On the Activation Energy of Termination in Radical Polymerization, as Studied at Low Conversion

Abstract

The chain-length-dependent nature of the termination reaction in radical polymerization (RP) renders the overall termination rate coefficient, <k_t>, a complex parameter in the usual situation where the radical chain-length distribution is non-uniform. This applies also for the activation energy of termination, E_a(<k_t>), which we subject to detailed mechanistic investigation for the first time. The experimental side of this work measures E_a(<k_t>) for the dilute-solution, low-conversion, chemically initiated homopolymerization of styrene (ST), methyl methacrylate (MMA), butyl methacrylate, and dodecyl methacrylate. Values of 25–39 kJ mol⁻¹ are obtained, consistent with strong chain-length-dependent termination (CLDT) for short chains. On other hand, the reanalysis of analogous bulk polymerization data for ST and MMA finds E_a(<k_t>) values of 18–24 kJ mol⁻¹, consistent with weak CLDT for long chains. Both these results are as expected from the so-called composite model for CLDT. A simple analytic framework for understanding and predicting E_a(<k_t>) values is presented for the standard RP situation of continuous initiation. All the results of this work can be rationalized via this framework, which clearly establishes that E_a(<k_t>) is determined by far more than just the E_a of radical diffusion. This framework is extended to activation energy for the number-average degree of polymerization, E_a(DP_n), which we measure and successfully scrutinize via our CLDT model. In the final section of this work, we make interesting, testable predictions about E_a(<k_t>) and/or E_a(DP_n) in various RP systems of different natures to those studied here, most notably, systems involving acrylates, continuous photoinitiation, or dominant chain transfer.

1. Introduction

As it is used for approximately half of the total synthetic polymer production, the high importance of radical polymerization (RP) is very well known. Accordingly, the study of RP kinetics has been a topic of keen interest for three quarters of a century [1]. Common knowledge is that the three fundamental reactions in this process are initiation, propagation, and termination. Given how pivotal this process is, it is astounding that fundamental aspects of it are still not grasped. For example, what is understood about the activation energy of termination, the fundamental reaction in which two propagating radicals react together so that the activity of each is lost and thus cease growing? Being a diffusion-controlled reaction, most workers would just say that the activation energy must be that of the relevant diffusion process, and indeed this approach is commonly used in modelling [2]. However, that it cannot be this simple is immediately clear from the well-known results [3,4].

(1)$⟨k_{\mathrm{t}}⟩=\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{k}_{\mathrm{t}}^{i,j}{\displaystyle \frac{c_{{\mathrm{R}}^i}}{c_{\mathrm{R}}}}{\displaystyle \frac{c_{{\mathrm{R}}^j}}{c_{\mathrm{R}}}}$

This equation, which holds at all times, expresses the overall rate coefficient for termination, <k_t>, in terms of the underlying microscopic termination rate coefficients, k_t^i,j, and radical concentrations, $c_{{\mathrm{R}}^i}$, where the former denotes the rate coefficient for termination between a radical of chain length i and another of chain length j, the latter is the concentration of radicals of chain length i, and c_R is the total radical concentration. The origin of Equation (1) resides in the termination being chain-length-dependent in rate, something that has long been suspected [3] and, in recent decades, has been definitively proven experimentally, as summarized in recent reviews [4,5]. This work is concerned with the activation energy of <k_t>, E_a(<k_t>), because <k_t> is the quantity that generally is measured and deployed in process modelling.

Looking at Equation (1), it is evident that the variation of <k_t> with temperature, T, cannot be straightforward. Most obviously, this is the case because multifarious k_t^i,j go into determining <k_t>, and one cannot expect that all of them have the same E_a, since the nature of the diffusion process that determines the value of k_t^i,j must change with the chain length. But, even if one assumes that all k_t^i,j have the same E_a, there are still the terms $c_{{\mathrm{R}}^i}$/c_R in Equation (1). This ratio is the fraction of polymerizing radicals of chain length i, i.e., the (living) radical chain-length distribution (RCLD). That there is extensive variation in this distribution with T is most obviously evidenced by the major variation in the number-average degree of polymerization, DP_n, of dead polymer produced by RP. Any factor that alters the RCLD must alter <k_t> and thus influence E_a(<k_t>). Such important factors include the initiator efficiency, f, rate coefficient for initiator decomposition, k_d, rate coefficient for propagation, k_p, and, of course, termination rate coefficients themselves. In this work, it is these subtle effects that will be the focus, and it will be seen that they have a compelling influence on E_a(<k_t>).

The diffusion-controlled nature of termination has led to the recognition of many factors that may influence <k_t>, including viscosity, solvent interactions, chain flexibility, dynamics of entanglements, polymer weight fraction, and—of particular relevance—chain length [6]. However, the effect of the RCLD on <k_t>, as captured in Equation (1), is relatively sparingly recognized, even though it is arguably responsible for a lot of the confounding complexities in RP kinetics [4,7]. Barely grasped at all is the effect of RCLD on the variation of <k_t> with T, which is why we substantiate some earlier musings [8,9] in this work. We do this by measuring <k_t> in continuously initiated polymerizations and by then seeking to understand the obtained E_a in terms of the latest understanding of chain-length-dependent termination (CLDT).

Because CLDT is central to this work, it is appropriate to summarize the state of play in this regard. Currently ascendant is the so-called composite model [10]:

(2a)$k_{\mathrm{t}}^{i,i}=k_{\mathrm{t}}^{\mathrm{1,1}}{i}^{-e_{\mathrm{S}}},i\le i_{\mathrm{c}}$

(2b)$k_{\mathrm{t}}^{i,i}=k_{\mathrm{t}}^{\mathrm{1,1}}{\left(i_{\mathrm{c}}\right)}^{(-e_{\mathrm{S}}+e_{\mathrm{L}})}{i}^{-e_{\mathrm{L}}},i>i_{\mathrm{c}}$

This model posits that to achieve a good approximation, there are two distinct regimes in the variation in the homotermination rate coefficient, k_t^i,i, with i, namely a power-law variation with exponent e_S for short chains (Equation (2a)), while beyond a crossover chain length, i_c, there is a different power-law variation, exponent e_L, for long chains (Equation (2b)). Consistent with the meaning of k_t^i,i, the fourth parameter above, k_t^1,1, is the rate coefficient for termination between monomeric radicals, which have i = 1. In the event of e_S = e_L, Equation (2) simplifies to one of the original models for CLDT [4] as follows:

(3)$k_{\mathrm{t}}^{i,i}=k_{\mathrm{t}}^{\mathrm{1,1}}{i}^{-e}$

As will be seen, Equation (3) is still of importance, even if it has been superseded.

Subsequent to its proposal, Equation (2) has been extensively verified for a considerable number of monomers thanks to the development of advanced techniques such as single-pulse pulsed-laser polymerization combined with EPR spectroscopy (SP-PLP-EPR) [11,12,13,14,15], SP-PLP performed with a reversible addition-fragmentation chain-transfer agent combined with NIR spectroscopy (SP-PLP-NIR-RAFT) [16,17,18], and steady-state RAFT polymerization (RAFT-CLD-T) [19,20]. There have been several reviews of this work [4,5,21]. In particular, these techniques have been deployed for examining the monomers methyl methacrylate (MMA) [22,23,24] and styrene (ST) [25,26] at low conversion. For long chains, there is a wealth of data supporting e_L ≈ 0.16–0.20 [4,7,11,22,27,28]. For small chains in ST and methacrylate systems, it has been found that e_S ≈ 0.5–0.65, which also agrees with the theoretical expectations [4,10]. These values will be of importance in this work, as will the measured value i_c ≈ 50–200 [4].

The aforementioned techniques through which Equation (2) has found verification all involve the narrowing of the RCLD to such an extent that the observed <k_t> at any instant is essentially a k_t^i,i value. However, the standard situation for producing polymer involves continuous initiation; for example, via thermally decomposing chemical initiator, which means that the RCLD is broad. How do the <k_t> values obtained in such circumstances shed light on CDLT, i.e., on the underlying k_t^i,j values, and are such findings consistent with results from the niche techniques used to determine k_t^i,i? In this work, we investigate these questions via the measurement and analysis of E_a(<k_t>) values, as obtained from straightforward determinations of the rate of polymerization, R_p, in thermally induced systems. As well as contributing to the building up of a comprehensive database of steady-state <k_t> values, we aim to show that variations in E_a(<k_t>) can be explained in terms of CLDT concepts, thereby providing further insight into the mechanism of termination. We have a particular focus on systems with DP_n < i_c, because the behaviour of such systems should be significantly influenced by Equation (2a) [10], for which e is higher than for long chains, as already explained. In our thermally induced polymerizations, we achieve DP_n < 100 ≈ i_c through employing relatively high T and through adding solvent to attain a low monomer concentration, c_M.

2. Materials and Methods

2.1. Materials

2,2′-Azobisisobutyronitrile (AIBN) was purified via recrystallisation. Bis(3,5,5-trimethylhexanoyl) peroxide (BTMHP) (Akzo Nobel, Amersfoort, The Netherlands) was utilized as received. Styrene (ST; Sigma-Aldrich, Auckland, New Zealand), methyl methacrylate (MMA; Sigma-Aldrich), n-butyl methacrylate (BMA; Sigma-Aldrich), and dodecyl methacrylate (DMA; Sigma-Aldrich) were obtained as indicated. In order to remove the inhibitor, all these monomers were purified chromatographically. The solvents trifluorotoluene (TFT; Sigma-Aldrich, ≥99.0%) and ethylbenzene (EBz, Fluka, Buchs, Switzerland) were utilized as received.

2.2. Polymerizations

A series of radical homopolymerizations with different monomers were conducted isothermally in solution at atmospheric pressure, employing either AIBN or BTMHP as the initiator and either EBz (for ST) or TFT (all three methacyrlates) as the solvent. Details about these solvent choices will be presented in due course. In order to identify the impact of temperature on the termination rate coefficient, polymerization was studied at temperatures ranging from 50 to 90 °C. Prior to use, reaction mixtures were thoroughly deoxygenated by purging with nitrogen gas for 15 min. The reaction mixtures were then divided into several samples, typically 5–10. The polymerization was carried out for each sample and stopped at a different time by immersing the sample in ice water. The stoppage times were chosen so as to maintain dilute-solution conditions; specifically, the conversion of the monomer into a polymer was kept below 20% (which, due to the high level of solvent, still results in a dilute solution). A negligible change in volume upon heating to the reaction temperature was assumed for all polymerizations, meaning that c_M and initiator concentration, c_I, were taken as being unchanged from the preparation conditions. The conversion for each sample was determined using gravimetric analysis, after evaporating the residual monomer and solvent and then drying the sample in a vacuum oven.

2.3. Molar Mass Measurement

Size exclusion chromatography (SEC) analysis was conducted on selected samples. The instrument was equipped with a refractive index (RI) detector, and the system included 2 × Polypore 300 × 7.5 mm columns with a nominal particle size of 5 μm. Tetrahydrofuran (THF) was used as the eluent at a flow rate of 1.0 mL min⁻¹, and the analysis was conducted at 35 °C with a polymer concentration of approximately 5 mg mL⁻¹. Calibration was via polystyrene standards; for poly(MMA), poly(BMA), and poly(DMA) samples, universal calibration was employed with Mark–Houwink parameters, as in Table S1.

Electrospray ionization mass spectrometry (ESI-MS) was conducted as detailed in the Supporting Information.

