Kinetics and Modeling of Aqueous Phase Radical Homopolymerization of 3-(Methacryloylaminopropyl)trimethylammonium Chloride and its Copolymerization with Acrylic Acid

: The radical homopolymerization kinetics of 3-(methacryloylaminopropyl) trimethylammonium chloride (MAPTAC) and its batch copolymerization with nonionized acrylic acid (AA) in aqueous solution are investigated and modeled. The drift in monomer composition is measured during copolymerization by in situ NMR over a range of initial AA molar fractions and monomer weight fractions up to 0.35 at 50 ◦ C. The copolymer becomes enriched in MAPTAC for monomer mixtures containing up to 60 mol% MAPTAC, but is enriched in AA for MAPTAC-rich mixtures; this azeotropic behavior is dependent on initial monomer content, as electrostatic interactions from the cationic charges inﬂuence the system reactivity ratios. Models for MAPTAC homopolymerization and AA-MAPTAC copolymerization are developed to represent the rates of monomer conversion and comonomer composition drifts over the complete range of experimental conditions.


Introduction
Recent investigations have led to an improved understanding of the radical aqueous-phase polymerization kinetics of nonionized and ionizable water-soluble monomers. These water-soluble monomers are utilized to synthesize polyelectrolytes with tailor-made properties for a wide range of consumer and industrial applications such as flocculants in wastewater treatments, and personal care and pharmaceutical products [1][2][3][4][5]. However, monomer-solvent interactions during polymerization in aqueous phase result in complexities not encountered in organic phase, with the rate coefficients dependent on monomer concentration, ionic strength, and pH. Application of specialized techniques such as pulsed laser polymerization (PLP) used in combination with size exclusion chromatography (SEC), near infrared (NIR) spectroscopy, and electron paramagnetic resonance (EPR) spectroscopy has resulted in tremendous progress in understanding the polymerization kinetics of these water-soluble monomers [6,7].
PLP investigations have revealed that both the propagation (k p ) and termination (k t ) rate coefficients of water-soluble monomers such as methacrylic acid (MAA) [8][9][10][11] and acrylic acid (AA) [12][13][14][15] decrease with increasing monomer content in aqueous solution when the monomer is in its nonionized form, effects that have been captured in mechanistic models formulated to represent their batch and semi-batch radical homopolymerization under natural pH conditions [16][17][18]. Both rate coefficients decrease by orders of magnitude as pH is increased to ionize the monomer, due to the influence electrostatic effects resulting from the increased distance of the charged moiety from the reactive double bond in MAPTAC compared to TMAEMC (7 bond length vs 6 bond length) [33].   [27] and MAPTAC (this work) based on PLP-SEC experimentation [26].
Under continuous initiation, a decreased rate of monomer conversion was found for both monomers when the initial monomer content was increased, with MAPTAC batch homopolymerization occurring at a lowered rate compared to TMAEMC [26]. In the present study we extend the comparison to copolymerization, applying the in situ NMR technique to investigate the influence of initial monomer composition, total monomer content (wmon,0 between 0.05 and 0.40) and added salt (NaCl) on the copolymerization of AA-MAPTAC. The results are compared and contrasted to the recent AA-TMAEMC study conducted under a similar range of conditions [29] to provide insights on the influence of the cationic monomer structure on reactivity, and to determine whether the modeling strategy developed previously for TMAEMC homo-and copolymerization can also be used represent MAPTAC homopolymerization and its copolymerization with AA.
Under continuous initiation, a decreased rate of monomer conversion was found for both monomers when the initial monomer content was increased, with MAPTAC batch homopolymerization occurring at a lowered rate compared to TMAEMC [26]. In the present study we extend the comparison to copolymerization, applying the in situ NMR technique to investigate the influence of initial monomer composition, total monomer content (wmon,0 between 0.05 and 0.40) and added salt (NaCl) on the copolymerization of AA-MAPTAC. The results are compared and contrasted to the recent AA-TMAEMC study conducted under a similar range of conditions [29] to provide insights on the influence of the cationic monomer structure on reactivity, and to determine whether the modeling strategy developed previously for TMAEMC homo-and copolymerization can also be used represent MAPTAC homopolymerization and its copolymerization with AA.
