a bis-tempo adduct. For this reaction, there is also strong spectroscopic and quantitative evidence from several different laboratories. 25,36,48 At 100 8C and a moderate initial TEMPO concentration (0.05 M), this reaction is reported to account for only 10% of the TEMPO molecules consumed. 36 At 130 8C and a moderate TEMPO concentration, the bis-tempo adduct is also a minor component in the product, 25 for which

5 6966 SALDÍVAR-GUERRA ET AL. 5 constitutes the major part. However, it has been reported that at very large concentrations of TEMPO (50/50 mol/mol styrene/tempo) at 120 8C, this reaction may be a dominant source of radicals. 48 This is presumably due to the decreased concentration of styrene under these conditions, besides possible effects associated with the activation energy for this reaction. Additionally, it has been reported that the reaction of the 1,1,3,3- tetramethyl-2,3-dihydroisoindol-2-yloxyl nitroxide radical with styrene at 110 8C yields an analogous bisnitroxide adduct. 49 In this work, we assume that the reactions in Scheme 1 are predominant in the mechanism, and we explore how well they can explain in a quantitative way the available experimental data during and after the induction period. To simplify the kinetic scheme to be used for the calculations, under the assumption of a fast first propagation of the different initiating radicals, it is possible to group together the primary radicals and the growing polymeric radicals of Scheme 1, yielding the following simplified mechanism: Thermal initiation (Mayo scheme) M þ M! k dim D þ M! k i k 1 D 2P Reaction of the Mayo dimer with TEMPO Propagation D þ N! k h P þ M! k p Capping/Decapping Scheme 2 P þ N! k c Irreversible Termination k d P þ P! k t PþHN P P N Z where M is the monomer, D is a Diels Alder adduct (i.e., Mayo dimer 1), HN is a hydroxylamine (4), P is a free radical (either initiating or polymeric), N is a nitroxide radical, P N is a polymeric alkoxyamine, Z is a dead polymer, and k h is the kinetic constant of hydrogen abstraction. k dim, k 1, k i, k p, k c, k d, and k t are the kinetic constants for dimerization, reverse dimerization, Mayo initiation, propagation, capping, de-capping, and irreversible termination, respectively. In this version of the model, we neglect the generation of radicals via the direct addition of TEMPO to styrene; later on, we include that reaction as well. It has been experimentally observed that this system shows a clear induction period with no conversion (or conversions below ca. 2%), whose length is proportional to the initial amount of nitroxide in the system. After that period, the polymerization rate (at least below ca. 40% conversion) proceeds as if no nitroxide were present, and the rate of the autothermal initiation of styrene returns to its normal level. During the induction period, all radicals generated by different mechanisms are trapped by TEMPO until this reaches some form of quasi-equilibrium with the dormant species (alkoxyamines). 50 Rate of Dimer Formation As mentioned before, Kothe and Fischer 38 found a kinetic law for the consumption of nitroxide (eq 1) that has been reported to be valid at TEMPO concentrations above 0.05 M and is second-order with respect to the monomer concentration and zero-order with respect to the TEMPO concentration. This law is dominant until TEMPO reaches a quasi-equilibrium: d½nš ¼ 2k dim ½MŠ 2 ð1þ where k dim is exp[ 93,500/(RT)] L mol 1 s 1 (R is the universal gas constant and is given in J mol 1 K 1 ; T is the temperature in 8K). 38 The controlling kinetic step is the dimer formation and its rate is half that of the nitroxide consumption (see Fig. 2). As we will use this expression in our calculations, we first review in some detail the validity and assumptions behind the derivation of this law. Kothe and Fischer assumed Scheme 1 to be dominant and, on the basis of previous experimental evidence, 36 concluded that most TEMPO (ca. 90%

6 THERMAL AUTOPOLYMERIZATION OF STYRENE 6967 Length of the Induction Period An immediate result of the validity of eq 1 is that it can be used to predict the length of the induction period in the TEMPO-mediated thermal autopolymerization of styrene, assuming that this is the only mechanism generating radicals. It is clear that during the induction period, the right-hand side of eq 1 is nearly constant, and so the integration of eq 1 with limits at time t ¼ 0 and [N] ¼ [N] 0 and at time t ¼ T and [N] & 0 (where T is the length of the induction period) leads to Figure 2. Chemical paths for Mayo dimer 1 in the thermal polymerization of styrene in the presence of TEMPO. ½N Š 0 ¼ 2 Therefore Z T 0 k dim ½MŠ 2 ðtþ 2 Z T 0 k dim ½MŠ 2 0 ¼ 2k dim ½MŠ 2 0 T ð2þ at 100 8C) must be consumed by the chemical path of hydrogen abstraction of TEMPO from 1, leading to products 4 and 5. Implicitly, they neglected the radical generation and