*2.3. 1H-NMR Characterization Method*

1H-nuclear magnetic resonance (1H-NMR) spectra were obtained using a Varian 300 MHz spectrometer (Varian, Santa Clara, CA, USA) using deuterated chloroform (CDCl3) as a solvent and tetramethylsilane (TMS) as an internal standard, at room temperature. Sample preparation and measurement proceeded as reported in our previous studies. [14] Data analysis for determination of copolymer composition and sequence distribution of monomer units from 1H-NMR characteristic signals was carried out using the appropriate equations [23].

#### *2.4. Copolymerization Reactions*

Bulk copolymerizations of St and GMA, at 103 ◦C, proceeded in a 1-L high-pressure stainless-steel jacketed reactor (Parr Model 4523, Moline, IL, USA) with temperature control, as well as pressure and stirring sensors. Ultra-high-purity nitrogen was used to provide an inert environment. Appropriate amounts of GMA and St were then added to the reactor, followed by the initiator and RAFT agent. A mixing rate of 150 rpm was used. The polymerizations proceeded under nitrogen atmosphere, at 4.14 bar. Then, temperature was increased to 103 ◦C using a Huber Unistat 815w thermoregulator, in cascade mode. The thermal oil flew directly to the reactor jacket and to the coil, which ensured temperature control with a precision of ±1 ◦C. Sample withdrawal, preparation and analyses proceeded as detailed in one of our earlier studies [19].

Monomer conversion was not measured directly. We measured polymer yield, which was calculated gravimetrically, as the ratio of mass of produced polymer to mass of initial total monomer. Therefore, although we refer to monomer conversion in the figures of this contribution, when referring to experimental data, it is strictly polymer yield.

#### **3. Model Development**

A kinetic mathematical model was developed to calculate polymerization rate, evolution of molar mass averages, and copolymer composition. The model contains the following assumptions: (a) penultimate effects were neglected; (b) while in some reactions the intermediate macroradical species formed during the additional step of the RAFT process may be stable enough to delay polymerization (maximum lifetime of 1 s), it is not considered here to initiate new species and terminate [24–27]; and (c) branching has also been neglected.

The starting polymerization scheme is shown in Table 1. Three polymer populations are involved: propagating radical or active (living) polymer molecules with terminal units A or B (Pn and Qn), dormant polymer molecules with terminal units A or B (TPn and TQn), and dead polymer molecules (*M*n), where subscript *n* is the number of monomeric units in the macromolecule. A and B represent St and GMA terminal units, respectively. T is the RAFT agent.


**Table 1.** Detailed polymerization scheme used in this study.

The mathematical model developed in this contribution is based on the method of moments. The definitions of moments of several polymer species are shown in Table 2.

**Table 2.** Definition of moments of the several polymer species.

$$\begin{array}{c} \mathbf{Y}\_{\mathrm{l}}^{\mathsf{a}} = \sum\_{n=0}^{\infty} \mathbf{n}^{\mathsf{i}} \begin{bmatrix} \mathbf{P}\_{\mathrm{n}} \end{bmatrix} \qquad \mathbf{Z}\_{\mathrm{l}}^{\mathsf{a}} = \sum\_{n=0}^{\infty} \mathbf{n}^{\mathsf{i}} \begin{bmatrix} \mathrm{TP}\_{\mathrm{n}} \end{bmatrix} \qquad \mathbf{D}\_{\mathrm{l}} = \mathbf{\sum}\_{n=0}^{\infty} \mathbf{n}^{\mathsf{i}} \begin{bmatrix} M\_{\mathrm{n}} \end{bmatrix} \\\ \mathbf{Y}\_{\mathrm{l}}^{\mathsf{b}} = \sum\_{n=0}^{\infty} \mathbf{n}^{\mathsf{i}} \begin{bmatrix} \mathbf{Q}\_{\mathrm{n}} \end{bmatrix} \qquad \mathbf{Z}\_{\mathrm{l}}^{\mathsf{b}} = \sum\_{n=0}^{\infty} \mathbf{n}^{\mathsf{i}} \begin{bmatrix} \mathrm{T} \mathbf{Q}\_{\mathrm{n}} \end{bmatrix} \end{array}$$

The detailed kinetic equations for low molar mass and polymer species are summarized in Table 3.