Under continuous initiation, a decreased rate of monomer conversion was found for both monomers when the initial monomer content was increased, with MAPTAC batch homopolymerization occurring at a lowered rate compared to TMAEMC [26]. In the present study we extend the comparison to copolymerization, applying the in situ NMR technique to investigate the influence of initial monomer composition, total monomer content (w mon,0 between 0.05 and 0.40) and added salt (NaCl) on the copolymerization of AA-MAPTAC. The results are compared and contrasted to the recent AA-TMAEMC study conducted under a similar range of conditions [29] to provide insights on the influence of the cationic monomer structure on reactivity, and to determine whether the modeling strategy developed previously for TMAEMC homo-and copolymerization can also be used represent MAPTAC homopolymerization and its copolymerization with AA.

Materials and Methods
All in situ NMR experiments were carried out in deuterium oxide (D 2 O, 99.9%, Cambridge Isotope Laboratories Inc., Montreal, QC, Canada) as solvent, while samples for physical measurements were prepared in deionized water. Other materials used as received from Sigma Aldrich (Oakville, ON, Canada) include: 3-(methacryloylaminopropyl)trimethylammonium chloride (MAPTAC, 50 wt% solution in H 2 O), acrylic acid (AA, 99%), propionic acid (PA, ≥99.5%), sodium chloride BioXtra (NaCl, ≥99.5%, AT), sodium hydroxide reagent grade (NaOH, ≥98%, anhydrous pellets), 2,2-azobis(2-methylpropionamidine) dihydrochloride (V-50, 97%). Hydrochloric acid (HCl, reagent, ACS-PUR, Fisherbrand, Mississauga, ON, Canada) was used for adjusting solution pH. Solution preparation for bench-scale and in situ batch polymerizations as well as the subsequent data analyses were carried out using the methods developed for the TMAEMC and AA-TMAEMC studies [27,29]. Figure 2 shows the NMR spectrum at 50 • C for an equimolar mixture of AA and MAPTAC (f MAPTAC,0 = 0.5) with a monomer weight fraction (w mon,0 ) of 0.05, and 0.80 wt% V-50 initiator in solution. The decrease in intensities of peaks from AA and MAPTAC terminal double bond protons relative to the HOD peak (4.71 ppm) was followed over time and used to calculate monomer conversion profiles and the change in the MAPTAC molar fraction, as detailed in Supporting Information. Repeat experiments carried out for AA-MAPTAC copolymerization under selected conditions showed good reproducibility, as shown in Figure S1. Solution preparation for bench-scale and in situ batch polymerizations as well as the subsequent data analyses were carried out using the methods developed for the TMAEMC and AA-TMAEMC studies [27,29]. Figure 2 shows the NMR spectrum at 50 °C for an equimolar mixture of AA and MAPTAC (fMAPTAC,0 = 0.5) with a monomer weight fraction (wmon,0) of 0.05, and 0.80 wt% V-50 initiator in solution. The decrease in intensities of peaks from AA and MAPTAC terminal double bond protons relative to the HOD peak (4.71 ppm) was followed over time and used to calculate monomer conversion profiles and the change in the MAPTAC molar fraction, as detailed in Supporting Information. Repeat experiments carried out for AA-MAPTAC copolymerization under selected conditions showed good reproducibility, as shown in Figure S1. Polymer samples used for physical characterization were synthesized under the same reaction conditions with the in situ NMR method using a mixed 20 mL lab reactor, as the previous TMAEMC homopolymerization study revealed good agreement in polymer properties and reaction rates between the two techniques [27]. An Orion™ Versa Star Pro™ pH/ISE/Conductivity/Dissolved Oxygen Multiparameter Benchtop Meter was used to measure pH, and a calibrated Cannon-Fenske viscometer was utilized to obtain the dynamic viscosity of the samples.

Experimental Results
While there are a few investigations of the copolymerization of acrylamide with MAPTAC [5,34], no study on the copolymerization of nonionized AA with MAPTAC could be found. Thus, this section provides first results for AA-MAPTAC copolymerization, comparing them to the recent AA-TMAEMC copolymerization study [29]. Addition of AA to the MAPTAC system greatly lowers system pH to ~2 such that AA remains in its nonionized form, as it also was for copolymerization with TMAEMC [29]. Batch copolymerization experiments with wmon,0 of 0.10 and initial MAPTAC molar fractions (fMAPTAC,0) varied between 0.3 and 0.9 were investigated at 50 °C and 0.80 wt% V-50 in D2O using in situ NMR to measure polymerization rates and copolymer composition. As shown by the conversion profiles in Figure 3a, the rate of polymerization systematically decreases with an Polymer samples used for physical characterization were synthesized under the same reaction conditions with the in situ NMR method using a mixed 20 mL lab reactor, as the previous TMAEMC homopolymerization study revealed good agreement in polymer properties and reaction rates between the two techniques [27]. An Orion™ Versa Star Pro™ pH/ISE/Conductivity/Dissolved Oxygen Multiparameter Benchtop Meter was used to measure pH, and a calibrated Cannon-Fenske viscometer was utilized to obtain the dynamic viscosity of the samples.