the corresponding TEMPO consumption via the complete Mayo mechanism leading to radicals 2 and 3, further trapped by TEMPO, presumably because the reaction rate between the monomer and 1 is much slower than that of TEMPO hydrogen abstraction from 1, at least at TEMPO concentrations above 0.05 M and at temperatures between 100 and 125 8C. Figure 2 summarizes the chemical paths and stoichiometry for the main reactions of the Mayo dimer. This kinetic law also implicitly neglects the reaction of dimer cleavage back to the monomer. Importantly, as a result of the fast reaction of the dimer with TEMPO, the quasi-steady-state assumption (QSSA) for dimer 1 is adopted during the induction period. There seems to be enough spectroscopic 36 and kinetic evidence 38,51 indicating that eq 1 is plausible and nearly quantitative for TEMPO initial concentrations above about 0.05 M, and as we will show, it is an excellent approximation for initial TEMPO concentrations even as low as about 0.02 M. To establish an upper bound for the initial TEMPO concentration at which eq 1 is valid, the direct addition of TEMPO to styrene should be considered. It is expected that this last reaction becomes increasingly important as the initial TEMPO concentration increases; as we will see, this is consistent with the experimental data. T ¼½N Š 0 =ð2k dim ½MŠ 2 0 Þ ð3þ To further test the validity range of eq 1, we show (Fig. 3) a comparison between the theoretical values of T given by eq 3 with the expression of k dim given by Kothe and Fischer 38 and the experimental data reported in the literature from different laboratories (including our own) for reactions both in bulk and in miniemulsions. It is important to realize that the experimentally determined induction period is subject to some uncertainty as it is not easy to precisely measure. In data from some of the laboratories (including ours), a small conversion (below ca. Figure 3. Experimental data from different laboratories and the model prediction (eq 3) for the length of the induction period as a function of the initial TEMPO concentration in the thermal polymerization of styrene in bulk and miniemulsions. The continuous line (model) and full symbols (experiments) are at 120 8C; the broken line (model) and empty symbols (experiments) are at 125 8C.

7 6968 SALDÍVAR-GUERRA ET AL. 2%) is observed during what seems to be predominantly an induction period (see Fig. 1), and this adds to the difficulty of accurately establishing the length of that period. The success of the detection of this small conversion seems to be associated to a large extent with the technique used for measuring the conversion (dry solids and NMR). We have also observed these small conversions with dilatometry, but these results will be published in the future. It is also possible that in the miniemulsion data, additional factors (e.g., transport limitations) not present in bulk experiments may influence the induction period, so these data should be considered with caution. Although there are apparent differences between the data coming from different laboratories, it seems clear that the best agreement of eq 3 with the experimental data can be obtained at low temperatures and at low TEMPO initial concentrations. At 125 8C, eq 3 overestimates the induction period. A likely reason for the deviation is the exclusion of the addition reaction of TEMPO to styrene for radical generation. Equation 1 does not exclude the complete Mayo mechanism as a source of radicals, but under the QSSA for 1, its contribution is indistinguishable from the other route of dimer consumption (either of the two paths consumes two TEMPO molecules). Despite the deviations, the data show that TEMPO consumption via the Mayo dimer formation and the subsequent reaction of 1 with TEMPO or with the monomer can account for most of the TEMPO consumed, explaining most of the duration of the induction period; therefore, this reaction should not be overlooked in any quantitative analysis of the induction periods. Zhang and Ray 40 simulated the induction period with the data of Devonport et al., 25 but they completely ignored this reaction and attributed the shortened induction period to a boost in what they considered the initiation efficiency due to the rapid capping of initiating radicals by TEMPO; these authors applied the concept of initiation efficiency to the radicals generated by the thermal autoinitiation of styrene and not to the radicals coming from a classical initiator, which is obviously absent. However, they had to multiply the traditional Hui and Hamielec 37 kinetic constant for thermal initiation by factors in the range of to fit the data of Devonport et al., implying an upper bound for the initiation efficiency as low as 0.02 for the postinduction period, which is rather unlikely. On the other hand, Ma et al. 33 explained the shortened periods of