**Table 3.** Kinetic equation for species.


Table 4 shows the obtained moment equations.

**Table 4.** Moment equations for the species present in the St-GMA copolymerization.


Overall monomer conversion, copolymer composition and average molar masses, Mn and Mw are calculated using Equations (26)–(32). M in Equations (26) and (27) stands for monomer content; subscripts 1, 2 and 0 stand for monomer 1, monomer 2, and initial conditions, respectively.

$$\text{Overall conversion} = (\text{M}\_{\text{lo}} + \text{M}\_{\text{2o}} - (\text{M}\_{\text{1}} + \text{M}\_{\text{2}})) / (\text{M}\_{\text{1o}} + \text{M}\_{\text{2o}}) \tag{26}$$

$$\text{Coplolymer composition: F}\_1 = (\text{M}\_{10} - \text{M}\_1) / ( (\text{M}\_{10} - \text{M}\_1) + (\text{M}\_{20} - \text{M}\_2) ) \tag{27}$$

$$\text{Number-average chain length}: \text{ r}\_{\text{N}} = \frac{\text{Y}\_1^\text{a} + \text{ Y}\_1^\text{b} + Z\_1^\text{a} + Z\_1^\text{b} + \text{ D}\_1}{\text{Y}\_0^\text{a} + \text{ Y}\_0^\text{b} + Z\_0^\text{a} + Z\_0^\text{b} + \text{ D}\_0} \tag{28}$$

$$\text{Weight-average chain length}: \text{ r}\_{\text{W}} = \frac{\text{Y}\_{2}^{\text{a}} + \text{ Y}\_{2}^{\text{b}} + \text{ Z}\_{2}^{\text{a}} + \text{ Z}\_{2}^{\text{b}} + \text{ D}\_{2}}{\text{Y}\_{1}^{\text{a}} + \text{ Y}\_{1}^{\text{b}} + \text{ Z}\_{1}^{\text{a}} + \text{ Z}\_{1}^{\text{b}} + \text{ D}\_{1}} \tag{29}$$

$$\text{Dispersity} : \mathcal{D} = \frac{\mathbf{r}\_{\text{W}}}{\mathbf{r}\_{\text{N}}} \tag{30}$$

$$\mathbf{M}\_{\rm n} = \mathbf{r}\_{\rm N} \left( \mathbf{F}\_1 \, \mathbf{P} \mathbf{M}\_1 + \mathbf{F}\_2 \, \mathbf{P} \mathbf{M}\_2 \right) \tag{31}$$

$$\mathbf{M\_W} = \mathbf{D} \,\mathbf{M\_n} \tag{32}$$

The kinetic rate constants and parameters required by the model are summarized in Table 5. The values of the reactivity ratios for the RAFT copolymerization of St and GMA were obtained using a weighted non-linear multivariable regression approach, using software RREVM [19]. These values are also provided in Table 5.

**Table 5.** Kinetic constants for RAFT copolymerization of St (1)-GMA (2).


(a) Assumed equal to the corresponding value for butyl acrylate (see [32]).

The mobility of high-molar-mass macromolecules is reduced at high conversions in FRP. Consequently, the rates of termination, propagation and RAFT reactions involving polymer molecules change throughout the reaction. In this study, diffusion-controlled (DC) effects were considered only for the termination reactions (auto-acceleration (AA) effect), using Equations (33) and (34), where kt is an effective kinetic rate constant, ko t is the corresponding intrinsic kinetic rate constant, Vfo and Vf are the initial and final free-volume fractions, respectively, and βkkt is a free-volume parameter. It was assumed that βktc = βktd, parameters to be evaluated as AA effect.