Experimental Results
While there are a few investigations of the copolymerization of acrylamide with MAPTAC [5,34], no study on the copolymerization of nonionized AA with MAPTAC could be found. Thus, this section provides first results for AA-MAPTAC copolymerization, comparing them to the recent AA-TMAEMC copolymerization study [29]. Addition of AA to the MAPTAC system greatly lowers system pH to~2 such that AA remains in its nonionized form, as it also was for copolymerization with TMAEMC [29]. Batch copolymerization experiments with w mon,0 of 0.10 and initial MAPTAC molar fractions (f MAPTAC,0 ) varied between 0.3 and 0.9 were investigated at 50 • C and 0.80 wt% V-50 in D 2 O using in situ NMR to measure polymerization rates and copolymer composition. As shown by the conversion profiles in Figure 3a, the rate of polymerization systematically decreases with an increase in initial MAPTAC fraction up to 90% mol. The polymerization rate at f MAPTAC,0 = 0.9 matches that of MAPTAC homopolymerization, despite the significant difference in solution pH, 2.8 for f MAPTAC,0 = 0.9 and 6.7 for MAPTAC homopolymerization, respectively. Figure 3b plots the change in f MAPTAC , the molar fraction of MAPTAC remaining in the comonomer solution, as a function of conversion. MAPTAC is preferentially incorporated into the copolymer (as indicated by the decrease in f MAPTAC ) for the experiments conducted with f MAPTAC,0 = 0.3 and 0.5, but there is a preferential incorporation of AA into the copolymer with f MAPTAC,0 = 0.7 and 0.9. Initial rates could not be captured by the in situ NMR technique at the higher V-50 level used for AA-MAPTAC copolymerization with f MAPTAC,0 = 0.10. increase in initial MAPTAC fraction up to 90% mol. The polymerization rate at fMAPTAC,0 = 0.9 matches that of MAPTAC homopolymerization, despite the significant difference in solution pH, 2.8 for fMAPTAC,0 = 0.9 and 6.7 for MAPTAC homopolymerization, respectively. Figure 3b plots the change in fMAPTAC, the molar fraction of MAPTAC remaining in the comonomer solution, as a function of conversion. MAPTAC is preferentially incorporated into the copolymer (as indicated by the decrease in fMAPTAC) for the experiments conducted with fMAPTAC,0 = 0.3 and 0.5, but there is a preferential incorporation of AA into the copolymer with fMAPTAC,0 = 0.7 and 0.9. Initial rates could not be captured by the in situ NMR technique at the higher V-50 level used for AA-MAPTAC copolymerization with fMAPTAC,0 = 0.10. It is informative to compare these results to those obtained for TMAEMC copolymerized with AA at identical conditions, although with a lowered V-50 level; the corresponding conversion profiles are shown in Figure 3c, and TMAEMC composition drift in Figure 3d. The influence of TMAEMC content on the conversion rates is minimal for AA-TMAEMC copolymerization, except for the slightly higher rate observed with fTMAEMC,0 = 0.10. In contrast, addition of AA at any level significantly It is informative to compare these results to those obtained for TMAEMC copolymerized with AA at identical conditions, although with a lowered V-50 level; the corresponding conversion profiles are shown in Figure 3c, and TMAEMC composition drift in Figure 3d. The influence of TMAEMC content on the conversion rates is minimal for AA-TMAEMC copolymerization, except for the slightly higher rate observed with f TMAEMC,0 = 0.10. In contrast, addition of AA at any level significantly increases the rate of monomer conversion for AA-MAPTAC compared to MAPTAC homopolymerization. The azeotropic behavior (i.e., minimal drift in comonomer composition) occurs at a lowered MAPTAC molar fraction (between 0.5 and 0.7) compared to that observed for TMAEMC at f TMAEMC,0 = 0.90 as initial monomer concentration was decreased from 10 to 5 wt% [29]. The relative reactivity of MAPTAC was also reduced from that of TMAEMC when copolymerized with acrylamide [5,34,35], a result consistent with MAPTAC's lowered homopolymerization rate and k p values (Figure 1), and its copolymerization with AA in this study relative to AA-TMAEMC.