3. Data Analysis and Results

The following form of the rate law for steady-state RP was utilized for analyzing conversion-time data [29]:

(4)${\displaystyle \frac{-\mathrm{d}\mathrm{l}\mathrm{n}(1-x)}{\mathrm{d}t}}=k_{\mathrm{p}}{c}_{\mathrm{R}}=k_{\mathrm{p}}{\left({\displaystyle \frac{f{k}_{\mathrm{d}}{c}_{\mathrm{I}}}{⟨k_{\mathrm{t}}⟩}}\right)}^{0.5}\equiv k_{\mathrm{o}}$

Here, x represents the fractional conversion of a monomer into a polymer, t denotes time, c_R is the overall radical concentration, and <k_t>, k_p, and k_d are the rate coefficients for termination, propagation, and initiator decomposition, respectively, while the initiator efficiency and concentration are f and c_I, respectively. Thus, x(t) data should be plotted as −ln(1 − x) versus t, a straight line fitted, and k_o for the polymerization obtained as the slope. Typical results are presented in Figure 1, in which it is evident that the expectation of linear behaviour is met. Furthermore, the well-known rate increase with temperature is immediately visible, and there is good experimental reproducibility (see the data at 50 and 85 °C). This gives confidence in the conclusions drawn from such experiments, and, once again, indicates that when carried out attentively, gravimetry is no less precise than many other methods for monitoring conversion and studying kinetics [9,29].

Results from every low-conversion solution polymerization carried out in this work are presented in Table 1. From each k_o, an experimentally measured value of <k_t> may be obtained using the following rearranged form of Equation (4):

(5)$⟨k_{\mathrm{t}}⟩=f{k}_{\mathrm{d}}{c}_{\mathrm{I}}{\left({\displaystyle \frac{k_{\mathrm{p}}}{k_{\mathrm{o}}}}\right)}^2$

Using this equation requires knowledge of f, k_d, and k_p, which, in our case, are needed as a function of temperature and for a variety of monomers and initiators. These variations were stipulated via Arrhenius expressions:

(6)$k=A\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{-E_{\mathrm{a}}}{RT}}\right)$

The employed values of A and E_a for all RP rate parameters for all systems studied here are given in Table 2, and the resulting <k_t> values are given in Table 1.

The values for k_p and k_d in Table 2 are well established, although further will be said about k_d (AIBN) below. However, f is rarely measured properly, and, consequently, Arrhenius parameters for it are virtually unreported. For AIBN, we fitted experimentally obtained f(T) in bulk styrene [34] to Equation (6), which resulted in the Arrhenius parameters of Table 2. This expression generates f = 0.80 at 100 °C, which agrees well with another study [38]. However, our fit should not be used above 100 °C, because it will soon generate f > 1, which is physically impossible. It is stressed that these f values are for low conversion only, because f decreases with conversion [34]. It also decreases with an increasing viscosity [34], but, in fact, all the systems of this work have very similar viscosities because the monomers are diluted in either TFT (0.57 cP at 20 °C) or EBz (0.67 cP at 20 °C), both of which are similar to styrene (0.76 cP at 20 °C), in which the measurements of f were made [34]. Thus, these values should hold reasonably accurately for all our systems without adjustment. For BTMHP, there is an absence of information about the effect of temperature on f, so consequently, the reported value f = 0.53 [35] was utilized at all temperatures, resulting in the parameters of Table 2. Of course, E_a(f) = 0 is not realistic, but the value is likely to be very small and of the magnitude of the AIBN value of 5.7 kJ mol⁻¹, which means that the variation with temperature is minor in scale. Furthermore, it will be seen below that the <k_t> values obtained with the two different initiators are in good agreement, which justifies E_a(f) = 0 for BTMHP as a workable approximation.

It is noted that the reporting of k_o—which is essentially the raw experimental output—in Table 1 will allow for the simple reprocessing of data if more accurate parameter values for use in Equation (5) become available in the future. Indeed, an example of this in relation to our previous work [9] will be provided below.

The other values in Table 1 are DP_n. These were calculated using the well-known Mayo equation [39]:

(7)${\displaystyle \frac{1}{{DP}_{\mathrm{n}}}}=C_{\mathrm{t}\mathrm{r}\mathrm{M}}+C_{\mathrm{t}\mathrm{r}\mathrm{S}}{\displaystyle \frac{c_{\mathrm{S}}}{c_{\mathrm{M}}}}+{\displaystyle \frac{\left(1+\lambda \right){\left(f{k}_{\mathrm{d}}{c}_{\mathrm{I}}⟨k_{\mathrm{t}}⟩\right)}^{0.5}}{k_{\mathrm{p}}{c}_{\mathrm{M}}}}$

This equation assumes only steady-state and long chains, so it will be reasonably accurate for our work as long as the input parameter values are accurate, which we made every effort to achieve. Thus, we used f, k_d, and k_p as already detailed and <k_t> as measured. For the fraction of termination by disproportionation, λ, we used the accurately measured value 0.63 for MMA [35]. In the absence of any other good information, we made the reasonable assumption that this value also holds for BMA and DMA, as all these monomers are chemically similar. For ST, we used λ = 0.1 [40], consistent with it predominantly terminating by combination. Although it is expected that λ changes with temperature [40], there is no good information in the literature on this, and, furthermore, the variation will be small in absolute terms because just like f, the values are constrained to be between 0 and 1. Thus, we utilized the stated values for all temperatures.

Transfer processes are incorporated into Equation (7) via the so-called transfer constants, with C_trM for monomers and C_trS for solvents, at a certain concentration c_S. Although these will not be zero, they can be assumed to be negligible for our conditions of low c_M and high T, which will result in relatively small DP_n and the so-called termination-control of dead-chain formation [7]. Although our high c_S potentially brings the transfer to solvent into play, we note that this is suppressed by the widespread use of TFT, which lacks labile C–H bonds for transfer [41]. Thus, we used C_trM = C_trS = 0 for calculations with Equation (7), and we note that if any significant dead-chain formation by transfer actually did occur, it would result in an even lower DP_n than those reported in Table 1.

Even if the DP_n are indicative calculations rather than precise measurements, it is evident that for the most part, we succeeded in our aim of establishing experimental conditions such that DP_n was of order i_c or lower, so that we could investigate the variation in <k_t> with T, where e_S—see Equation (2a)—plays a significant role in shaping the narrative.

Before proceeding, we present in Figure 2 all our <k_t> results for MMA. The scatter in the data is actually very low by historical standards for <k_t> [6,9]. Furthermore, the agreement between the two sets of results using different initiators, viz. AIBN and BTMHP, is excellent, where it should be noted that the different concentrations of each (0.05 and 0.018 mol L⁻¹, respectively) were chosen so as to give very similar rates of initiation at each T, thereby eliminating any potential difference in <k_t> due to this effect of CLDT [7]. The data of Figure 2 can therefore be said to evidence once again the reproducibility of our experimental results. Furthermore, these data generate confidence in the parameter values in Table 2 tha, which have been used to derive <k_t> values from our x(t) measurements.

Reanalysis of Previous Results

Equation (5) makes clear that <k_t> ~ fk_d in the processing of k_o values. Thus, any aberrant trend in fk_d values that are used in such processing will be carried over into the resulting <k_t>. Since carrying out our earlier work on the variation in MMA and ST <k_t> with c_I, c_M, and T for long-chain conditions [9], it came to our attention that the Arrhenius expression we used for fk_d (AIBN) was sub-optimal [41]. This is illustrated in Figure 3, which is an Arrhenius plot showing the fk_d [8] from Berger [42] that we previously used [9], k_d for AIBN from an AkzoNobel catalogue [33], and te literature values of k_d [43,44,45,46,47,48,49,50,51,52] and fk_d [53] for AIBN in various media. Despite the large number of different solvent media, it is evident from Figure 3 that the AkzoNobel Arrhenius parameters provide a highly accurate description of the literature data for k_d (AIBN). Because of this, and because one would expect a supplier to have characterized their product meticulously, we have switched to using the AkzoNobel k_d values in the present work (see Table 2). Furthermore, the data of Figure 3 give confidence that the AkzoNobel fit will hold well in the TFT- and EBz-dominated monomer solutions of our work because it shows at most a minor variation in k_d amongst solvents of this type.

In our prior study [9], we used the Berger results [42] because they account for both AIBN decomposition and efficiency under actual polymerization conditions. These fk_d values turn out to be in precise agreement with the product of f [34] and k_d [33] from Table 2 at 40 °C, as well as the Fukuda et al. [53] values at the same temperature. It was for this reason that the Berger results—for which E_a(fk_d) = 123.5 kJ mol⁻¹—were adopted at all T values. However, Table 2 makes clear the problem with this, for it gives E_a(fk_d) = (130.2 + 5.7) = 135.9 kJ mol⁻¹. While this change in E_a may look to be relatively minor (e.g., it can hardly be discerned in Figure 3), it turns out to be highly significant for E_a(<k_t>). This is because Equation (5) shows that E_a(<k_t>) is a balance of different E_a, giving a final result that is much smaller than E_a(fk_d), and on which scale the difference of 12.4 kJ mol⁻¹ has a large impact. This is illustrated in Figure 4, which shows our previous <k_t> results as published [9] and as re-analyzed with the fk_d of the present work, which we believe to more accurate. Because E_a(fk_d) is larger, the obtained E_a(<k_t>) are larger, rising from 6 [9] to 18 kJ mol⁻¹ for MMA and from 12 (previously misreported as 14 [9]) to 24 kJ mol⁻¹ for ST. These updated values, which we regard as more accurate, will be dissected in the following section. Figure 4 makes clear that the higher E_a(<k_t>) arise because previously we used fk_d that were too low at high T, and hence the obtained <k_t> at these T values were too low. The important lesson from this re-analysis is that the accuracy of input data can be of utmost importance in the measurement of trends in <k_t>, such as its E_a.

4. Discussion

4.1. Theoretical Framework

Although, unfortunately, the following equation [10,54] is not widely appreciated, it is a tremendous tool for understanding RP kinetics [4,55]:

(8)$⟨k_{\mathrm{t}}⟩=k_{\mathrm{t}}^{\mathrm{1,1}}{\left[Г\left({\displaystyle \frac{2}{2-e}}\right)\right]}^{-2}{\left[{\displaystyle \frac{{\left(2R_{\mathrm{i}}{k}_{\mathrm{t}}^{\mathrm{1,1}}\right)}^{0.5}}{k_{\mathrm{p}}{c}_{\mathrm{M}}}}\left({\displaystyle \frac{2}{2-e}}\right)\right]}^{2e/\left(2-e\right)}$

In this equation, Γ is the gamma function, R_i = 2fk_dc_I is the rate of initiation, and all other parameters have been previously introduced. In particular, the e is that of Equation (3), which is the homotermination model used in deriving Equation (8). Also assumed are steady-state, long chains, negligible chain transfer, and the so-called geometric-mean model for k_t^i,j [4]. All these assumptions are necessary; otherwise, a closed expression for <k_t> is not possible. While this list may seem highly restrictive, in fact, steady state and long chains are assumptions that are standardly met, and we have already discussed how our experimental conditions were designed to result in negligible dead-chain formation by transfer. So, attention is speicifically focussed on the simple power-law model for k_t^i,i, Equation (3), and on the geometric-mean model for k_t^i,j. Neither of these are physically realistic [4,5]. However, the latter turns out to be of no consequence, because it has been shown that the trends in Equation (8) are quantitatively accurate regardless of the cross-termination model [56]. With the composite model, in Equation (2), the above result is not strictly valid, which is unfortunate. However, no analytic expression for <k_t> is possible with Equation (2) [10], leaving Equation (8) as the only option for gaining insight. Happily, it was shown that if DP_n » i_c, then Equation (8) with e = e_L is accurate, and alos analogously, when the average chain size is very short [10], results which make intuitive sense.