induction in a TEMPO-mediated miniemulsion system initiated by KPS only on the basis of the acid-catalyzed disproportionation of TEMPO in the presence of KPS. Although the factors suggested in previous studies may contribute to short induction periods in the presence of TEMPO, the evidence reviewed here seems to make clear that the dominant factor for the enhanced radical production rate is the reaction between dimer 1 and TEMPO. Mathematical Model Material balances of the chemical species according to Scheme 1, yield the following set of differential equations describing the time evolution of the species concentrations: d½mš d½dš ¼ 2k dim ½MŠ 2 þ 2k 1 ½DŠ k i ½DŠ½MŠ d½pš d½nš k p ½MŠ½P Š k p ½MŠ½P Š ¼ k dim ½MŠ 2 k 1 ½DŠ k i ½DŠ½MŠ k h ½DŠ½NŠ ¼ 2k i ½DŠ½MŠþk h ½DŠ½N Š k c ½N Š½P Š þ k d ½P NŠ k t ½P Š 2 ¼ k h ½DŠ½N Šþk d ½P NŠ k c ½N Š½P Š d½p NŠ The initial conditions are ¼ k c ½N Š½P Š k d ½P NŠ ð4þ ð5þ ð6þ ð7þ ð8þ ½P NŠð0Þ ¼0 and ½N Šð0Þ ¼½N Š 0 6¼ 0 ð9þ Limit-Case Analysis It is instructive to develop analytical expressions based on equations 4 8 that approximately describe the kinetics for some of the species. The rate of consumption of nitroxide radicals can be derived from equations 5 7 with the following assumptions during the induction period: (1) QSSA for dimer D, (2) QSSA for free radical P, (3) k t [P] 2 k c [P][N], and (4) k d [P N] k c [P][N]. This leads to

8 THERMAL AUTOPOLYMERIZATION OF STYRENE 6969 d½nš 8 9 ¼ 2k dim ½MŠ 2 k i ½MŠþk h ½NŠ >: >; ð10þ k i ½MŠþk 1 þ k h ½NŠ where [M] & [M] 0 during the induction period. This is reduced to the expression given by Kothe and Fischer 38 (eq 1) only if k 1 is much smaller than the rest of the denominator. Another more restrictive condition leading to the same reduction is that k i [M] 0 and k 1 are both k h [N]. This last condition seems to be approximately true, given the values of the kinetic constants from previous measurements and estimations. At 120 8C, k i is of the order of or L mol 1 s 1 ; [M] 0 is about 8 mol L 1. For [N] > 0.05 M, an estimation of k h [N] > s 1 can be made (k h was estimated to be 0.01 L mol 1 s 1 by Boutevin and Bertin 26 ); on the basis of the fitting by Hui and Hamielec, 37 we estimate k 1 to be of the order of s 1 and k i [M] 0 to be in the range of (0.8 4) 10 7 s 1. With respect to the third assumption, k c is about L mol 1 s 1 ; 53 therefore, k c [N] is larger than s 1, whereas k t [P] is about [P] s This requires [P] to be about mol L 1 to make this term comparable to the term corresponding to radical trapping by TEMPO, and this is clearly not the case. On the other hand, according to Scheme 1, the net rate of radical generation at any time is given by R i ¼ 2k i ½DŠ½MŠþk h ½DŠ½NŠ From the QSSA of dimer D, we get ð11þ 8 9 k dim ½MŠ 2 ½DŠ ¼>: >; ð12þ k i ½MŠþk 1 þ k h ½NŠ Replacing (12) in (11) yields 8 9 R i ¼ k dim ½MŠ 2 2k i ½MŠþk h ½NŠ >: >; ð13þ k i ½MŠþk 1 þ k h ½NŠ Given the order of magnitude of the constants discussed before, during the induction period, eq 13 reduces to This implies that R i ¼ k dim ½MŠ 2 0 d½nš 2R i ð14þ ð15þ As we see later with a more detailed model, these expressions are better approximations at the initial and intermediate stages of the induction period than at other stages. After the period of induction, if the QSSA is still valid for dimer D, then eq 13 is still valid, but under those conditions, [N]? 0, and given the order of magnitude of the remaining quantities, eq 13 becomes approximately 8 9 R i ¼ k dim ½MŠ 2 2½MŠ >: >; ¼ 2k dimk i ½MŠ 3 ð16þ k 1 k 1 which is the well-known cubic expression by Hui and Hamielec 37 for thermal styrene autoinitiation. In summary, eq 13 represents in one extreme R i during the induction period, in which it can be reduced to eq 14, which is consistent with eq 1 of Kothe and Fischer. 38 In the other extreme, after the induction period, it is reduced to the well-known cubic law for thermal styrene autoinitiation with the lumped kinetic constant k th, which is defined as follows: k th ¼ k dimk i k 1 ð17þ The starting point, eq 11, assumes that the only radical sources are those depicted in Scheme 1. This is true for nitroxides such as TEMPO with a relatively low equilibrium constant (<10 11 mol L 1 ); for more efficient nitroxides (those with equilibrium constants of the order of 10 9 mol L 1 or greater), the in situ created alkoxyamine will also contribute with radicals that will significantly increase the reaction rate above that of the thermal styrene autopolymerization. 50 Moreover, the rate of dimer generation is k dim [M] 2, which is the same as the rate of radical generation.thisisbecausethechemicalpathmarkedly favored by dimer D, manifested by a significant difference in the kinetic rates, is the second one, which yields one radical per dimer molecule. This also explains why the rate of radical generation is much faster in the presence of TEMPO than without it (as experimentally reported in several publications, e.g., refs. 25 and 36); the second path is much faster than the first one and is the one prevailing in the presence of TEMPO. Simulation with Model Switching From the previous discussion, it is evident that there is a switch of the dominant mechanism