Both the counterion concentration (C Cl − ) and the charge density of the cationic polyelectrolyte formed during polymerization depends on initial MAPTAC molar fraction and total monomer concentration [36]. Thus, AA-MAPTAC copolymerizations were also carried out at w mon,0 levels of 0.05, 0.20 and 0.40 to study the combined influence of initial monomer composition and total monomer concentration (both affecting C Cl − ) on rates of polymerization and comonomer composition drifts.
The polymerization rates are grouped according to w mon,0 in Figure S2 and f MAPTAC,0 in Figure 4, which also includes a plot of the comonomer composition drifts for the complete set of experiments. Figure S2 demonstrates that there is a decrease in the polymerization rate with increased MAPTAC fraction at all monomer contents, as seen for w mon,0 = 0.10 in Figure 3a. The decrease in polymerization rate with increasing w mon,0 from 0.05 to 0.40 is more pronounced as f MAPTAC,0 increases from 0.3 ( Figure 4a) to 0.9 (Figure 4c), with the relative decrease smaller for all comonomer compositions than observed for MAPTAC homopolymerization [26]. increases the rate of monomer conversion for AA-MAPTAC compared to MAPTAC homopolymerization. The azeotropic behavior (i.e., minimal drift in comonomer composition) occurs at a lowered MAPTAC molar fraction (between 0.5 and 0.7) compared to that observed for TMAEMC at fTMAEMC,0 = 0.90 as initial monomer concentration was decreased from 10 to 5 wt% [29]. The relative reactivity of MAPTAC was also reduced from that of TMAEMC when copolymerized with acrylamide [5,34,35], a result consistent with MAPTAC's lowered homopolymerization rate and kp values (Figure 1), and its copolymerization with AA in this study relative to AA-TMAEMC. Both the counterion concentration (CCl−) and the charge density of the cationic polyelectrolyte formed during polymerization depends on initial MAPTAC molar fraction and total monomer concentration [36]. Thus, AA-MAPTAC copolymerizations were also carried out at wmon,0 levels of 0.05, 0.20 and 0.40 to study the combined influence of initial monomer composition and total monomer concentration (both affecting CCl−) on rates of polymerization and comonomer composition drifts. The polymerization rates are grouped according to wmon,0 in Figure S2 and fMAPTAC,0 in Figure 4, which also includes a plot of the comonomer composition drifts for the complete set of experiments. Figure S2 demonstrates that there is a decrease in the polymerization rate with increased MAPTAC fraction at all monomer contents, as seen for wmon,0 = 0.10 in Figure 3a. The decrease in polymerization rate with increasing wmon,0 from 0.05 to 0.40 is more pronounced as fMAPTAC,0 increases from 0.3 ( Figure  4a) to 0.9 (Figure 4c), with the relative decrease smaller for all comonomer compositions than observed for MAPTAC homopolymerization [26]. The drift in f TMAEMC for AA-TMAEMC copolymerization was observed to be more pronounced as w mon,0 increased from 0.05 to 0.40 [29], and the same is found for the AA-MAPTAC system, as seen for f MAPTAC in Figure 4d. However, the influence of w mon,0 is weaker for the MAPTAC copolymerization system, with little difference seen between the curves for w mon,0 at 0.20 and 0.40. There is no observable composition drift with f MAPTAC,0 = 0.5 and w mon,0 = 0.05, indicating that these conditions lead to an azeotropic copolymerization. The difference in azeotropic behavior of MAPTAC ( Figure 3b) and TMAEMC (Figure 3d and [29]) is likely related to the lowered k p of MAPTAC relative to TMAEMC, as well as its lowered sensitivity to C Cl − (Figure 1), as will be further illustrated in the modeling section of this study. AA-MAPTAC copolymerizations were also carried out with 1.0 mol·L −1 added NaCl and w mon,0 of 0.10 at f MAPTAC,0 = 0.3, 0.5 and 0.9, to explore how manipulating C Cl − by addition of NaCl affects copolymerization kinetics. Figure 5a-c demonstrates that there is a consistent slight decrease in conversion rates with added NaCl. As shown in Figure 5d, addition of NaCl increases the relative rate of incorporation of MAPTAC into the copolymer (i.e. lowers f MAPTAC compared to the cases without salt), although to a lesser extent compared to AA-TMAEMC copolymerization [29].  The drift in fTMAEMC for AA-TMAEMC copolymerization was observed to be more pronounced as wmon,0 increased from 0.05 to 0.40 [29], and the same is found for the AA-MAPTAC system, as seen for fMAPTAC in Figure 4d. However, the influence of wmon,0 is weaker for the MAPTAC copolymerization system, with little difference seen between the curves for wmon,0 at 0.20 and 0.40. There is no observable composition drift with fMAPTAC,0 = 0.5 and wmon,0 = 0.05, indicating that these conditions lead to an azeotropic copolymerization. The difference in azeotropic behavior of MAPTAC ( Figure 3b) and TMAEMC (Figure 3d and [29]) is likely related to the lowered kp of MAPTAC relative to TMAEMC, as well as its lowered sensitivity to CCl− (Figure 1), as will be further illustrated in the modeling section of this study.