$$\mathbf{k}\_{\rm tc} = \mathbf{k}\_{\rm tc}^{o} \exp\left[-\beta \mathbf{k}\_{\rm tc} \left(\frac{1}{\mathbf{V}\_{\rm f}} - \frac{1}{\mathbf{V}\_{\rm fo}}\right)\right] \tag{33}$$

$$\mathbf{k}\_{\rm td} = \mathbf{k}\_{\rm td}^{o} \exp\left[-\beta \mathbf{k}\_{\rm td} \left(\frac{1}{\mathbf{V}\_{\rm f}} - \frac{1}{\mathbf{V}\_{\rm fo}}\right)\right] \tag{34}$$

The free-volume fraction, Vf, is calculated using Equation (35) [33].

$$\mathbf{V\_{f}} = \left[0.025 + \mathfrak{a}\_{\mathrm{P}} \left(\mathrm{T} - \mathrm{T}\_{\mathrm{g},\mathrm{P}}\right)\right] \ \mathfrak{o}\_{\mathrm{P}} + \left[0.025 + \mathfrak{a}\_{\mathrm{M\_{1}}} \left(\mathrm{T} - \mathrm{T}\_{\mathrm{g},\mathrm{M\_{1}}}\right)\right] \ \mathfrak{o}\mathcal{M}\_{\mathrm{I}} + \left[0.025 + \mathfrak{a}\_{\mathrm{M\_{2}}} \left(\mathrm{T} - \mathrm{T}\_{\mathrm{g},\mathrm{M\_{2}}}\right)\right] \ \mathfrak{o}\mathcal{M}\_{\mathrm{2}}\tag{35}$$

α in Equation (35) is the thermal expansion coefficient, ϕ is the volume fraction, and Tg is the glass transition temperature. Subscripts p and Mi denote polymer and monomer i, respectively. Tgp is estimated using the Fox expression, given by Equation (36) [34].

$$\mathbf{T\_{g,P}} = 1/\left[\frac{\mathbf{f\_{P1}}}{\mathbf{T\_{g,P\_1}}} + \frac{\mathbf{f\_{P2}}}{\mathbf{T\_{g,P\_2}}}\right] \tag{36}$$

fp in Equation (36) is the weight fraction of the polymer. Table 6 shows the physical properties of monomers and polymers used.

**Table 6.** Physical properties of monomers and polymer for calculation of fractional free volume.


(a) Assumed equal to the corresponding value for butyl acrylate (see [36]).

Although there are many mathematical models for DC effects in FRP and step-growth polymerization processes available in the literature, it is difficult to adequately describe the performance of different monomers under wide ranges of operating conditions using a single model with a single set of parameters. One such model is the Marten-Hamielec (MH) model [43,44], but it has the disadvantage of being discontinuous and requires an onset trigger criterion. Attempts to remove the trigger criterion resulted in a simpler, but less accurate model [45]. Therefore, in this study, we used the simplified version of the MH model [45], with a simpler onset trigger criterion, which causes it to be closer to the original model.

The assumptions summarized in Table 7 allow the determination the kinetic rate constants of RAFT activation and transfer for homopolymerizations of St and GMA. The assumptions indicated in the columns of Table 7 (e.g., kaa1 = kat1) are necessary, due to the absence of experimental data to isolate the contributions of the two RAFT cycles to the properties of the produced polymer. Some authors have argued that the kinetic constants of the RAFT activation and RAFT transfer cycles may be different [46–48]; in RAFT polymerization modeling work, the equality of the kinetic constants of the RAFT cycles is supported [49–52].

**Table 7.** Simplifications made about the RAFT activation and transfer kinetic rate constants.


Another assumption is that the kinetic rate constants associated with the dormant species [TPn] and [TQn] (kat3, kft3, kat4, kft4) can be approximated from the Mayo-Lewis terminal model [53]. This is achieved by considering the four reactions present in the RAFT transfer cycle and performing only consumption balances for the [TPn] and [TQn] species, which are complemented with consumption balances for the [TPr] and [TQr] species to complete the cycle. By calculating this, Equations (37)–(42) were obtained.