9 6970 SALDÍVAR-GUERRA ET AL. Figure 4. Simulations with model switching (lines) compared with the experimental data (points) of Devonport et al. 25 for the thermal polymerization of styrene with different TEMPO concentrations at 125 8C. The model during the induction period follows Scheme 1; after the induction period, the radical generation is replaced by the cubic expression (eq 16; see the appendix for the complete model equations). Simulations are shown with the literature value of k dim ¼ L mol 1 s 1 and with a fitted value of k dim ¼ L mol 1 s 1. The rest of the kinetic values are those in the second column of Table 1. that generates radicals from the induction period to the period after the onset of the polymerization regime. To simulate the full course of the reaction, including this switch of mechanisms, we solved in one continuous integration two modified forms of equations 4 9, starting with one for the induction period, for which the radical-generation rate (R i ) is given by eq 14, and then switching to the other after the onset of polymerization, for which R i is represented by the cubic law of Hui and Hamielec 37 (eq 16). The switching of models was triggered automatically when the concentration of free TEMPO dropped below [N] 0. The detailed equations are given in the appendix. Simulations using this model are compared with the experimental data of Devonport et al. 25 in Figure 4. The values for the kinetic constants were taken from literature sources in all cases (see Table 1), but the value of k dim was also varied with respect to the literature value just for visualization, as explained later. Simulations are shown with the literature value of k dim ¼ L mol 1 s 1 and with a fitted value of k dim ¼ L mol 1 s 1. This value was fitted with the criterion of optimizing the match of the model predictions with the experimental induction period length so the slope of the monomer conversion versus time could be more easily compared with the experimental data. Figure 4 shows that up to about 40% conversion, the slopes of the model curves follow closely the experimental curves. This agreement is kept up to about 60% conversion for the higher TEMPO concentrations. Also, as shown in Figure 5, when these curves are shifted by their corresponding induction time and compared with a simulated thermal auto-

10 THERMAL AUTOPOLYMERIZATION OF STYRENE 6971 Table 1. Values of the Kinetic Constants Taken from the Literature and Used in the Simulations (125 8C) Kinetic Constant Model with Switching Global Model Reference k th L 2 mol 2 s 1 37 k p 2314 L mol 1 s L mol 1 s 1 55 k t L mol 1 s L mol 1 s 1 54 k d s s 1 56 K a mol L mol L 1 22 k dim b L mol 1 s 1 38 k i L mol 1 s 1 43 k s 1 This work a K is the equilibrium constant of TEMPO. b See Figure 4. polymerization with no nitroxide added, the curves overlap except at high conversions. At low conversions, the curves are identical to those of traditional thermal autopolymerization, as expected from the original explanation by Fukuda et al., 22 applicable to relatively low-equilibrium-constant nitroxides, such as TEMPO. However, at high conversions, and because of the decreased monomer concentration, the active radical concentration is no longer controlled by the thermal radical generation, which becomes slower, but it is given instead by the nitroxide equilibrium; 50 this explains why in Figure 5 at high conversions the rate of polymerization is faster at higher initial TEMPO concentrations. These results lead to the following conclusions, some of them advanced by different groups: 22,25,40 (1) the induction period can be viewed as a prolongation of the nonstationary regime for the nitroxide radicals, until enough radicals are generated, so the quasi-equilibrium is reached; (2) the kinetics after the nonstationary period and up to moderate conversions are almost identical to those of traditional thermal autopolymerization (for TEMPO-like nitroxides); and (3) the length of the induction period is explained to a large extent by the kinetic law of eq 1. Simulation with a Global Model The forced switching of models in the previous section is a crude representation of the kinetics of the system. For deeper comprehension, it is better to look for a global model that encompasses the full range of the reaction, at least throughout the transition to nitroxide quasiequilibrium and up to moderate conversions. To further test the viability of Scheme 1 and investigate its deviations from the experimental data and to obtain initial estimates of unknown kinetic constants, we