AA-MAPTAC copolymerizations were also carried out with 1.0 mol·L −1 added NaCl and wmon,0 of 0.10 at fMAPTAC,0 = 0.3, 0.5 and 0.9, to explore how manipulating CCl− by addition of NaCl affects copolymerization kinetics. Figure 5a-c demonstrates that there is a consistent slight decrease in conversion rates with added NaCl. As shown in Figure 5d, addition of NaCl increases the relative rate of incorporation of MAPTAC into the copolymer (i.e. lowers fMAPTAC compared to the cases without salt), although to a lesser extent compared to AA-TMAEMC copolymerization [29]. As shown in Figure S3a, the pH of MAPTAC monomer in aqueous solution is close to 7 and relatively independent of w mon , while the values are below 3 for AA-MAPTAC comonomer mixtures and decrease with increasing AA content; the pH of the comonomer mixture also decreases with an increase in w mon,0 , as shown in Figure S3b. This suggests that polymerization rate is controlled by the influence of C Cl − on MAPTAC reactivity, as the AA remains in nonionized form under all reaction conditions; the same conclusion was reached for the AA-TMAEMC system [29]. To further study the specific influence of AA on kinetics, conversion profiles were compared for experiments that substituted propionic acid (PA) as a non-reactive version of AA while maintaining the total MAPTAC and acid weight fraction at 0.10; the copolymerization of MAPTAC with AA at f AA,0 = 0.7 and 0.5 was compared to MAPTAC homopolymerization with PA molar fraction of 0.7 and 0.5 (replacing AA monomer), and also MAPTAC homopolymerization at the natural pH. The faster reaction rate of the copolymerization reaction relative to the homopolymerization cases (with and without PA) shown in Figure S4 highlights that it is the relative reactivity of the comonomer that influences copolymerization rate under the nonionized conditions of AA, not the decreased pH of the solution.

Model Development
With a better understanding of the influence of the reaction environment on kinetics, the approach to represent the copolymerization behavior of cationic monomers can be generalized. To this end, the modeling framework previously implemented in the mechanistic modeling package Predici ® (version 11) for TMAEMC homopolymerization [27,37] and AA-TMAEMC copolymerization [29] is used here for the MAPTAC systems, with adjustments to account for weaker influence of counterion concentration on the kinetic rate coefficients. The mechanisms included are initiation, propagation (terminal model for copolymerization), termination (geometric mean for copolymer cross-termination), transfer to monomer and reactions associated with AA backbiting, as summarized in Table 1. The influence of C Cl − on MAPTAC propagation and termination rate coefficients are established using a combination of the current experimental results and k p values obtained from PLP-SEC studies [26].
Other treatments of rate coefficients are the same as implemented for MAPTAC and AA [16] homopolymerization models, as summarized in Table 2. The value of the AA homotermination rate coefficient is expressed as a function of the initial monomer/solvent viscosity. Viscosity measurements of AA-MAPTAC mixtures carried out at the same overall monomer contents (w mon,0 = 0.025, 0.05 and 0.10) were constant over the range of comonomer compositions ( Figure S5). Thus, there was no need to adjust the treatment of the AA k t value to account for the effect of MAPTAC on solution viscosity of the comonomer mixture in water. As also seen in the studies on TMAEMC homopolymerization and AA-TMAEMC copolymerization, the viscosity of pMAPTAC solution is significantly higher than that of the monomer solutions ( Figure S6); although the viscosity of the systems increases substantially during polymerization, the increase does not influence termination kinetics which are dominated by electrostatic interactions [27,29].