In consequence of all this, Equation (8) is a powerful lens for the analysis of RP kinetic data. For example, it has been successfully used to quantitatively explain deviations from the classical rate law for RP that are due to variations of <k_t> with c_I and c_M [4,9]. Apropos of which, it is clear from Equation (8) that if one is seeking to understand the variation in <k_t> with T, then a sensible experimental design involves keeping both c_I and c_M constant, because this eliminates two of the many factors that cause <k_t> to vary. This is why we kept these concentrations constant across all temperatures in our experiments (see Table 1), and the fact that workers have not historically paid attention to this is one of several reasons for the notorious scatter in Arrhenius plots of <k_t> found in the literature [6,9], making it almost impossible to identify trends in E_a(<k_t>).

Given constant c_I and c_M, one has from Equation (8) that

(9)$E_{\mathrm{a}}\left(⟨k_t⟩\right)=\left(1+a\right)E_{\mathrm{a}}\left(k_{\mathrm{t}}^{\mathrm{1,1}}\right)+a{E}_{\mathrm{a}}\left(f\right)+a{E}_{\mathrm{a}}\left(k_{\mathrm{d}}\right)-2a{E}_{\mathrm{a}}\left(k_{\mathrm{p}}\right),\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} a={\displaystyle \frac{e}{2-e}}$

This equation is a paradigm-shifting result, since it indicates that E_a(<k_t>) is not just a function of the activation energy of a termination process, viz. E_a(k_t^1,1), but, due to CLDT, it is also greatly affected by how the rates of initiation and propagation vary with temperature because these processes change the RCLD, which then changes <k_t>, as earlier explained using Equation (1). Thus, one sees the power of Equation (8); it furnishes a relatively simple quantitative means for analyzing E_a(<k_t>) values, viz. Equation (9). This is as opposed to large-scale kinetic simulations, the complexity of which tends to obscure any simple patterns of behaviour in the output. For these reasons, we will use Equation (9) to interrogate our kinetic data.

We start with Figure 5, which presents evaluations of Equation (9) for the four monomers of this work using the parameter values of Table 2, all of which are based on the most accurate experimental data available. As shown, E_a(<k_t>) = E_a(k_t^1,1) for chain-length-independent termination, e = 0, as expected. However, the interesting thing is that E_a(<k_t>) ≠ E_a(k_t^1,1) when there is any CLDT (e > 0). Specifically, E_a(<k_t>) increases as the chain-length dependence of termination increases. Furthermore, the variation with e depends on the monomer type, e.g., compare ST with the methacrylates in Figure 5. From Equation (9), it is clear that this is due to E_a(k_p) being different for ST and the methacrylate family (see Table 2), which evidently means that ST’s RCLD changes differently as T is varied. These findings immediately explain the variability and apparent lack of pattern in the literature values of E_a(<k_t>) and, in particular, their common inequality with E_a of diffusion processes. Even the methacrylates, which often show family-type commonality in E_a, can be seen from Figure 5 to have different E_a(<k_t>) for a particular e, this stemming from E_a(k_t^1,1) variation, e.g., this is why the results for DMA stand apart—due to its large size and high viscosity, this monomer has a significantly higher E_a(k_t^1,1) than MMA and BMA (see Table 2).

4.2. Long Chains

In our previous work [9], we carried out bulk polymerisations of ST and MMA, as opposed to the relatively dilute-solution (c_M = 0.67 mol L⁻¹) polymerisations of the present work. Furthermore, we used c_AIBN = 0.0005, 0.005, and 0.05 mol L⁻¹, as opposed to only the highest of these values here. For these reasons, our DP_n values were much larger in our previous work, as per Equation (7) with higher c_M and lower c_I. Specifically, we had DP_n » i_c. This means that one would expect <k_t> behaviour to be dictated by Equation (2b), i.e., the long-chain portion of the composite model. For E_a(<k_t>), one may use Equation (9) to see whether this is the case. It rearranges to

(10)$e={\displaystyle \frac{2a}{1+a}}, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} a={\displaystyle \frac{E_{\mathrm{a}}\left(⟨k_{\mathrm{t}}⟩\right)-E_{\mathrm{a}}\left(k_{\mathrm{t}}^{\mathrm{1,1}}\right)}{{E_{\mathrm{a}}\left(k_{\mathrm{t}}^{\mathrm{1,1}}\right)+E}_{\mathrm{a}}\left(f\right)+E_{\mathrm{a}}\left(k_{\mathrm{d}}\right)-2E_{\mathrm{a}}\left(k_{\mathrm{p}}\right)}}$

This equation provides a novel and relatively easily executed procedure for obtaining the chain-length dependence of termination from steady-state measurements of E_a(<k_t>), where we stress that such measurements must be carried out with constant c_M, constant c_I, and the same initiator.

Using Equation (10) with the values of Table 2, our long-chain MMA value of E_a(<k_t>) = 18 kJ mol⁻¹ from Figure 4A gives e = 0.17, which is in remarkably precise agreement with experimentally measured [4,24] values and with the theoretically predicted [4,57] value of e_L = 0.16. For ST, the situation is not quite so remarkable, with the long-chain value of E_a(<k_t>) = 24 kJ mol⁻¹ from Figure 4B giving e = 0.31, which is above the theoretical prediction for e_L (also 0.16), as well as other experimental measurements (e_L ≈ 0.2 [4]). This may be due to the low k_p and high <k_t> of ST systems, both of which result in smaller DP_n values (see Equation (7)) that are closer to the crossover chain length i_c. Hence, there may be some influence from smaller chains, for which e = e_S is higher than e_L (see Section 1), thus making E_a(<k_t>) higher (see Figure 5). Another thing to note is the influence of E_a(k_t^1,1). For example, if, instead of 9 kJ mol⁻¹ (Table 2), one uses the value 11 kJ mol⁻¹ in Equation (10), as measured for toluene diffusion coefficients [58] and for styrene fluidity [26], one obtains e = 0.27 for ST, which is closer to the known e_L. If nothing else this illustrates just how sensitive the value of E_a(<k_t>) is to many underlying factors.

The higher than expected value of E_a(<k_t>) for ST should not obscure two really important and novel accomplishments from the preceding paragraph of work: (1) the measurement of E_a(<k_t>) > E_a(k_t^1,1) for steady-state polymerization has been fully explained in terms of CLDT; and (2) experimental values of low-conversion E_a(<k_t>) have been shown to be in broad quantitative agreement with what CLDT predicts for long chains (see Figure 5 with e ≈ 0.2).

It is worth mentioning that our long-chain E_a(<k_t>) for MMA and ST are actually very close to the values obtained from considering the large number of <k_t> values from the literature that are presented in our previous work [9]: Figure S1 yields E_a(<k_t>) = 14 kJ mol⁻¹ for literature MMA, as compared with 18 kJ mol⁻¹ here, while for ST (Figure S2), these values are 21 and 24 kJ mol⁻¹, respectively. Of course, the literature <k_t> values are highly scattered, and the underlying experiments involved a wide variety of initiators, c_I and c_M, as opposed to the deliberate design of our experiments recognizing the impacts of CLDT (see above). However, one can expect effects from such variability to cancel out when <k_t> values from multiple and diverse data sets are combined and fitted, which evidently is the case. This provides enhanced confidence in our experimental procedures, and thus the interpretations attributed to our results.

Lastly, there is something important about the results of this section that may easily go unappreciated. It is that the theory behind Equation (8) and the value e_L ≈ 0.2 are established beyond doubt [4], and hence, there is no conjecture about the finding E_a(<k_t>) ≈ 20 kJ mol⁻¹ for the chemically initiated, long-chain polymerization of MMA, ST, and other similar monomers (see Figure 5). Notably, this value is approximately double that expected from E_a(k_t^1,1), i.e., by viewing E_a(<k_t>) as arising purely from diffusion. We contend that this is an important paradigm shift in understanding and experimentally addressing RP kinetics.

4.3. Short Chains

Figure 5 shows that E_a(<k_t>) is expected to increase as CLDT becomes stronger. How can this be tested? One cannot just dial up the e of a system in the way that one can choose a value of c_M or c_I, for example. However, if one shifts the RCLD to short enough chain lengths, then Equation (2a) should start to play a more prominent role in determining <k_t> behaviour, and because e_S > e_L, one should therefore observe higher E_a(<k_t>). That is why we designed our present experiments to have low c_M and relatively high c_I, thus producing chains with low DP_n, in many cases, far less than i_c (see Table 1). As far as we are aware, ours is the first deliberate determination of E_a(<k_t>) for thermally induced polymerization in the short chain-length regime, DP_n ≤ i_c. That this was achieved by using a high level of solvent is not of mechanistic consequence for termination, because bulk polymerizations at very low conversion—as considered in the preceding section—also have dilute-solution conditions, where the solvent is the monomer. Thus, termination in each case is the same mechanistically, meaning that e_S and e_L are much the same.

Our <k_t> values for AIBN-initiated experiments are presented in Figure 6. We do not include the relatively small number of BTMHP data (see Table 1 and Figure 2) because we are investigating E_a(<k_t>), which Equation (9) reveals as depending on the initiator. So, by restricting our analysis to AIBN results, we are controlling the number of variables as much as possible, just as with our use of constant c_I and c_M across all T.

There are several ways in which the results of Figure 6 are qualitatively pleasing: (1) Our MMA and ST results of Figure 6 are clearly higher in value than those of Figure 4 at the same T. This is plainly a manifestation of CLDT: smaller DP_n results in higher <k_t>. (2) The stark trend <k_t>(ST) > <k_t>(MMA) is also most plausibly explained as originating in CLDT [8,9]: the diffusion coefficients of MMA and ST are very similar, which means their k_t^1,1 values will also be very similar, and hence, their difference in <k_t> likely stems from ST’s slower propagation, which results in its RCLD being more weighted towards small chains and therefore <k_t> being higher, as captured in Equation (8). (3) This reasoning also helps to explain the observation, as also made for longer chains [59], that <k_t>(MMA) > <k_t>(BMA) > <k_t>(DMA), for k_p increases in this order for these monomers (see Table 2). However, there is also a reinforcing effect here from k_t^1,1 because it becomes smaller as the methacrylate size becomes bigger. (4) The E_a(<k_t>) values derived from Figure 6, which are presented in Table 3, are indeed higher than those observed for long chains, as discussed in the previous section. This is consistent with the expectation from the composite model.

Before proceeding, it is important to stress that none of the trends identified in the preceding paragraph can be an artefact of f and k_d values because all data analysis has been with identical f(T) and k_d(T): were these values changed, then all <k_t> values would be identically changed, and thus all the noted trends would still be obtained. Furthermore, we do not believe that any of these trends are artefacts of noise in our data because it seems clear from Figure 4 and Figure 6 that the quantitative differences underlying these trends are larger than the experimental error.