performed simulations solving the set of equations 4 9 for the full course of the reaction (induction and polymerization), without any external model switching, and compared the predictions with the experimental data of Devonport et al. 25 at 125 8C. Most of the values for kinetic constants were taken from the literature and are shown in Table 1. The only fitted parameters were k 1 and k h. To estimate k 1 from eq 17 and the values of the other constants in that expression, we first estimated that a lower bound at 125 8C should fall in the range of s 1, in which the value at 120 8C lies. Then, the parameter was jointly fitted with the value of k h to better match the experimental trends. The fitting of the two parameters could be made almost independently because each one affected different features of the conversion time curves, as concluded from simulation results not presented here for spacesaving reasons. k 1 especially affects the slope of the curve at both moderate and high conversions, whereas k h determines the length of the induction period and to some extent the slope at low conversions. From an examination of eq 10, it can be seen that large values of k 1 will make the induction period dependent on k h up to some lower bound of this constant, above which the induction period is independent of k h. Therefore, k 1 was fitted first, attending the mentioned features of the conversion time curves, and then the value of k h was fitted. Figure 6 shows the sensitivity of the model predictions to the value of k h for TEMPO from dimer 1. Very low values of this constant, such as 0.01 L mol 1 s 1 (gross estimation by Boutevin and Bertin 26 for this

11 6972 SALDÍVAR-GUERRA ET AL. Figure 5. Simulations with model switching shifted left by the corresponding induction time (continuous line) compared with simulated thermal autopolymerization with no nitroxide added (broken line). The values for the simulated kinetic constants and conditions are the same as those in Figure 4. The lines for the two values of k dim are overlapped. [Color figure can be viewed in the online issue, which is available at parameter at 120 8C), do considerably enlarge the induction period over the estimation of eq 3 and are therefore not likely. Toward a value of k h ¼ 0.5 L mol 1 s 1, the length of the induction period reaches asymptotically the value predicted by eq 3. Larger values of this kinetic constant simply cause a slight increase in the polymerization rate, and this is noticeable at conversions starting around 0.2. From this figure, it is reasonable to assume that a lower bound for k h is around 0.5 L mol 1 s 1 at 125 8C. We could have fitted the value of k dim to get a better agreement of the simulations with the experimental data, but we think that we must keep the value determined by independent experiments for that constant so that deviations ascribed to the lack of other possible reactions (for which strong spectroscopic evidence is available) are detected. Moreover, we give heavy weight to the experimental value of k dim because its careful experimental determination by Kothe and Fischer 38 gave an excellent fitting in the Arrhenius plot of this constant at temperatures in the ample range of C, and their values also agree very well with experimental values of k dim determined in that temperature range in two other laboratories. On the other hand, we have allowed the fitted value of k 1 at 125 8C to vary more than 1 order of magnitude with respect to the estimated value of the same parameter at 120 8C ( s 1 ); however, this is a rough, indirect estimate coming from the compound value of k th and therefore is more subject to uncertainty. On the other hand, it is conceivable that the corresponding reaction step has a relatively large activation energy, which further justifies the variation in the value of the rate constant. Effect of the Inclusion of TEMPO Addition to the Monomer It is evident, as expected, that for the largest concentration of TEMPO (0.156 M), the induc-

12 Figure 6. Simulations with a global model (lines) representing the mechanism of Scheme 1 compared with the experimental data (points) of Devonport et al. 25 for the kinetics of the thermal polymerization of styrene with different TEMPO concentrations at 125 8C. The plots show the sensitivity of the model to the value of k h. The rest of the values for the kinetic constants are given in column 3 of Table 1. Figure 7. Simulations with a global model (lines) representing the mechanisms of Schemes 1 and 2 compared with the experimental data (points) of Devonport et al. 25 for the kinetics of the thermal polymerization of styrene with different TEMPO concentrations at 125 8C. The value fitted for k ad was L mol 1 s 1. The plots show the sensitivity of the model to the value of k h. The rest of the values for the kinetic constants are given in column 3 of Table 1.