Reactions Related to Backbiting
Backbiting

Cross Termination MCR-SPR
→ D n + D m * Termination occurs by combination to yield product chains of length n + m, or by disproportionation. α and (1 − α) represent the fraction of termination by disproportionation and combination respectively, with subscripts introduced for secondary propagating radicals (s) and midchain radicals (t).    [26] w m refers to AA + MAPTAC monomer content on a polymer free basis.
The systematic decrease of rate of conversion (and thus k p /<k t > 0.5 ) observed with increased MAPTAC content [26], also observed in TMAEMC homopolymerization [27], implies that <k t > increases more strongly with C Cl − in the system than k p . As there are no independent studies of MAPTAC termination kinetics, it is assumed that k MAPTAC t follows the same functional form as used to represent k TMAEMC t , both in terms of temperature and chain-length dependencies [27,39] as well as the dependence on C Cl − . The parameters for the latter relationship were determined by fitting the conversion profiles of the MAPTAC homopolymerizations measured at 50 • C [26] to the empirical relation k t (1, 1) = a + bC Cl − + cC 2 Cl − using the parameter estimation capabilities of Predici. As shown in Figure 6, and in contrast to the TMAEMC homopolymerization system, the increase in k t (1,1) values is linear with total C Cl − in solution. The best-fit values obtained for the k t (1,1) parameters are 0.19 ± 0.04, 1.37 ± 0.13, and 0.002 ± 0.08 for a, b and c, respectively; thus, the quadratic term was not used in the k t (1,1) representation. These values indicate a higher termination of charged MAPTAC macroradicals at lower monomer contents compared to TMAEMC up to w mon,0 of 0.20 with the opposite observed at higher monomer contents. Figure 7 shows that the representation of the MAPTAC homopolymerization conversion profiles obtained for w mon,0 between 0.05 and 0.35 at 50 • C with 0.80 wt% V-50 is excellent over the full range of polymerizations.
MAPTAC content [26], also observed in TMAEMC homopolymerization [27], implies that <kt> increases more strongly with CCl− in the system than kp. As there are no independent studies of MAPTAC termination kinetics, it is assumed that kt MAPTAC follows the same functional form as used to represent kt TMAEMC , both in terms of temperature and chain-length dependencies [27,39] as well as the dependence on CCl−. The parameters for the latter relationship were determined by fitting the conversion profiles of the MAPTAC homopolymerizations measured at 50 °C [26] to the empirical relation (1,1) = + + using the parameter estimation capabilities of Predici. As shown in Figure 6, and in contrast to the TMAEMC homopolymerization system, the increase in kt (1,1) values is linear with total CCl− in solution. The best-fit values obtained for the kt(1,1) parameters are 0.19 ± 0.04, 1.37 ± 0.13, and 0.002 ± 0.08 for a, b and c, respectively; thus, the quadratic term was not used in the kt (1,1) representation. These values indicate a higher termination of charged MAPTAC macroradicals at lower monomer contents compared to TMAEMC up to wmon,0 of 0.20 with the opposite observed at higher monomer contents. Figure 7 shows that the representation of the MAPTAC homopolymerization conversion profiles obtained for wmon,0 between 0.05 and 0.35 at 50 °C with 0.80 wt% V-50 is excellent over the full range of polymerizations.  (1,1) as a function of counterion concentration from fits to TMAEMC (♦) [27] and MAPTAC () [26] homopolymerizations carried out at 50 °C and varying initial monomer content in aqueous solution. Conversion Time (s) Figure 6. Estimated variation of k t (1,1) as a function of counterion concentration from fits to TMAEMC ( ) [27] and MAPTAC ( ) [26] homopolymerizations carried out at 50 • C and varying initial monomer content in aqueous solution.