Turning now to quantitative analysis, we have applied Equation (10) to our E_a(<k_t>) values and listed the resulting values of e in Table 3. Importantly, all are between e_S and e_L in value, as one would expect. The e obtained for MMA and BMA are rather amazing in that they are only marginally below the range e_S = 0.5–0.65, which is what has been obtained in experimental measurements of k_t^i,i for methacrylates [4,36], with such values being held to arise from centre-of-mass diffusion being the rate-determining step for termination [4,10]. Of course, one would not expect to obtain e values quite this large in the present experiments because DP_n are not so small that the entire RCLD has i < i_c, and thus there will still be some long-chain influence on the obtained value of e. Hence, one can take our MMA and BMA results as being fully consistent with composite-model parameter values, and so the perhaps surprisingly high E_a(<k_t>) values are resoundingly explained.

While the DMA and ST results of Table 3 are at least qualitatively in line with the notions of the previous paragraph (higher e and elevated E_a(<k_t>)), it must equally be admitted that the quantitative boosts are not as large as anticipated. Plausible explanations for this may be offered. In the case of DMA, i_c is in the range 50–70, and so it is evident from Table 1 that the accessed DP_n values simply do not meet the criterion of being far less than i_c in value. This is simply because the high k_p and low <k_t> of DMA mean that it is not possible to achieve DP_n < i_c across the temperature range of this work with the same c_M and c_I as in other systems. Consequently, one would expect the behaviour of DMA in our present experiments to be far more dictated by the long-chain value e ≈ 0.16 (see above). Furthermore, the value of e obtained from E_a(<k_t>) is dependent on the value of E_a(k_t^1,1), for which the one available measurement of 20 kJ mol⁻¹ is rather high compared with other methacrylate values (see Table 2). If, for example, one changes this to 15 kJ mol⁻¹, then e = 0.27 is obtained with Equation (10).

With regard to ST, the first thing to say is that our observation of E_a(<k_t>,MMA) > E_a(k_t,ST) is at least consistent with Figure 5, where it stems from E_a(k_p) being higher for ST. However, it is equally clear from Figure 5 that one does not expect the difference to be as large as 38 vs. 25 kJ mol⁻¹, respectively. One explanation for this could be a contribution in the case of ST from the transfer to EBz. This solvent was employed because it is the saturated analogue of ST, and thus it furnishes a solution environment that is nearly physically identical to bulk ST. However, EBz has C–H bonds, which are labile in terms of chain transfer. On the other hand, the solvent used for methacrylate work, TFT, does not. Indeed, in previous work, we have demonstrated via ESI-MS that transfer to methyl isobutyrate definitely occurs when it is used as a solvent in MMA polymerization, whereas this reaction is eliminated when TFT is used instead [41]. If any transfer to EBz did occur in our ST experiments, it would have elevated the value of <k_t> by generating more short radicals. This effect would be more pronounced at lower temperatures due to the much stronger decline in R_i with temperature, meaning that the ‘competition’ between transfer and initiation in dead-chain formation somewhat evens out [7]. This would explain that our measured E_a(<k_t>) is lower than expected on the basis of dead-chain formation by termination alone.

Another observation about our ST experiments is that the ESI-MS analysis of the generated polymer indicated the presence of O₂ in some chains—see Figure S3. This was not found with methacrylate systems [41], and even with ST, it was a surprise because there was no evidence of inadequate deoxygenation in our x(t) measurements in the form of inhibition (delayed polymerization start) or retardation (rate acceleration as residual O₂ is consumed). It is also not clear why a deoxygenation procedure that worked perfectly adequately for the three methacrylates should not have been so effective for ST, unless there were high levels of O₂ in the EBz. Furthermore, it is clear from Figure S3 that some O₂ was present in our ST experiments, and this must have given rise to an elevation of <k_t>. Since O₂ solubility increases as T decreases, this retardation effect must have been greater at lower T values, which also plausibly explains E_a(<k_t>) being slightly lower than expected.

Although these potential issues with our ST systems are not ideal, it seems helpful for other workers that we draw attention to them.

4.4. Investigations Involving Degree of Polymerization

A CLDT relationship that is simpler and therefore better known than Equation (8) is

<k_t> = G k_t^1,1 DP_n^−e(11)

Although this equation is challenging to derive [10,54], it is intuitive in that it says that if i = DP_n is used in Equation (3), then the k_t^i,i obtained is essentially equal to <k_t>. This is because the value of G, which depends only on e and λ, is close to 1 [10,54]. The assumptions behind Equation (11) are the same as those for Equation (8), except for one important difference: Equation (11) holds also when there is dead-chain formation by transfer [7,56]. This confers on it the additional advantage of being more widely applicable. A further advantage in this regard is that variables such as c_M and c_I are not explicitly present in Equation (11) because they affect both <k_t> and DP_n in ways that are implicitly captured by the relationship. This means that one does not need to be careful about controlling these variables in a set of experiments.

Due to these advantages, Equation (11) has been the most commonly used vehicle for probing CLDT [4]. Mostly, this has been carried out via the log–log plotting of simultaneously determined <k_t> and DP_n data, with the slope of the plot yielding e and the intercept k_t^1,1. Such experiments have always been carried out at a constant temperature. But, because T is the variable of interest in the present work and because Equation (11) must also hold as T is varied, a new way of looking at it is prompted, for in terms of E_a, it yields

E_a(<k_t>) = E_a(k_t^1,1) − e × E_a(DP_n)(12)

This result assumes negligible variation in G and e with temperature. The former is reasonable because, as mentioned, G is always close to 1 in value. The latter finds experimental support from the many SP-PLP-EPR investigations that have been carried out, for they always find negligible variation in e_S and e_L with T [14,24,26,36].

It is well known that DP_n decreases as T increases in chemically initiated RP, meaning E_a(DP_n) < 0. For e = 0 (chain-length-independent termination), Equation (12) shows that this has no effect on E_a(<k_t>), which is simply equal to E_a(k_t^1,1), as it must be. However, for the reality of CLDT (e > 0), E_a(DP_n), < 0 means E_a(<k_t>) > E_a(k_t^1,1), which is fully consistent with Figure 5, our experimental results, and the new understanding of E_a(<k_t>) that we have herein developed. It seems worthwhile to push the envelope and see if Equation (12) is quantitatively successful. To do this, we must introduce our measurements of molar mass, M.

Figure 7 presents SEC results for our MMA experiments. It is evident that reproducibility is good and that M decreases with increasing T, as expected. The corresponding values of DP are in line with expectations (see Table 1) when it is remembered that (1) the SEC peak is close in value to DP_w, the weight-average DP, which will be about double the value of DP_n [60], and (2) the effect of chain-length-dependent propagation, which has been ignored in using Equation (7), will be to add about 3–5 to the value of DP_n [61,62] because of the initial rapid growth of chains. Thus, for example, the 90 °C MMA value of DP_n ≈ 10 in Table 1 is consistent with peak DP ≈ 25 for this system in Figure 7. These are only back-of-the-envelope calculations and should not be taken literally—they are merely to establish that there is decent agreement between our SEC results and what we expected for DP_n.

Values of DP_p, the peak DP, from all our SEC measurements are presented in Figure 8. We opted to use DP_p rather than DP_n for quantitative analysis because the latter is prone to significantly higher error than the former due to the difficulty of converting w(logM) into accurate number-molar mass distribution, n(M), at very low M, where w(logM) is small in value but n(M) is large. Of course, Equation (11) calls for DP_n, but recall that our interest is in E_a(DP_n), which is obtained from the variation in log(DP_n) with T⁻¹. This variation will be the same as that for log(DP_p), providing that DP_p ~ DP_n, which should be the case to good approximation. We note that in previous work investigating Equation (7), it was shown that the approach of using DP_p is indeed more reliable [60].

All systems in Figure 8 show decreasing DP_p with increasing T, as expected (see above). This gives rise to negative values of E_a(DP_p), as tabulated in Table 4. It is evident that values for all four monomers are comparable, as is also visually apparent from Figure 8. Note, however, that the absolute values of DP_p show the expected variation from monomer to monomer, with DMA > BMA > MMA > ST, due to k_p decreasing in the same order (see Table 2) and <k_t> increasing in this order (see Figure 6).

How can the E_a(DP_p) values of Table 4 be put to use? If E_a(<k_t>) is not known, then it could be estimated using Equation (12) with E_a(DP_p) ≈ E_a(DP_n) combined with known values of e and E_a(k_t^1,1), an approach with the advantage of requiring less input data than that of Equation (9). For example, using E_a(k_t^1,1) from Table 2 together with e ≈ 0.6 for short-chain MMA [24], one obtains E_a(<k_t>) = 40 kJ mol⁻¹, which is very close to the value measured here for MMA, viz. 38 kJ mol⁻¹ (see Table 3). Equally, from measured E_a(<k_t>), one could estimate E_a(DP_n). Alternatively, where both E_a(<k_t>) and E_a(DP_p) ≈ E_a(DP_n) have been measured from the same experiments, one can obtain an estimate of e for the prevailing conditions using the following rearranged from of Equation (12):

(13)$e={\displaystyle \frac{E_{\mathrm{a}}\left(⟨k_{\mathrm{t}}⟩\right)-E_{\mathrm{a}}\left(k_{\mathrm{t}}^{\mathrm{1,1}}\right)}{-E_{\mathrm{a}}\left({DP}_{\mathrm{n}}\right)}}$

This approach is the varying-T analogue of the aforementioned approach of plotting log(<k_t>) versus log(DP_n), noting that the latter approach assumes constant k_t^1,1 and thus can only be used for constant-T data. Equation (13) is a curious result in that it sets no constraint on the value of E_a(DP_n) when termination is chain-length-independent (e = 0, meaning E_a(<k_t>) = E_a(k_t^1,1)), but when e ≠ 0, this equation rigidly stipulates the relationship between these three activation energies, even though E_a(DP_n) must also be consistent with the Mayo equation (Equation (7)).

Table 4 also presents values of e obtained using Equation (13) with E_a(k_t^1,1) from Table 2, E_a(<k_t>) from Table 3, and E_a(DP_n) = E_a(DP_p) from Table 4. Not surprisingly the results are a mixed bag, exactly along the lines of the earlier E_a(<k_t>) results, and for the same reasons. Namely, for MMA and BMA, the obtained e values are in very good agreement with expectation for e_S, while for DMA and ST, the obtained values are somewhat lower. For DMA, this is plausibly due to the chains being more in the long-chain regime, consistent with Figure 8. Indeed, it would be interesting to investigate E_a(DP_n) for long chains, as discussed further in the following section. The purpose of the present section has primarily been to draw attention to the novel result that is Equation (12), discussing what it means and illustrating how it can be used with experimental data.

4.5. Further Consequences and Future Avenues

The results of this paper make many predictions worthy of further experimental investigation. Some examples that occur are as follows:

(i) E_a(<k_t>) for long-chain methacrylates. The predictions of Figure 5 for long-chain (e ≈ 0.2) systems have been probed only for ST and MMA in this work. It obviously would be of interest to extend this to other methacrylates, e.g., BMA and DMA. This requires experimental design as in the present work, viz. constant c_M, c_I, and initiator. Of particular interest is whether E_a(<k_t>) ≈ 32 kJ mol⁻¹ for long-chain DMA (see Figure 5) on account of E_a(k_t^1,1) ≈ 20 kJ mol⁻¹. Indeed, we note that if the value of e is confidently known, then a measurement of E_a(<k_t>) can be used to estimate the value of E_a(k_t^1,1).