13 6974 SALDÍVAR-GUERRA ET AL. tion period is overestimated by the kinetics of Scheme 1, regardless of the value of k h. This suggests that an additional mechanism of radical generation besides that in Scheme 1, in particular the reaction of TEMPO addition to the monomer (Scheme 2), may be activated at this concentration. We must emphasize that this election is not arbitrary but is based on the strong spectroscopic evidence from several laboratories of the presence of the bis-tempo adduct in the reaction products. 25,36,48 To test the inclusion of this mechanism, we added the term k ad [N][M] to eq 6 (positive) and eq 7 (negative). When the modified equations (eqs 4 9) are solved, an estimation of the value of k ad can be obtained by the fitting of the induction period of the highest TEMPO concentration (0.156 M). This gives an estimation of k ad of L mol 1 s 1. The simulation result with the added mechanism is shown in Figure 7 for all TEMPO concentrations and for different values of k h. The value fitted for k ad is not sensitive to the value of k h. On the other hand, the best value for k h, based on the match of conversion values up to about 0.4, is around L mol 1 s 1. The deviations for the lower TEMPO concentrations at conversions higher than 0.4 can be ascribed to diffusion control, as these conditions will lead to higher molecular weights of the polymer, causing chain entanglement and consequently a gel effect. This seems not to occur at the highest TEMPO concentrations at which lower molecular weights are obtained. Some deviations still of unknown origin are apparent at the lowest TEMPO concentration, but the complete model seems to correctly and almost quantitatively capture some of the essential features of the kinetics, as judged by the length of the induction periods and the conversion curves up to moderate conversions. QSSA of Dimer D Once we have order-of-magnitude estimates for the values of the kinetic parameters of Scheme 1, it is important to verify if the QSSA for dimer D, which has been assumed for the derivation of eqs 14 and 16, is valid. We can easily do this by solving eqs 4 9 with no simplification and comparing the solution with that obtained by making the QSSA for dimer D, which in mathematical terms implies replacing the differential equation for dimer D (eq 5) with the corresponding algebraic equation (12), in which the time derivative has been made zero. The result is quite interesting and is shown in Figure 8. The kinetic parameters used are those in Table 1, with k h ¼ 0.5 L mol 1 s 1 and k ad ¼ L mol 1 s 1, and the simulated conditions are the same as those of Devonport et al. 25 Although a specific value of k h was used for the simulations in Figure 8, the same qualitative behavior was observed for other values of this parameter when it was varied in the range of L mol 1 s 1. Also, this qualitative behavior is insensitive to the variation of the value of k 1 in the range of different estimated values discussed in this work. The QSSA during the induction period started to fail at values of k h < 0.1 L mol 1 s 1 for the lowest TEMPO concentration ( M). For all the initial TEMPO concentrations, there is an almost perfect match of the two solutions during the induction period, confirming the validity of the QSSA during that period (this is also confirmed by the plotting of the dimer concentration on a linear scale, not shown here); however, just at the end of the induction period, the two solutions start to diverge, and they differ by amounts as large as about 1 order of magnitude for the rest of the polymerization (postinduction) regime, except for a couple of points at which they cross each other. All the curves show a sharp peak in the concentration of dimer D exactly at the end of the induction period; this is indicative of a sharp change in the mechanism at this point. The behavior of the curves clearly indicate that the concentration of dimer D is not in the quasi-steady state (QSS) after the induction period. The same result was suggested by Kothe and Fischer 38 on the basis of previous physicochemical data, but this numerical finding adds to the evidence. This result is striking when we consider that the Hui and Hamielec 37 expression for the rate of thermal initiation (eq 16), based on this apparently inaccurate assumption, has been used for more than 30 years in polymer science and technology and has at least 116 citations in journals in the last 9 years alone. Although that expression has undoubtedly proved useful, its value seems to be more of a practical, rather than scientific, nature. Focusing on the main subject of this article, we conclude that, out of the paths for dimer 1 consumption (see Fig. 2), the path of the traditional Mayo mechanism has a relatively slow reaction rate that does not allow the dimer to reach the