increases more strongly with CCl− in the system than kp. As there are no independent studies of MAPTAC termination kinetics, it is assumed that kt MAPTAC follows the same functional form as used to represent kt TMAEMC , both in terms of temperature and chain-length dependencies [27,39] as well as the dependence on CCl−. The parameters for the latter relationship were determined by fitting the conversion profiles of the MAPTAC homopolymerizations measured at 50 °C [26] to the empirical relation (1,1) = + + using the parameter estimation capabilities of Predici. As shown in Figure 6, and in contrast to the TMAEMC homopolymerization system, the increase in kt (1,1) values is linear with total CCl− in solution. The best-fit values obtained for the kt(1,1) parameters are 0.19 ± 0.04, 1.37 ± 0.13, and 0.002 ± 0.08 for a, b and c, respectively; thus, the quadratic term was not used in the kt (1,1) representation. These values indicate a higher termination of charged MAPTAC macroradicals at lower monomer contents compared to TMAEMC up to wmon,0 of 0.20 with the opposite observed at higher monomer contents. Figure 7 shows that the representation of the MAPTAC homopolymerization conversion profiles obtained for wmon,0 between 0.05 and 0.35 at 50 °C with 0.80 wt% V-50 is excellent over the full range of polymerizations.  (1,1) as a function of counterion concentration from fits to TMAEMC (♦) [27] and MAPTAC () [26] homopolymerizations carried out at 50 °C and varying initial monomer content in aqueous solution. The rate coefficients used to represent AA [16] and MAPTAC homopolymerization are combined for the representation of AA-MAPTAC copolymerization following the approach used to develop the model of AA-TMAEMC copolymerization [29]. The model includes the backbiting mechanisms that result from presence of AA monomer and radicals in the copolymerization system. The previous study by Wittenberg et al. [16] provides the rate coefficients for AA backbiting and the subsequent reactions involving the midchain radicals. Parameter estimation in Predici is used to estimate k cross t from the copolymer conversion profiles after first fitting the comonomer composition drifts to determine the system reactivity ratios.

Estimation of Reactivity Ratios from Comonomer Composition Drifts
As the pH of the system indicates that AA does not ionize under polymerization conditions, it can be assumed that only the addition of MAPTAC to a radical with a MAPTAC terminal unit is influenced by electrostatic effects. Thus, r AA is assumed to be constant, while r MAPTAC varies with C Cl − following the same functional form as k MAPTAC p : The experimental comonomer composition drifts obtained over a range of w mon,0 and f MAPTAC,0 levels (Figures 3b and 4d) are used to estimate reactivity ratios assuming terminal model kinetics, with the instantaneous copolymer composition given by: The These parameters were estimated by the model to fit the change in comonomer composition with conversion using the direct numerical integration (DNI) method [41]: with initial condition f MAPTAC = f MAPTAC,0 at x = 0. This technique has been previously used for the estimation of reactivity ratios for copolymerization of AA with both non-ionized AM [42] and cationic TMAEMC [29] monomers from experimental results obtained by the in situ NMR technique across a wide range of initial conditions. The only additional modification for charged systems is to express the reactivity ratio of the cationic component as a function of monomer content according to Equation (1). Estimation using the experimental results shown in Figures 3b and 4d results in estimated values of r AA = 0.36 ± 0.006 and a = 0.46 ± 0.006, with the fit compared to the experimental data in Figure 8. The faster relative incorporation of MAPTAC into the copolymer at higher w mon,0 (and thus C Cl − ) is well captured by the model. Furthermore, excellent representations are obtained (without additional fitting) for the increased relative consumption of MAPTAC with added NaCl (Figure 8c) as well as the azeotropic behavior beyond f MAPTAC,0 = 0.5. Therefore, the effects of initial monomer concentration and added NaCl on the MAPTAC composition drifts are accurately captured using the same functional representation that was developed for AA-TMAEMC copolymerization.  Table 2.

Model Fit of Conversion Profiles
Using the Predici software, the set of reaction mechanisms and rate coefficients listed in Tables  1 and 2 are combined with the expressions and values obtained for reactivity ratio estimations to develop a full kinetic model to simulate the conversion profiles obtained experimentally using the assumptions made to model AA-TMAEMC copolymerization [29]. It is informative to examine the terminal model propagation kinetics that leads to the following expression for the compositionaveraged kp cop :  Table 2.