(ii) E_a(DP_n) for long chains. Equation (12) rearranges to

E_a(DP_n) = [E_a(k_t^1,1) − E_a(<k_t>)]/e (14)

It has been seen in this work that E_a(k_t^1,1) ≈ 10 kJ mol⁻¹, E_a(<k_t>) ≈ 20 kJ mol⁻¹, and e ≈ 0.2 for long-chain MMA. Using these in Equation (14) predicts E_a(DP_n) ≈ −50 kJ mol⁻¹ for chemically initiated, long-chain MMA. Is this the case? Such predictions could be investigated, whether approximately (as in this example) or precisely, noting that <k_t> and DP_n measurements from the same experiments are required, as opposed to picking values from separate experiments out of the literature.

(iii) Polymerization with continuous photoinitiation. Chemical initiation relies on thermal energy to bring about the homolysis of the initiator, and thus the process has a high E_a. With photoinitiation, the homolysis is induced by incident radiation, and hence the rate of primary radical generation will be independent of temperature, to good approximation at least. Thus, with (continuous) photoinitiation, one should switch to using E_a(k_d) = 0 in Equation (9). For this reason, we have carried out evaluations of this equation in which all else is the same as before (i.e., the values of Table 2 are used) aside from E_a(k_d). The results are presented in Figure 9. Of course, the value of E_a(f) for a photoinitiator will not be the same as for AIBN; however, there will still be initiator inefficiency that can be expected to vary with T in the same qualitative manner as for AIBN. Furthermore, this value has only a very minor effect on results. For these reasons, it seems appropriate to stick with the Table 2 value of E_a(f) for the present illustrative purposes.

The remarkable effect seen in Figure 9 is that E_a(<k_t>) with photoinitiation is predicted to decrease with increasing e, opposite to what is observed in Figure 5 for chemical initiation. This is because initiation no longer has a strongly temperature-dependent effect on the RCLD, and thus other factors play a more noticeable role. Specifically, increasing k_p weights the RCLD towards longer chains, an effect which counterbalances that of increasing k_t^i,j. Indeed, Figure 9 shows that at high enough e, this effect of k_p becomes so strong that E_a(<k_t>) will become negative in value for monomers like ST and MMA, as first pointed out long ago [8].

Obviously, it would be of interest to investigate the predictions of Figure 9 by measuring E_a(<k_t>) for polymerizations with continuous photoinitiation. Encouragement can be found in the literature in the form of E_a(<k_t>) = 5.6 kJ mol⁻¹, having been obtained for MMA in SP-PLP experiments [63]. Remarkably, 5.6 kJ mol⁻¹ is exactly what Figure 9 predicts for long-chain (e ≈ 0.2) MMA! However, this result should be taken as being indicative only because the photoinitiation in SP-PLP is not continuous. That said, the Olaj group conducted a lot of theoretical work showing that trends in <k_t> measured by PLP are the same as those with continuous initiation [64], so there is good reason to think that Buback and Kowollik’s SP-PLP result should reflect what would be found with continuous photoinitiation.

An important take-home message here is that E_a(<k_t>) measured with laser-based techniques will be significantly lower than when applied for chemical initiation. Specifically, photoinitiation will result in E_a(<k_t>) < E_a(k_t^1,1) (see Figure 9), while for chemical initiation, one has E_a(<k_t>) > E_a(k_t^1,1) (Figure 5).

(iv) Acrylate polymerization. In terms of termination kinetics, acrylates differ from methacrylates in two major ways: they have lower E_a(k_p) and higher e_S. Specifically, E_a(k_p) = 17.3 and 17.9 for chain-end methyl acrylate (MA) [65] and butyl acrylate (BA) [66], respectively, while e_S ≈ 0.8 for these monomers [67]. Both these effects serve to elevate E_a(<k_t>) in chemically initiated polymerization, as can be seen in Figure 10. This shows evaluations of Equation (9) for MA and BA with the just-given E_a(k_p) but with all else, as in Table 2 for MMA and BMA, respectively. Of course, E_a(k_t^1,1) for an acrylate will be different to that for the corresponding methacrylate, but measurements [67] indicate that these differences are very small (of order 1–2 kJ mol⁻¹ at most), which is as one would expect on the basis of small-molecule diffusion, e.g., an MA monomer is physically very similar to MMA. So, for illustrative purposes, it was decided to make E_a(k_p) the only altered value in performing acrylate calculations. For comparison, the MMA and BMA results of Figure 5 are also presented in Figure 10.

Figure 10 makes the prediction that E_a(<k_t>) will be slightly higher for long-chain (e = e_L ≈ 0.2 [67]) acrylates than methacrylates. It additionally predicts that this difference will become quite appreciable if very short chain lengths are accessed because of the higher value of acrylate e_S coming into play. Of course, these findings apply only for very low temperatures, at which a negligible fraction of mid-chain radicals can be expected. Once backbiting becomes significant, the kinetics of acrylate polymerization are greatly complicated [68], and it is not clear what should be expected of E_a(<k_t>) in chemically initiated systems that constantly have both mid-chain and end-chain radical populations present.

(v) Dominant transfer. Up until now, this work has assumed dead-chain formation predominantly by termination. The opposite limit to this is dominant transfer [7]. For this situation, the analogue of Equation (9) is [7]

(15)$E_{\mathrm{a}}\left(⟨k_t⟩\right)=E_{\mathrm{a}}\left(k_{\mathrm{t}}^{\mathrm{1,1}}\right)+e{\times E}_{\mathrm{a}}\left(C_{\mathrm{t}\mathrm{r}\mathrm{M}}+C_{\mathrm{t}\mathrm{r}\mathrm{S}}{\displaystyle \frac{c_{\mathrm{S}}}{c_{\mathrm{M}}}}\right)$

Here, the notation of Equation (7) has been used for transfer terms. Note that, once again, there is the phenomenon that E_a(<k_t>) > E_a(k_t^1,1) due to the RCLD favouring shorter chains—which have higher k_t^i,j—as the temperature increases due to the C_tr ratios having positive E_a.

It would be of interest to test the predictions of Equation (7). Experiments with a constant ratio of transfer agent to monomer concentration across different T values are recommended. If this is the case and there is a dominant transfer with E_a(C_tr) = 20 kJ mol⁻¹ (a typical value for transfer to labile solvent [69]), then E_a(k_t^1,1) = 10 kJ mol⁻¹ and e = 0.2 (long chains) results in a prediction of E_a(<k_t>) = 14 kJ mol⁻¹, while e = 0.5 (short chains) results in E_a(<k_t>) = 20 kJ mol⁻¹. Note that these values are independent of the rate of initiation, so another interesting prediction is that photoinitiation and chemical initiation should give the same E_a(<k_t>), which is opposite to the situation with the termination control of dead-chain formation (see above). That said, with chemical initiation, it is hard to remain in the limit of dominant transfer as T is raised because of E_a(k_d) being so large.

Where there is significant dead-chain formation by both transfer and termination, no analytic expressions exist for <k_t> in the event of CLDT, so all one can say is that E_a(<k_t>) will be in between the two limits of Equation (9) and Equation (15). For this reason, it is advisable to carry out experimental investigations of termination in one of these limits. The parameter values obtained can then be used in computer-based predictions for the situation of mixed termination and transfer. If one tries to investigate termination in this mixed-mode situation, then the only option is computer-based modelling with a plethora of adjustable parameters, a practice which is not ideal if one’s aim is to obtain mechanistic insight.

5. Conclusions

Understanding how <k_t> varies with T is of utmost importance for predicting how the rate of polymerization and DP_n vary with T in RP. Where a reaction has a complex mechanism, it is well known in physical chemistry that its E_a may be complicated in that it can be a function of several individual E_a for elementary steps in the mechanism. However, there is no such expectation for termination in RP because it is a facile, single-step reaction. Prima facie, it is therefore a surprise that no clear understanding of E_a(<k_t>) has emerged over many decades of RP study. In this work, we have addressed this uncomfortable situation, finding that it is explained by the chain-length-dependent nature of termination, which in fact bestows complicated and indeed fascinating behaviour on E_a(<k_t>), as encapsulated in Equation (9). While this result may strike many workers as being overly elaborate and unnecessarily complicated, in fact, it contains only the bare minimum detail, for it ignores chain transfer, chain-length-dependent propagation, and composite-model termination, all of which can exert further influences.

Another objection to Equation (9) is that it applies for the broad RCLDs from continuous initiation, and termination is better studied in designer experiments with narrow RCLD, most notably time-resolved SP-PLP [5]. While this is indeed beyond dispute for obtaining values of k_t^i,i, it does not change that the overwhelming majority of commercial RP is via chemically initiated systems without a reversible-deactivation RP agent. What are E_a(<k_t>) for such systems? They are necessary to know, and a major point of this work has been to make clear that they do not follow in a simple way from E_a(k_t^i,i) measurements. Rather, we have argued that Equation (9) is an invaluable lens for understanding chemically initiated E_a(<k_t>) in all its complexity. We have conducted this by demonstrating that measurements of chemically initiated E_a(<k_t>) using a simple technique are consistent with the predictions of Equation (9) for different monomers and different DP_n. CLDT parameters from SP-PLP-EPR experiments are indispensable inputs in this process, which shows that both types of investigation are essential for establishing a total picture of termination kinetics in RP. We have also pointed out interesting predictions from Equation (9) that should be investigated in further work.

CLDT is an indisputable reality that renders RP kinetics complicated, but we are adamant that these realities can be conquered, as we have shown here for E_a(<k_t>), hopefully in an esthetically pleasing way.

Author Contributions

Conceptualization, M.M.A. and G.T.R.; methodology, M.M.A. and G.T.R.; validation, M.M.A. and G.T.R.; formal analysis, M.M.A.; investigation, M.M.A.; data curation, M.M.A.; writing—original draft preparation, M.M.A.; writing—review and editing, G.T.R.; visualization, M.M.A. and G.T.R.; supervision, G.T.R. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article and Supplementary Materials; further inquiries can be directed to the corresponding author.

Acknowledgments

M.M.A. extends his appreciation to the Scientific Council at King Khalid University for funding his studies in New Zealand. Both authors express their gratitude to Neil Edmonds of the University of Auckland for hosting M.M.A. during the experimental part of these endeavours.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Fractional conversion (x)-time (t) data from the radical polymerization of styrene (ST) at various temperatures as indicated, where cAIBN = 0.05 mol L−1 and cST = 0.67 mol L−1 with ethyl benzene (EBz) as the solvent were used. Points: experimental measurements; lines: best fit for each set of results. Note that the two lines and sets of symbols at 50 and 85 °C represent duplicate runs.

Figure 2. Arrhenius plot for the variation in the overall termination rate coefficient, <kt>, with temperature, T, for MMA polymerizations initiated by AIBN or BTMHP, as indicated (see Table 1 for further details on polymerization conditions).

Figure 3. Arrhenius plot for variation in the initiator decomposition rate coefficient, kd, with temperature, T, for AIBN in various solvents by various workers as follows (from top of legend to bottom): in bulk styrene, by Berger [42] (presented as fkd from [8], where f is initiator efficiency); chlorobenzene, AkzoNobel [33]; MMA and DMF, Szafko and Feist [49]; in MMA and in DMF, Moroni [48]; ethyl acetate, Bawn and Mellish [43]; ST, Breitenbach and Schindler [52]; ST/toluene, Moad et al. [47]; Tâlat-Erben and Bywater, toluene [50]; in toluene and in benzene, Van Hook and Tobolsky [51]; benzene, Bawn et al. [43,44]; benzene, Krstina et al. [46]; in cyclohexane and in dichlorobenzene, Moroni [48]; fkd in MMA and in ST, Fukuda et al. [53]; and dodecane, Charton et al. [45].