14 THERMAL AUTOPOLYMERIZATION OF STYRENE 6975 Figure 8. Testing of the QSSA for dimer 1. Curves are shown for the evolution of the dimer with and without the QSSA. The two curves match during the induction period and diverge after it. Simulations were performed for the conditions of the experiments of Devonport et al. 25 The kinetic values used are those of the global model in Table 1 with the additional parameters k ad ¼ L mol 1 s 1 and k h ¼ 0.5 L mol 1 s 1. [Color figure can be viewed in the online issue, which is available at QSS; on the other hand, the reaction of the dimer with TEMPO seems to be fast enough for the QSS to be attained. Discussion of the Concentration Profiles A more complete picture of the underlying mechanisms during and after the induction period is obtained by an examination of the concentration profiles of the nitroxide radicals, dimer 1, and P living radicals, as shown in Figures 9 11, respectively. The kinetic parameters used for these simulations are those of the global model in Table 1, in addition to a value of k ad of L mol 1 s 1 and variable values for k h, as indicated in the figures. The conditions are also those in the experiments of Devonport et al., 25 although the trends are better appreciated if we look at the curves for the highest TEMPO concentration (0.156 M). During a good portion of the induction period, the dominant behavior derives directly from eq 1 and gives a smooth decrease in the TEMPO concentration (in fact linear, although not evidenced by the logarithmic scale used), in agreement with the zero order in eq 1. Initially, there is a slow buildup of P, explained by the QSS for P, whose instantaneous value depends on a decreasing concentration of TEMPO. Similar reasoning explains the slow rise of the dimer concentration (see eq 12). As time increases during the induction period, the idealized behavior of N, represented by eq 1, starts to show some downward curvature due to the complex interaction between the concentrations of P and N and the increasing importance of the nonlinear term k t [P] 2 in eq 6. Near the end of the induction period, there are sharp changes in the species concentrations that show a feedback effect: the decreasing concentration of TEMPO leads to an increasing dimer concentration until TEMPO is not enough to consume all the generated dimer, which starts accu-

15 Figure 9. Simulated evolution of the concentration of nitroxide radical N in the thermal polymerization of styrene with different TEMPO concentrations at 125 8C. Simulations were performed for the conditions of the experiments of Devonport et al. 25 The kinetic values used are those of the global model in Table 1 with additional parameters k ad ¼ L mol 1 s 1 and the values of k h in the legend in L mol 1 s 1. Figure 10. Simulated evolution of the concentration of dimer 1 in the thermal polymerization of styrene with different TEMPO concentrations at 125 8C. Simulations were performed for the conditions of the experiments of Devonport et al. 25 The kinetic values used are those of the global model in Table 1 with additional parameters k ad ¼ L mol 1 s 1 and the values of k h in the legend in L mol 1 s 1.

16 THERMAL AUTOPOLYMERIZATION OF STYRENE 6977 Figure 11. Simulated evolution of the concentration of living radical P in the thermal polymerization of styrene with different TEMPO concentrations at 125 8C. Simulations were performed for the conditions of the experiments of Devonport et al. 25 The kinetic values used are those of the global model in Table 1 with additional parameters k ad ¼ L mol 1 s 1 and the values of k h in the legend in L mol 1 s 1. mulating and eventually loses its QSS condition (see also Fig. 8). The rapid accumulation of the dimer increases sharply the concentration of P, which in turn precipitates the fall in the N concentration. The entire process leads to a quasiequilibrium condition for the N concentration. After that, the dimer concentration decreases because of the lower monomer concentration as the polymerization proceeds, and the concentrations of P and N follow the well-known behavior described by the persistent radical effect. 