$$\frac{\text{d}\left[\text{TP}\_{\text{n}}\right]}{\text{d}\left[\text{TQ}\_{\text{n}}\right]} = \frac{\text{k}\_{\text{ft3}}[\text{TP}\_{\text{n}}](\text{r}\_{3}[\text{P}\_{\text{r}}] + [\text{Q}\_{\text{r}}])}{\text{k}\_{\text{ft4}}[\text{TQ}\_{\text{n}}](\text{r}\_{4}[\text{Q}\_{\text{r}}] + [\text{P}\_{\text{r}}])} \tag{37}$$

where:

$$\mathbf{r\_3} = \frac{\mathbf{k\_{ft1}}}{\mathbf{k\_{ft3}}} \tag{38}$$

$$\mathbf{r}\_4 = \frac{\mathbf{k}\_{\rm ft2}}{\mathbf{k}\_{\rm ft4}} \tag{39}$$

$$\frac{\text{d}\left[\text{TP}\_{\text{r}}\right]}{\text{d}\left[\text{TQ}\_{\text{r}}\right]} = \frac{\text{k}\_{\text{at4}}\left[\text{TP}\_{\text{r}}\right](\text{r}\_{5}\left[\text{P}\_{\text{n}}\right] + \left[\text{Q}\_{\text{n}}\right])}{\text{k}\_{\text{at3}}\left[\text{TQ}\_{\text{r}}\right](\text{r}\_{6}\left[\text{Q}\_{\text{n}}\right] + \left[\text{P}\_{\text{n}}\right])}\tag{40}$$

where:

$$\mathbf{r\_5} = \frac{\mathbf{k\_{at1}}}{\mathbf{k\_{at4}}} \tag{41}$$

$$\mathbf{r}\_6 = \frac{\mathbf{k}\_{\text{at2}}}{\mathbf{k}\_{\text{at3}}} \tag{42}$$

Additionally, r1 = r3 = r5 and r2 = r4 = r6. This is due to the application of the terminal model to the RAFT-transfer cycle. The RAFT-activation and -transfer kinetic parameters corresponding to homopolymerization of styrene and GMA were estimated using homopolymerization data for each monomer.

RAFT-related kinetic rate constants (optimization A) and AA effect parameters (optimization B) were estimated from overall conversion (X)-time and Mn-time experimental results, using a weighted non-linear multivariable regression procedure where the residual variance was minimized. The objective function is defined in Equation (43).

$$\text{Objective function} = \min \left[ \sum\_{i}^{\text{n}} \frac{1}{\sigma\_{\text{X}}^{2}} (\mathbf{X}\_{i}^{\text{e}} - \mathbf{X}\_{i}^{\text{c}})^{2} + \sum\_{i}^{\text{n}} \frac{1}{\sigma\_{\text{M}}^{2}} (\mathbf{M}\_{i}^{\text{e}} - \mathbf{M}\_{i}^{\text{c}})^{2} \right] \tag{43}$$

Superscripts e and c in Equation (43) stand for experimental and calculated values, respectively; σ<sup>X</sup> <sup>2</sup> and σ<sup>M</sup> <sup>2</sup> are variances of conversion and molar mass data, respectively; and n is the number of data points in each experimental data set. However, for simplicity, both variances were assumed equal to one. Optimization A was carried out using St and GMA homopolymerization data only, using the model without the AA terms. Optimization B was conducted for each copolymerization data set. The parameters obtained from each data set were regressed to obtain the final estimates. The flow chart that describes the modeling and parameter estimation strategies used in this contribution is shown in Figure 1. The model equations were solved using an in-house Fortran code. The optimization procedure for parameter estimation was carried out with the subroutine UWHAUS [54]. The system of ordinary differential equations was solved using subroutine DDASSL [55].

**Figure 1.** Flow chart for the estimation of kinetic and AA effect parameters.

### **4. Results**

RAFT Synthesis and Characterization of Reactive Copolymers

St-GMA copolymers of different compositions (fGMA = 0.10, 0.15, 0.30, and 0.40) were synthesized by RAFT bulk copolymerization of the monomers, at 103 ◦C, according to Figure 2. A Mn of ~30,000 g mol−<sup>1</sup> was sought for all polymers. Final overall monomer conversions in a range of 85–90% were obtained. The experimental conditions used in this study are reported in Table 8.