Figure 4. Arrhenius plots for the variation in the overall termination rate coefficient, <kt>, with temperature, T, for AIBN-initiated bulk polymerisations of MMA (A, left) and ST (B, right). Squares: previously reported values [9] obtained using fkd from Berger [42]; half-filed circles: updated values using fkd from Table 2; lines: best fits for each data set.

Figure 5. Predicted activation energy of the overall termination rate coefficient, Ea(<kt>), for steady-state MMA, BMA, DMA, and ST polymerization as a function of e, the strength of the chain-length dependence of termination (under conditions of constant cM, constant cAIBN and negligible transfer).

Figure 6. Arrhenius plot for the variation in the overall termination rate coefficient, <kt>, with temperature, T, for the AIBN-initiated, dilute-solution polymerization of ST, MMA, BMA, and DMA, as indicated. Points: experimental values from Table 1; lines: best fits for each data set.

Figure 7. SEC distributions of polymer from MMA polymerizations at various temperatures, as indicated, with cM = 0.67 mol L−1, TFT as solvent, and cAIBN = 0.05 mol L−1. Distributions are presented as w(logM), where w is weight fraction and M is molar mass. For ease of comparison, all peak heights have been set equal to 1.

Figure 8. Arrhenius plot for variation in the peak degree of polymerization, DP, with temperature, T, for AIBN-initiated, dilute-solution polymerizations of ST, MMA, BMA, and DMA, as indicated. Points: experimental values; lines: best fits for each data set.

Figure 9. As for Figure 5, except that Ea(kd) = 0 in order to mimic photoinitiation.

Figure 10. As for Figure 5, except that Ea(kp) = 17.3 kJ mol−1 [65] was used to obtain methyl acrylate (MA) estimates, and Ea(kp) = 17.9 kJ mol−1 [66] for butyl acrylate (BA).

Table 1

Results from low-conversion, solution polymerizations of methyl methacrylate (MMA), n-butyl methacrylate (BMA), dodecyl methacrylate (DMA), and styrene (ST), employing c_M = 0.67 mol L⁻¹ in trifluorotoluene (TFT) for MMA, BMA, and DMA, and in EBz for ST; and c_I = 0.05 mol L⁻¹ for AIBN or 0.018 mol L⁻¹ for BTMHP, as indicated. Values of DP_n are those calculated using Equation (7) with the parameter values as per the text.

Monomer (Initiator)	Temperature/°C	ko/s−1	<kt>/(L mol−1 s−1)	DP n
MMA (AIBN)	50	1.93 × 10−5	8.85 × 107	102
	50	1.96 × 10−5	8.60 × 107	103
	60	5.07 × 10−5	9.64 × 107	59
	60	5.32 × 10−5	8.75 × 107	62
	70	9.89 × 10−5	1.69 × 108	27
	70	1.30 × 10−4	9.81 × 107	36
	71	1.14 × 10−4	1.54 × 108	27
	85	3.81 × 10−4	1.62 × 108	14
	85	2.03 × 10−4	5.70 × 108	8
	85	3.46 × 10−4	1.96 × 108	13
	85	2.33 × 10−4	4.35 × 108	9
	85	3.00 × 10−4	2.61 × 108	11
	85	2.47 × 10−4	3.85 × 108	9
	85	2.45 × 10−4	3.91 × 108	9
	90	5.23 × 10−4	1.98 × 108	11
	90	3.74 × 10−4	3.88 × 108	8
MMA (BTMHP)	70	6.86 × 10−5	1.84 × 108	36
	70	7.93 × 10−5	1.38 × 108	42
	85	1.63 × 10−4	4.14 × 108	13
	85	1.58 × 10−4	4.44 × 108	13
	90	2.66 × 10−4	3.41 × 108	12
BMA (AIBN)	50	2.96 × 10−5	5.07 × 107	157
	50	3.57 × 10−5	3.52 × 107	188
	50	2.80 × 10−5	5.74 × 107	147
	60	6.51 × 10−5	8.07 × 107	75
	60	7.97 × 10−5	5.39 × 107	92
	70	1.77 × 10−4	7.44 × 107	49
	70	1.65 × 10−4	8.51 × 107	45
	70	1.70 × 10−4	8.01 × 107	47
	70	1.44 × 10−4	1.12 × 108	40
	85	4.67 × 10−4	1.53 × 108	18
	85	4.20 × 10−4	1.89 × 108	16
	87	4.74 × 10−4	2.09 × 108	14
	90	5.74 × 10−4	2.35 × 108	11
	90	5.59 × 10−4	2.45 × 108	11
DMA (AIBN)	50	5.53 × 10−5	2.62 × 107	292
	60	1.72 × 10−4	1.98 × 107	199
	70	3.86 × 10−4	2.55 × 107	107
	85	9.94 × 10−4	5.25 × 107	37
	85	1.03 × 10−3	4.91 × 107	39
	90	1.07 × 10−3	1.03 × 108	21
ST (AIBN)	50	3.26 × 10−6	4.11 × 108	26
	50	3.74 × 10−6	3.13 × 108	29
	60	1.02 × 10−5	3.95 × 108	18
	70	2.24 × 10−5	6.8 × 108	9
	85	8.66 ×10−5	8.76 × 108	5
	85	9.34× 10−5	7.51 × 108	5

Table 2

Arrhenius parameters E_a and A used in Equation (6) to calculate the propagation rate coefficients, k_p, initiator decomposition rate coefficients, k_d, and initiator efficiencies, f, in this work. Also included are E_a for the monomeric radical termination rate coefficients, k_t^1,1, that are used for calculations with Equation (9).

Quantity	Material	Ea/(kJ mol−1)	A	Reference
k p	MMA	22.36	2.673 × 106 L mol−1 s−1	[30]
k p	ST	32.51	4.266 × 107 L mol−1 s−1	[31]
k p	BMA	22.9	3.802 × 106 L mol−1 s−1	[32]
k p	DMA	21.00	2.512 × 106 L mol−1 s−1	[32]
k d	AIBN	130.23	2.89 × 1015 s−1	[33]
f	AIBN	5.70	5.04	[34]
k d	BTMHP	128.34	2.84 × 1015 s−1	[33]
f	BTMHP	0	0.53	[35]
k t 1,1	MMA	9	–	[24]
k t 1,1	ST	9	–	[26]
k t 1,1	BMA	10	–	[14]
k t 1,1	DMA	20	–	[36,37]

Table 3

Arrhenius parameters E_a and A for small-chain <k_t>, as derived from Figure 6, together with the corresponding values of the CLDT parameter e, as obtained using Equation (10) with parameter values from Table 2.

Monomer	Ea(<kt>)/(kJ mol−1)	A(<kt>)/(L mol−1 s−1)	e
MMA	38 ± 12	8.3 × 1013	0.45
BMA	39 ± 7	8.1 × 1013	0.45
DMA	32 ± 12	2.3 × 1012	0.19
ST	25 ± 6	2.3 × 1012	0.34

Table 4

Activation energy E_a for the peak degree of polymerization, DP_p, as derived from Figure 8, together with the resulting values of the CLDT parameter e, as obtained using Equation (13) with E_a(k_t^1,1) from Table 2 and E_a(<k_t>) from Table 3.

Monomer	Ea(DPp)/(kJ mol−1)	E
MMA	−52 ± 6	0.56
BMA	−57 ± 1	0.51
DMA	−54 ± 4	0.22
ST	−55 ± 2	0.29

Supplementary Materials

The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/polym16223225/s1: Table S1: Mark-Houwink parameters K and α employed for universal calibration in size exclusion chromatography; Figure S1: Arrhenius plot for variation of experimental values of overall termination rate coefficient, <k_t>, with temperature, T, for bulk, low-conversion polymerization of MMA; Figure S2: Arrhenius plot for variation of experimental values of overall termination rate coefficient, <k_t>, with temperature, T, for bulk, low-conversion polymerization of ST; Figure S3: ESI-MS spectrum of polystyrene (PST) obtained from radical polymerization in EBz at 90 °C with c_AIBN = 0.05 mol L⁻¹ and c_ST = 0.67 mol L⁻¹. References [70,71,72,73,74,75,76,77,78,79] are cited in the supplementary materials.

References

1. Norrish, R.G.W.; Smith, R.R. Catalyzed polymerization of methyl methacrylate in the liquid phase. Nature; 1942; 150, pp. 336-337. [DOI: https://dx.doi.org/10.1038/150336a0]

2. Pirman, T.; Ocepek, M.; Likozar, B. Radical Polymerization of Acrylates, Methacrylates, and Styrene: Biobased Approaches, Mechanism, Kinetics, Secondary Reactions, and Modeling. Ind. Eng. Chem. Res.; 2021; 60, pp. 9347-9367. [DOI: https://dx.doi.org/10.1021/acs.iecr.1c01649]

3. Allen, P.E.M.; Patrick, C.R. Diffusion-Controlled Reactions in Free Radical Polymerization. Makromol. Chem.; 1961; 47, pp. 154-167. [DOI: https://dx.doi.org/10.1002/macp.1961.020470114]

4. Barner-Kowollik, C.; Russell, G.T. Chain-length-dependent termination in radical polymerization: Subtle revolution in tackling a long-standing challenge. Prog. Polym. Sci.; 2009; 34, pp. 1211-1259. [DOI: https://dx.doi.org/10.1016/j.progpolymsci.2009.07.002]

5. Buback, M.; Russell, G.T. Detailed analysis of termination kinetics in radical polymerization. Polym. Int.; 2023; 72, pp. 869-880. [DOI: https://dx.doi.org/10.1002/pi.6501]

6. Buback, M.; Egorov, M.; Gilbert, R.G.; Kaminsky, V.; Olaj, O.F.; Russell, G.T.; Vana, P.; Zifferer, G. Critically evaluated termination rate coefficients for free-radical polymerization, 1: The current situation. Macromol. Chem. Phys.; 2002; 203, pp. 2570-2582. [DOI: https://dx.doi.org/10.1002/macp.200290041]

7. Smith, G.B.; Russell, G.T. The cutthroat competition between termination and transfer to shape the kinetics of radical polymerization. Macromol. Symp.; 2007; 248, pp. 1-11. [DOI: https://dx.doi.org/10.1002/masy.200750201]

8. Russell, G.T. The kinetics of free radical polymerizing systems at low conversion, 3. On the variation of the termination rate coefficient with monomer and with temperature. Macromol. Theory Simul.; 1995; 4, pp. 549-576. [DOI: https://dx.doi.org/10.1002/mats.1995.040040310]

9. Taylor, D.R.; Van Berkel, K.Y.; Alghamdi, M.M.; Russell, G.T. Termination rate coefficients for radical homopolymerization of methyl methacrylate and styrene at low conversion. Macromol. Chem. Phys.; 2010; 211, pp. 563-579. [DOI: https://dx.doi.org/10.1002/macp.200900668]

10. Smith, G.B.; Russell, G.T.; Heuts, J.P.A. Termination in dilute-solution free-radical polymerization: A composite model. Macromol. Theory Simul.; 2003; 12, pp. 299-314. [DOI: https://dx.doi.org/10.1002/mats.200390029]

11. Buback, M.; Egorov, M.; Junkers, T.; Panchenko, E. Free-radical termination kinetics studied using a novel SP-PLP-ESR technique. Macromol. Rapid Commun.; 2004; 25, pp. 1004-1009. [DOI: https://dx.doi.org/10.1002/marc.200400050]