31 Although this detailed picture resembles somewhat the previously discussed idealized models (analytical expressions and model switching), the main difference of this model is that it predicts a more gradual transition from linear behavior to nonlinear behavior in N (with respect to time) in an advanced stage of the induction period, just before the sharp transition occurs. The behavior of the other variables is directly related to this more gradual change. It is clear that further testing of the mechanisms proposed here would benefit from a comparison of model predictions with experimental data for the molecular weight distribution of the polymers produced in these reactions. For example, the bis-tempo adduct produced in Scheme 2 would tend to produce long chains because of the double capping of the molecule; in fact, Devonport et al. 25 isolated the adduct and used it as an initiator for the controlled polymerization of TEMPO, obtaining a polymer with somewhat high polydispersities (1.55). They attributed this to the fact that the two alkoxyamines in the molecule have different efficiencies as initiators/ controllers. We restrict our work here to the analysis of the induction period and the time conversion data; the analysis of molecular weight data will be the subject of future work. CONCLUSIONS A more quantitative picture of the dominant kinetic mechanisms of thermal styrene autopolymerization in the presence of different concentrations of a nitroxide radical (TEMPO in particular) emerges from this study. According to previous evidence, in the absence of TEMPO, and presumably at very low concentrations of it, the most sig-

17 6978 SALDÍVAR-GUERRA ET AL. nificant source of radicals will be the traditional Mayo mechanism. At C and moderate TEMPO concentrations (from 0.02 M or less up to ca M), the data and simulations suggest that the dominant mechanism is that of Scheme 1, in which the main source of radicals is the hydrogen abstraction of dimer 1 by TEMPO. At higher TEMPO concentrations (starting at ca. 0.1 M), the mechanism of Scheme 1 still is dominant, but another radical generating reaction (direct attack against styrene by TEMPO; Scheme 2) becomes relevant and has to be included in a quantitative description of the kinetics. From the experimental data of Devonport et al. 25 and simulations with the kinetics of Scheme 1 and 2, rough estimates for key values of the kinetic constants at 125 8C have been obtained: k h & 1 L mol 1 s 1 and k ad & L mol 1 s 1. The simulations show that from the induction period to the polymerization regime, there is an abrupt change in the dominant mechanism generating radicals, mainly due to the sudden decrease in the nitroxide radicals and the relative magnitude of the relevant kinetic constants. No artificial switching of models is required to represent this transition as suitable values of the kinetic parameters lead to a smooth representation of this phenomenon. Finally, under the conditions of the induction period, the simulations confirm the validity of the QSSA for 1 in this regime; however, after the induction period, the QSSA for 1 is not valid, and this brings into question the scientific basis of the expression R i ¼ k th [M] 3 for the initiation rate of autothermal styrene polymerization. APPENDIX Model with Switching for the Radical-Generation Mechanism A model that uses different expressions for radical generation during and after the induction period has been derived as follows. Induction Period In this regime, the radical-generation terms of eq 6 can be replaced by the right-hand side of eq 14. In the nitroxide radical balance equation (eq 7), the QSS value of the concentration of dimer D given by eq 12 can be used. The resulting expression can be simplified with arguments similar to those employed when we go from eq 13 to eq 14. This allows us to eliminate the explicit calculation of the concentration of dimer D, making it unnecessary to solve eq 5. The resulting system of equations during the induction period is as follows: d½p Š d½n Š d½mš ¼ k p ½MŠ½P Š ða1þ ¼ k dim ½MŠ 2 k c ½N Š½P Šþk d ½P NŠ k t ½P Š 2 ¼ k dim ½MŠ 2 þ k d ½P NŠ k c ½N Š½P Š d½p NŠ ¼ k c ½N Š½P Š k d ½P NŠ ða2þ ða3þ ða4þ Period after the Onset of Polymerization In this case, the right-hand side of eq 16, the cubic law of Hui and Hamielec, 37 is used as the radical-generation term in eq 6. This regime is akin to a controlled thermal autopolymerization once the nitroxide is in equilibrium with alkoxyamine moieties. The resulting equations are d½mš d½p Š ¼ 2k th ½MŠ 3 k p ½MŠ½P Š k p ½MŠ½P Š ða5þ ¼ 2k th ½MŠ 3 k c ½N Š½P Šþk d ½P NŠ k t ½P Š 2 d½n Š d½p NŠ ¼ k d ½P NŠ k c ½N Š½P Š ¼ k c ½N Š½P Š k d ½P NŠ REFERENCES AND NOTES ða6þ ða7þ ða8þ 1. Hawker, C. J.; Wooley, K. L. Science 2005, 309, Hawker, C. J.; Bosman, A. W.; Harth, E. Chem Rev 2001, 101, Wegrzyn, J. K.; Stephan, T.; Lau, R.; Grubbs, R. B. J Polym Sci Part A: Polym Chem 2005, 43, Lohmeijer, B. G. G.; Schubert, U. S. J Polym Sci Part A: Polym Chem 2005, 43, Miura, Y.; Dote, H. J Polym Sci Part A: Polym Chem 2005, 43,