Molar compositions of the St-GMA copolymers synthesized in this study were determined from the relative areas of the 1H NMR characteristic signals [23,30]. 1H NMR spectra for some of the obtained St-GMA copolymers are shown in Figure 2. Chemical shifts from phenyl protons in the region of 6.6–7.3 ppm, and methylene oxy (–OCH2–) protons and methyl protons of GMA units at 3.5–4.5 and 0.5–1.2 ppm, respectively, are

observed in Figure 3. The mole fraction of GMA in the copolymer was calculated as: F2 =5A3/(5 A3 +3A2), where A2 and A3 are peak areas of phenyl and methyl protons, respectively. This method was used in this work due to the distinct NMR resonance of the GMA methyl group even at low GMA mole fractions in the copolymer [30].

**Figure 2.** Simplified polymerization scheme.

**Table 8.** Summary of experimental conditions used in this study; T = 103 ◦C; [St + GMA]o:[CPDT]o: [ACHN]o = R3:R2:R1.


The St-GMA copolymers were characterized by SEC. They had Mn ~22,200–26,300 g mol−<sup>1</sup> and Ð~1.21–1.28, which suggests that no side reactions took place and that most of the active polymer molecules remained living until the end of the polymerization.

Several kinetic models have been developed for RAFT homo- [56–58] and copolymerization of a few monomers [59–62]. As stated earlier, in our polymerization scheme we assumed that no branches to the adduct were produced, making it easier to model our RAFT copolymerization system using the terminal model [53], which is given by Equations (37)–(42). Therefore, the RAFT homo- and cross-propagation kinetic rate constants for the copolymerization system were determined by the corresponding values of RAFT homopolymerizations of St and GMA, and from reported values of r1 y r2 for the same copolymerization system [19].

Even though the RAFT polymerization mechanism is well-established and accepted [63–67], the parameters involved, such as addition, fragmentation, and termination kinetic rate constants, are not always reliable even in well-known systems, such as the RAFT homopolymerizations of methyl methacrylate and St [68]. The activation and transfer kinetic rate constants evaluated in this study for RAFT copolymerization of St and GMA are summarized in Table 9.

The profiles obtained with the parameters reported in Table 9 are not included due to space restrictions, but very good agreement is obtained in the low-conversion region, where DC effects are not observed. Although it has been reported that DC effects are important in all the reactions where polymer molecules are involved, in RAFT polymerizations [64], we considered DC termination only [69], to capture the phenomenon without adding too many additional parameters that required estimation. The AA termination parameters evaluated are provided in Table 10. As observed in Table 10 the higher the content of St in the copolymer, the higher the value of the AA termination parameter.

**Figure 3.** 1H NMR spectrum of a RAFT synthesized St-GMA copolymer with (**a**) 55% and (**b**) 20% mole fractions of GMA in the feed mixture, using CPDT and ACHN. (f) phenyl protons of styrene; (m, n, o) methylene oxy (–OCH2–) protons of GMA; (p, q) methyl protons of GMA.

Figure 4 shows a first order behavior plot. A comparison of experimental data and calculated profiles of conversion, Mn, Mw, and dispersity versus time (or conversion, in one case) is shown in Figures 5–10.

**Figure 4.** First-order of different St-GMA samples. Symbols represent experimental data, whereas solid lines correspond to model predictions.

**Figure 5.** Profiles of overall conversion versus time for the different St-GMA samples. Symbols represent experimental data, whereas solid lines correspond to model predictions.

**Figure 6.** Copolymer composition versus time profiles for the different St-GMA samples. Symbols represent experimental data, whereas solid lines correspond to model predictions.


**Table 9.** Activation and transfer kinetic rate constants evaluated in this study.


**Figure 7.** Profiles of Mn versus time for St-GMA samples. Symbols and solid lines correspond to experimental and calculated profiles, respectively.

**Figure 8.** Profiles of Mw versus time for the different St-GMA samples. Symbols and solid lines correspond to experimental and calculated profiles, respectively.