12. Buback, M.; Egorov, M.; Junkers, T.; Panchenko, E. Termination Kinetics of Dibutyl Itaconate Free-Radical Polymerization Studied via the SP–PLP–ESR Technique. Macromol. Chem. Phys.; 2005; 206, pp. 333-341. [DOI: https://dx.doi.org/10.1002/macp.200400358]

13. Buback, M.; Müller, E.; Russell, G.T. SP-PLP-EPR study of chain-length-dependent termination in free-radical polymerization of n-dodecyl methacrylate, cyclohexyl methacrylate, and benzyl methacrylate: Evidence of “composite” behavior. J. Phys. Chem. A; 2006; 110, pp. 3222-3230. [DOI: https://dx.doi.org/10.1021/jp054606r]

14. Barth, J.; Buback, M.; Hesse, P.; Sergeeva, T. Chain-Length-Dependent Termination in n-Butyl Methacrylate and tert-Butyl Methacrylate Bulk Homopolymerizations Studied via SP-PLP-ESR. Macromolecules; 2009; 42, pp. 481-488. [DOI: https://dx.doi.org/10.1021/ma802078g]

15. Buback, M.; Kattner, H. Detailed analysis of radical polymerisation kinetics by pulsed-laser techniques. Mol. Phys.; 2021; 119, e1939452. [DOI: https://dx.doi.org/10.1080/00268976.2021.1939452]

16. Buback, M.; Junkers, T.; Vana, P. Laser single pulse initiated RAFT polymerization for assessing chain-length dependent radical termination kinetics. Macromol. Rapid Commun.; 2005; 26, pp. 796-802. [DOI: https://dx.doi.org/10.1002/marc.200500067]

17. Junkers, T.; Theis, A.; Buback, M.; Davis, T.P.; Stenzel, M.H.; Vana, P.; Barner-Kowollik, C. Chain Length Dependent Termination in Butyl Acrylate Free-Radical Polymerization Studied via Stationary and Pulsed Laser Initiated RAFT Polymerization. Macromolecules; 2005; 38, pp. 9497-9508. [DOI: https://dx.doi.org/10.1021/ma051485k]

18. Buback, M.; Hesse, P.; Junkers, T.; Theis, T.; Vana, P. Chain-Length-Dependent Termination in Acrylate Radical Polymerization Studied via Pulsed-Laser-Initiated RAFT Polymerization. Aust. J. Chem.; 2007; 60, pp. 779-787. [DOI: https://dx.doi.org/10.1071/CH07236]

19. Vana, P.; Davis, T.P.; Barner-Kowollik, C. Easy access to chain-length-dependent termination rate coefficients using RAFT polymerization. Macromol. Rapid Commun.; 2002; 23, pp. 952-956. [DOI: https://dx.doi.org/10.1002/1521-3927(200211)23:16<952::AID-MARC952>3.0.CO;2-F]

20. Johnston-Hall, G.; Monteiro, M.J. Bimolecular radical termination: New perspectives and insights. J. Polym. Sci. Part A Polym. Chem.; 2008; 46, pp. 3155-3173. [DOI: https://dx.doi.org/10.1002/pola.22684]

21. Barth, J.; Buback, M. SP-PLP-EPR–A Novel Method for Detailed Studies into the Termination Kinetics of Radical Polymerization. Macromol. React. Eng.; 2010; 4, pp. 288-301. [DOI: https://dx.doi.org/10.1002/mren.200900066]

22. Johnston-Hall, G.; Theis, A.; Monteiro, M.J.; Davis, T.P.; Stenzel, M.H.; Barner-Kowollik, C. Accessing chain length dependent termination rate coefficients of methyl methacrylate (MMA) via the reversible addition fragmentation chain transfer (RAFT) process. Macromol. Chem. Phys.; 2005; 206, pp. 2047-2053. [DOI: https://dx.doi.org/10.1002/macp.200500354]

23. Barth, J.; Buback, M.; Schmidt-Naake, G.; Woecht, I. Termination kinetics of free-radical polymerization in ionic liquids. Polymer; 2009; 50, pp. 5708-5712. [DOI: https://dx.doi.org/10.1016/j.polymer.2009.09.065]

24. Barth, J.; Buback, M. SP–PLP–EPR Investigations into the Chain-Length-Dependent Termination of Methyl Methacrylate Bulk Polymerization. Macromol. Rapid Commun.; 2009; 30, pp. 1805-1811. [DOI: https://dx.doi.org/10.1002/marc.200900335] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/21638457]

25. Johnston-Hall, G.; Monteiro, M.J. Diffusion controlled termination of linear polystyrene radicals in linear, 4-Arm, and 6-Arm star polymer matrices in dilute, semidilute, and concentrated solution conditions. Macromolecules; 2008; 41, pp. 727-736. [DOI: https://dx.doi.org/10.1021/ma702569m]

26. Kattner, H.; Buback, M. Chain-Length-Dependent Termination of Styrene Bulk Homopolymerization Studied by SP−PLP−EPR. Macromolecules; 2015; 48, pp. 309-315. [DOI: https://dx.doi.org/10.1021/ma5022665]

27. Mahabadi, H.K. Effects of chain length dependence of termination rate constant on the kinetics of free-radical polymerization. 1. Evaluation of an analytical expression relating the apparent rate constant of termination to the number-average degree of polymerization. Macromolecules; 1985; 18, pp. 1319-1324. [DOI: https://dx.doi.org/10.1021/ma00148a048]

28. Olaj, O.F.; Vana, P. Chain-length dependent termination in pulsed-laser polymerization, 6: The evaluation of the rate coefficient of bimolecular termination kt for the reference system methyl methacrylate in bulk at 25 °C. Macromol. Rapid Commun.; 1998; 19, pp. 533-538. [DOI: https://dx.doi.org/10.1002/marc.1998.030191006]

29. Barner-Kowollik, C.; Buback, M.; Egorov, M.; Fukuda, T.; Goto, A.; Olaj, O.F.; Russell, G.T.; Vana, P.; Yamada, B.; Zetterlund, P.B. Critically evaluated termination rate coefficients for free-radical polymerization: Experimental methods. Prog. Polym. Sci.; 2005; 30, pp. 605-643. [DOI: https://dx.doi.org/10.1016/j.progpolymsci.2005.02.001]

30. Beuermann, S.; Buback, M.; Davis, T.P.; Gilbert, R.G.; Hutchinson, R.A.; Olaj, O.F.; Russell, G.T.; Schweer, J.; Herk, A.M.v. Critically evaluated rate coefficients for free-radical polymerization, 2. Propagation rate coefficients for methyl methacrylate. Macromol. Chem. Phys.; 1997; 198, pp. 1545-1560. [DOI: https://dx.doi.org/10.1002/macp.1997.021980518]

31. Buback, M.; Gilbert, R.G.; Hutchinson, R.A.; Klumperman, B.; Kuchta, F.-D.; Manders, B.G.; O’Driscoll, K.F.; Russell, G.T.; Schweer, J. Critically evaluated rate coefficients for free-radical polymerization, 1. Propagation rate coefficient for styrene. Macromol. Chem. Phys.; 1995; 196, pp. 3267-3280. [DOI: https://dx.doi.org/10.1002/macp.1995.021961016]

32. Beuermann, S.; Buback, M.; Davis, T.P.; Gilbert, R.G.; Hutchinson, R.A.; Kajiwara, A.; Klumperman, B.; Russell, G.T. Critically evaluated rate coefficients for free-radical polymerization, 3. Propagation rate coefficients for alkyl methacrylates. Macromol. Chem. Phys.; 2000; 201, pp. 1355-1364. [DOI: https://dx.doi.org/10.1002/1521-3935(20000801)201:12<1355::AID-MACP1355>3.0.CO;2-Q]

33. Initiators for High Polymers (Product Catalog); AkzoNobel Chemicals: Amersfoort, The Netherlands, 2006.

34. Buback, M.; Huckestein, B.; Kuchta, F.D.; Russell, G.T.; Schmid, E. Initiator efficiencies in 2,2′-azoisobutyronitrile-initiated free-radical polymerizations of styrene. Macromol. Chem. Phys.; 1994; 195, pp. 2117-2140. [DOI: https://dx.doi.org/10.1002/macp.1994.021950620]

35. Buback, M.; Günzler, F.; Russell, G.T.; Vana, P. Determination of the mode of termination in radical polymerization via mass spectrometry. Macromolecules; 2009; 42, pp. 652-662. [DOI: https://dx.doi.org/10.1021/ma801732r]

36. Riemann, L. Termination Kinetics of Radical Homo- and Copolymerizations Studied via Time-Resolved EPR Spectroscopy. Ph.D. Thesis; Georg-August-Universität Göttingen: Göttingen, Germany, 2021.

37. Soerensen, N. Kinetics and Mechanism of Cu-Catalyzed Atom Transfer Radical Polymerization. Ph.D. Thesis; Georg-August-Universität Göttingen: Göttingen, Germany, 2015.

38. Vogt, V.W.; Dulog, L. Die radikalausbeute beim thermischen zerfall von azoisobuttersäuredinitril in gegenwart von sauerstoff und in methylmethacrylat als lösungsmittel. Makromol. Chem.; 1969; 122, pp. 223-236. [DOI: https://dx.doi.org/10.1002/macp.1969.021220117]

39. Barson, C.A. Chain Transfer. Chain Polymerization I; Eastwood, G.C.; Ledwith, A.; Russo, S.; Sigwalt, P. Pergamon: Oxford, UK, 1989; Volume 3, pp. 171-183.

40. Moad, G.; Solomon, D.H. The Chemistry of Free Radical Polymerization; 1st ed. Pergamon: Oxford, UK, 1995.

41. Alghamdi, M.M.; Russell, G.T. Investigations into the Mass Spectrometric Method for the Determination of the Mode of Termination in Radical Polymerization. Macromol. Chem. Phys.; 2013; 214, pp. 1384-1395. [DOI: https://dx.doi.org/10.1002/macp.201200734]

42. Berger, K.C. Disproportionierung und Kombination als Abbruchmechanismen bei der radikalischen Polymerisation von Styrol, 2. Analyse der Temperaturabhängigkeiten. Makromol. Chem.; 1975; 176, pp. 3575-3592. [DOI: https://dx.doi.org/10.1002/macp.1975.021761205]

43. Bawn, C.E.H.; Mellish, S.F. A method of determination of the rate of molecular dissociation in solution. Parts I and II. The rate of dissociation of benzoyl peroxide and 2:2′-azo-bis(isobutyronitrile) in various solvents. Trans. Faraday Soc.; 1951; 47, pp. 1216-1227. [DOI: https://dx.doi.org/10.1039/TF9514701216]

44. Bawn, C.E.H.; Verdin, D. Molecular dissociation processes in solution. Part 5.-The decomposition of azo-bis-isobutyronitrile in the presence of αα-diphenyl-β-picryl hydrazyl. Trans. Faraday Soc.; 1960; 56, pp. 815-822. [DOI: https://dx.doi.org/10.1039/TF9605600815]

45. Charton, N.; Feldermann, A.; Theis, A.; Stenzel, M.H.; Davis, T.P.; Barner-Kowollik, C. Initiator efficiency of 2,2′-azobis(isobutyronitrile) in bulk dodecyl acrylate free-radical polymerizations over a wide conversion and molecular weight range. J. Polym. Sci. Part A Polym. Chem.; 2004; 42, 5559. [DOI: https://dx.doi.org/10.1002/pola.20518]