**Figure 9.** Profiles of dispersity versus time for the different St-GMA samples. Symbols and solid lines correspond to experimental and calculated profiles, respectively.

**Figure 10.** Profiles of Mn versus conversion for the different St-GMA samples. Symbols and solid lines correspond to experimental and calculated profiles, respectively.

The AA effect occurs at high conversions for the copolymerization reactions of St-GMA. Considering this, the determination of the kinetic rate constants associated with the RAFT cycles and the parameters associated with the termination reactions were carried out independently, which helped minimize possible correlations.

Except for sample St-GMA 00-100 where some discrepancies between experimental data and calculated profiles of conversion versus time were obtained (see Figure 5), the agreement is good in all other cases. Regarding copolymer composition, the agreement between calculated and experimental profiles of F1 versus conversion is good when St content is high, but some deviations are observed in low conversions when its content decreases (see Figure 6).

Calculated and experimental profiles of Mn and Mw versus time are compared in Figures 7 and 8, respectively. Once again, the agreement is good except for sample S-GMA 00-100, where some discrepancies are observed. Considering the results obtained with the adjustment in high conversions, it is not enough to consider only diffusioncontrolled effects on the termination kinetic rate constants for GMA homopolymerization (St-GMA 00-100).

Figure 9 shows how the dispersity of the produced copolymers evolves over time. Figures 9 and 10 show that the model can describe the controlled/living behavior of this RDRP system. Figure 10 shows the typical linear behavior of an Mn versus conversion profile for an RDRP system.

The effect of GMA content on the AA-termination parameter (βkt) is shown in Figure 11. A linear behavior of the AA termination parameter as a function of GMA content is observed, with an R2 correlation of 0.9237.

**Figure 11.** Linear regression of the AA-termination parameters vs. GMA content.

As observed from this modeling study, AA-termination parameters for each copolymerization case were needed to obtain good results, but as observed in Figure 11, a linear trend was obtained for the GMA homopolymerization case.

### **5. Conclusions**

Our model for the RAFT copolymerization of St and GMA agrees very well with the experimental data generated in our laboratory and also reported in this contribution. The reactivity ratios determined for this copolymerization system were used considering that the terminal model was also fulfilled in the RAFT activation and transfer reactions. DC-termination using a simple free-volume model was sufficient to capture the effect of DC reactions in this system.

Unlike other modeling studies where neglection of the intermediate adduct results in qualitatively correct, but quantitatively inaccurate, predictions of the behavior of a RAFT polymerization system (Model 3 of [62]), the use of activation and transfer RAFT reactions resulted in our case in both qualitatively and quantitatively correct representations of the RAFT copolymerization of St and GMA. This model can be applied to other homo- and copolymerizations, and also extended to other systems, such as photo-RAFT polymerizations [15,70–72].

**Author Contributions:** J.J.B.-T. conceived and designed experiments; J.A.T.-L. and J.J.B.-T. wrote the paper, carried out the modeling, analyzed and interpreted data; J.J.B.-T. and N.G.-N. performed the experiments; E.V.-L. wrote, reviewed the paper and also helped in data analysis and interpretation; P.C. reviewed and edited the paper; E.S.-G. reviewed and edited the paper, providing insightful technical comments. We are indebted to Alex Penlidis, from the University of Waterloo, for technical discussions and for allowing E.V.-L. to check the manuscript for similarity through his UW affiliation. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by: (a) Dirección General de Asuntos del Personal Académico (DGAPA), Universidad Nacional Autónoma de México (UNAM), México, Projects PAPIIT IV100119 and IG100122; (b) UNAM-UV collaboration agreement; and (c) CIQA-UV collaboration agreement.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article.

**Acknowledgments:** J.A.T.-L. acknowledges the Collaboration Agreement CIQA-UV; the financial support from UNAM within collaboration agreement UNAM-UV; and L324B-FQ-UNAM, CIQA and Queen's University for hosting a research visit. E.V.-L. acknowledges financial support from DGAPA, UNAM, Projects PAPIIT IV100119 and IG100122.

**Conflicts of Interest:** The authors declare no conflict of interest.