46. Krstina, J.; Moad, G.; Willing, I.R.; Danek, S.K.; Kelly, D.P.; Jones, S.L.; Solomon, D.H. Further studies on the thermal decomposition of AIBN–implications concerning the mechanism of termination in methacrylonitrile polymerization. Eur. Polym. J.; 1993; 29, pp. 379-388. [DOI: https://dx.doi.org/10.1016/0014-3057(93)90108-R]

47. Moad, G.; Rizzardo, E.; Solomon, D.H.; Johns, S.R.; Willing, R.I. Application of ¹³C-labelled initiators and ¹³C NMR to the study of the kinetics and efficiency of initiation of styrene polymerization. Die Makromol. Chem. Rapid Commun.; 1984; 5, pp. 793-798. [DOI: https://dx.doi.org/10.1002/marc.1984.030051203]

48. Moroni, A.F. Über den einfluß des lösungsmittels beim thermischen zerfall des azoisobuttersäuredinitrils. Makromol. Chem.; 1967; 105, pp. 43-49. [DOI: https://dx.doi.org/10.1002/macp.1967.021050104]

49. Szafko, J.; Feist, W. Solvation effect in thermal decomposition of 2,2′-Azoisobutyronitrile on the N,N-dimethylformamide/methyl methacrylate system. J. Polym. Sci. Part A Polym. Chem.; 1995; 33, pp. 1637-1642. [DOI: https://dx.doi.org/10.1002/pola.1995.080331010]

50. Talât-Erben, M.; Bywater, S. The Thermal Decomposition of 2,2′-Azo-bis-isobutyronitrile. Part II. Kinetics of the Reaction. J. Am. Chem. Soc.; 1955; 77, pp. 3712-3714. [DOI: https://dx.doi.org/10.1021/ja01619a011]

51. Van Hook, J.P.; Tobolsky, A.V. The Thermal Decomposition of 2,2′-Azo-bis-isobutyronitrile. J. Am. Chem. Soc.; 1958; 80, pp. 779-782. [DOI: https://dx.doi.org/10.1021/ja01537a006]

52. Breitenbach, J.W.; Schindler, A. Polymerisationskinetische Versuche mit α-Azoisobuttersäurenitril und Benzoylperoxyd an Styrol zur Aufklärung der Geschwindigkeit von Start- und Übertragungsreaktion. Monatshefte Für Chem. Und Verwandte Teile Anderer Wiss.; 1952; 83, pp. 724-730.

53. Fukuda, T.; Ma, Y.D.; Inagaki, H. Free-radical copolymerization. 3. Determination of rate constants of propagation and termination for styrene/methyl methacrylate system. A critical test of terminal-model kinetics. Macromolecules; 1985; 18, pp. 17-26. [DOI: https://dx.doi.org/10.1021/ma00143a003]

54. Olaj, O.F.; Zifferer, G.; Gleixner, G. Termination Processes in Free Radical Polymerization. 8. Complete Treatment of a Kinetic Scheme Comprising Chain Length Dependent Termination in Terms of Closed and Approximate Closed Expressions Based on the Geometric Mean Assumption. Macromolecules; 1987; 20, pp. 839-850. [DOI: https://dx.doi.org/10.1021/ma00170a024]

55. Smith, G.B.; Heuts, J.P.A.; Russell, G.T. New paradigms in free-radical polymerization kinetics. Macromol. Symp.; 2005; 226, pp. 133-146. [DOI: https://dx.doi.org/10.1002/masy.200550813]

56. Olaj, O.F.; Zifferer, G. Termination Processes in Free Radical Polymerization. 9. Derivation of Universal Relationships between Kinetic Quantities for Arbitrary Chain Length Dependence of the Termination Constant. Macromolecules; 1987; 20, pp. 850-861. [DOI: https://dx.doi.org/10.1021/ma00170a025]

57. Friedman, B.; O’Shaughnessy, B. Kinetics of Intermolecular Reactions in Dilute Solution Polymer Solutions and Unentangled Melts. Macromolecules; 1993; 26, pp. 5726-5739. [DOI: https://dx.doi.org/10.1021/ma00073a030]

58. Pickup, S.; Blum, F.D. Self-diffusion of toluene in polystyrene solutions. Macromolecules; 1989; 22, pp. 3961-3968. [DOI: https://dx.doi.org/10.1021/ma00200a025]

59. Plate, N.A.; Ponomarenko, A.G. Kinetic features of the radical polymerization of n-alkylmethacrylates. Polym. Sci. U.S.S.R.; 1974; 16, pp. 3067-3081. [DOI: https://dx.doi.org/10.1016/0032-3950(74)90299-8]

60. Heuts, J.P.A.; Davis, T.P.; Russell, G.T. Comparison of the Mayo and Chain Length Distribution Procedures for the Measurement of Chain Transfer Constants. Macromolecules; 1999; 32, pp. 6019-6030. [DOI: https://dx.doi.org/10.1021/ma990076j]

61. Heuts, J.P.A.; Russell, G.T. The nature of the chain-length dependence of the propagation rate coefficient and its effect on the kinetics of free-radical polymerization. 1. Small-molecule studies. Eur. Polym. J.; 2006; 42, pp. 3-20. [DOI: https://dx.doi.org/10.1016/j.eurpolymj.2005.09.022]

62. Willemse, R.X.E.; Staal, B.B.P.; van Herk, A.M.; Pierik, S.C.J.; Klumperman, B. Application of Matrix-Assisted Laser Desorption Ionization Time-of-Flight Mass Spectrometry in Pulsed Laser Polymerization. Chain-Length-Dependent Propagation Rate Coefficients at High Molecular Weight: An Artifact Caused by Band Broadening in Size Exclusion Chromatography?. Macromolecules; 2003; 36, pp. 9797-9803.

63. Buback, M.; Kowollik, C. Termination Kinetics of Methyl Methacrylate Free-Radical Polymerization Studied by Time-Resolved Pulsed Laser Experiments. Macromolecules; 1998; 31, pp. 3211-3215. [DOI: https://dx.doi.org/10.1021/ma971748d]

64. Olaj, O.F.; Kornherr, A.; Zifferer, G. Chain-length dependent termination in pulsed-laser polymerization, 4. The influence of the type of mean involved in the bimolecular termination step. Macromol. Theory Simul.; 1998; 7, pp. 501-508. [DOI: https://dx.doi.org/10.1002/(SICI)1521-3919(19980901)7:5<501::AID-MATS501>3.0.CO;2-U]

65. Barner-Kowollik, C.; Beuermann, S.; Buback, M.; Castignolles, P.; Charleux, B.; Coote, M.L.; Hutchinson, R.A.; Junkers, T.; Lacik, I.; Russell, G.T. et al. Critically evaluated rate coefficients in radical polymerization-7. Secondary-radical propagation rate coefficients for methyl acrylate in the bulk. Polym. Chem.; 2014; 5, pp. 204-212. [DOI: https://dx.doi.org/10.1039/C3PY00774J]

66. Asua, J.M.; Beuermann, S.; Buback, M.; Castignolles, P.; Charleux, B.; Gilbert, R.G.; Hutchinson, R.A.; Leiza, J.R.; Nikitin, A.N.; Vairon, J.-P. et al. Critically Evaluated Rate Coefficients for Free-Radical Polymerization, 5. Propagation Rate Coefficient for Butyl Acrylate. Macromol. Chem. Phys.; 2004; 205, pp. 2151-2160. [DOI: https://dx.doi.org/10.1002/macp.200400355]

67. Barth, J.; Buback, M.; Russell, G.T.; Smolne, S. Chain-Length-Dependent Termination in Radical Polymerization of Acrylates. Macromol. Chem. Phys.; 2011; 212, pp. 1366-1378. [DOI: https://dx.doi.org/10.1002/macp.201000781]

68. Ballard, N.; Asua, J.M. Radical polymerization of acrylic monomers: An overview. Prog. Polym. Sci.; 2018; 79, pp. 40-60. [DOI: https://dx.doi.org/10.1016/j.progpolymsci.2017.11.002]

69. Gregg, R.A.; Mayo, F.R. Chain transfer in the polymerisation of styrene iii. The reactivities of hydrocarbons toward the styrene radical. Discuss. Faraday Soc.; 1947; 2, pp. 328-337. [DOI: https://dx.doi.org/10.1039/df9470200328]

70. Hutchinson, R.A.; Beuermann, S.; Paquet, D.A., Jr.; McMinn, J.H. Determination of Free-Radical Propagation Rate Coefficients for Alkyl Methacrylates by Pulsed-Laser Polymerization. Macromolecules; 1997; 30, pp. 3490-3493. [DOI: https://dx.doi.org/10.1021/ma970176u]

71. Hutchinson, R.A.; McMinn, J.H.; Paquet, D.A., Jr.; Beuermann, S.; Jackson, C. A Pulsed-Laser Study of Penultimate Copolymerization Propagation Kinetics for Methyl Methacrylate/n-Butyl Acrylate. Ind. Eng. Chem. Res.; 1997; 36, pp. 1103-1113. [DOI: https://dx.doi.org/10.1021/ie9604031]

72. Sack-Kouloumbris, R.; Meyerhoff, G. Radikalpolymerisation von methylmethacrylat im gesamten umsatzbereich. Stationäre und instationäre experimente zur bestimmung von wachstums- und abbruchsgeschwindigkeit. Die Makromol. Chemie; 1989; 190, pp. 1133-1152. [DOI: https://dx.doi.org/10.1002/macp.1989.021900520]

73. Mahabadi, H.K.; O’Driscoll, K.F. Absolute Rate Constants in Free-Radical Polymerization. III. Determination of Propagation and Termination Rate Constants for Styrene and Methyl Methacrylate. J. Macromol.Sci. Part A-Chem.; 1977; 11, pp. 967-976. [DOI: https://dx.doi.org/10.1080/00222337708061301]

74. Matheson, M.S.; Auer, E.E.; Bevilacqua, E.B.; Hart, E.J. Rate Constants in Free Radical Polymerizations. I. Methyl Methacrylate. J. Am. Chem. Soc.; 1949; 71, pp. 497-504. [DOI: https://dx.doi.org/10.1021/ja01170a034]

75. Stickler, M. Kinetics of radical polymerization of methyl methacrylate initiated with dimethyl 2,2′-azodiisobutyrate. Makromol. Chem.; 1986; 187, pp. 1765-1775. [DOI: https://dx.doi.org/10.1002/macp.1986.021870720]

76. Yamada, B.; Kageoka, M.; Otsu, T. ESR study of the radical polymerization of styrene. Polym. Bull.; 1992; 29, pp. 385-392. [DOI: https://dx.doi.org/10.1007/BF00944835]

77. Buback, M.; Kuchta, F.-D. Termination kinetics of free-radical polymerization of styrene over an extended temperature and pressure range. Macromol. Chem. Phys.; 1997; 198, pp. 1455-1480. [DOI: https://dx.doi.org/10.1002/macp.1997.021980513]

78. Olaj, O.F.; Vana, P. Chain-length dependent termination in pulsed-laser polymerization, 5a: The evaluation of the rate coefficient of bimolecular termination kt for the reference system styrene in bulk at 25 °C. Macromol. Rapid Commun.; 1998; 19, pp. 433-439.

79. Matheson, M.S.; Auer, E.E.; Bevilacqua, E.B.; Hart, E.J. Rate Constants in Free Radical Polymerization. III. Styrene1. J. Am. Chem. Soc.; 1951; 73, pp. 1700-1706. [DOI: https://dx.doi.org/10.1021/ja01148a082]