Model Fit of Conversion Profiles
Using the Predici software, the set of reaction mechanisms and rate coefficients listed in Tables 1  and 2 are combined with the expressions and values obtained for reactivity ratio estimations to develop a full kinetic model to simulate the conversion profiles obtained experimentally using the assumptions made to model AA-TMAEMC copolymerization [29]. It is informative to examine the terminal model propagation kinetics that leads to the following expression for the composition-averaged k cop p : k MAPTAC p is two orders of magnitude lower than k AA p , and thus controls the averaged rate coefficient, as shown by the plot of log k kp MAPTAC is two orders of magnitude lower than kp AA , and thus controls the averaged rate coefficient, as shown by the plot of log kp cop against fMAPTAC in Figure 9. The kp cop values for MAPTAC gradually decrease as fMAPTAC increases in the system, consistent with experimental results and in contrast to the sharper decrease seen upon addition of TMAEMC to AA, with little change between fTMAEMC,0 of 0.1 and 1.0. Also, the terminal model treatment captures the lower influence of CCl − in MAPTAC compared to TMAEMC as seen in the smaller difference in the curves generated with wmon of 0.10 and 0.40. Various approaches to represent cross-termination of the TMAEMC and AA copolymerization system were investigated in the previous investigation [29], an important task due to the more than two order of magnitude difference between the termination rate coefficients for homopolymerization of AA and that of the charged cationic (either TMAEMC or MAPTAC) system. The best representation of the AA-TMAEMC conversion profiles was obtained using the geometric mean treatment [29], an assumption also used here to model AA-MAPTAC copolymerization: The complete set of copolymerization experiments run across different conditions (monomer concentration and composition) were used to perform parameter estimation of the single fitting parameter, with the best fit Φ value of 2.9 ± 0.2 obtained. The resulting representation of the conversion profiles is shown in Figure 10 grouped according to wmon,0. Except for slight underpredictions of conversion rates for wmon,0 = 0.05 (Figure 10a, fMAPTAC,0 = 0.3 and 0.5), the effects of both wmon,0 and fMAPTAC,0 on conversion rates are well captured by the model. A similar underprediction of conversion rates at low fTMAEMC,0 and low monomer concentrations was found in the AA-TMAEMC copolymerization study and was attributed to overprediction of the influence of AA backbiting on rate [29]. Despite this minor issue, the model, using the same assumptions and functional forms as developed for AA-TMAEMC, is able to describe the AA-MAPTAC copolymerization system over a wide range of monomer contents and the complete range of comonomer compositions, including MAPTAC homopolymerization (Figure 7). Various approaches to represent cross-termination of the TMAEMC and AA copolymerization system were investigated in the previous investigation [29], an important task due to the more than two order of magnitude difference between the termination rate coefficients for homopolymerization of AA and that of the charged cationic (either TMAEMC or MAPTAC) system. The best representation of the AA-TMAEMC conversion profiles was obtained using the geometric mean treatment [29], an assumption also used here to model AA-MAPTAC copolymerization: The complete set of copolymerization experiments run across different conditions (monomer concentration and composition) were used to perform parameter estimation of the single fitting parameter, with the best fit Φ value of 2.9 ± 0.2 obtained. The resulting representation of the conversion profiles is shown in Figure 10 grouped according to w mon,0 . Except for slight underpredictions of conversion rates for w mon,0 = 0.05 (Figure 10a, f MAPTAC,0 = 0.3 and 0.5), the effects of both w mon,0 and f MAPTAC,0 on conversion rates are well captured by the model. A similar underprediction of conversion rates at low f TMAEMC,0 and low monomer concentrations was found in the AA-TMAEMC copolymerization study and was attributed to overprediction of the influence of AA backbiting on rate [29]. Despite this minor issue, the model, using the same assumptions and functional forms as developed for AA-TMAEMC, is able to describe the AA-MAPTAC copolymerization system over a wide range of monomer contents and the complete range of comonomer compositions, including MAPTAC homopolymerization (Figure 7).  Tables 1 and 2, with the copolymer termination rate coefficient calculated using the geometric mean method (best fit Φ values = 2.9).

Conclusions
An extensive investigation of the synthesis of polyelectrolytes from the aqueous-phase copolymerization of the cationic monomer MAPTAC with nonionized AA was conducted, to broaden and generalize the knowledge of copolymerization kinetics of cationic monomers. The rates of polymerization and relative consumption rates of AA and MAPTAC as a function of conversion were measured across varying initial monomer contents (wmon,0 = 0.05 − 0.35) and initial comonomer compositions, with some experiments conducted with added salt to manipulate counterion concentration. Functional representations of MAPTAC propagation and termination rate coefficients were formulated to capture the effect of charge screening on the rate coefficients, extended to also represent the influence of CCl − on comonomer composition drifts. As found for the TMAEMC system, an increase in wmon,0 resulted in increased relative incorporation of MAPTAC into the copolymer, indicative of reduced electrostatic repulsion. The correctness of this interpretation is verified by the ability of the representation to also capture the influence of adding NaCl on the comonomer  Tables 1 and 2, with the copolymer termination rate coefficient calculated using the geometric mean method (best fit Φ values = 2.9).

Conclusions
An extensive investigation of the synthesis of polyelectrolytes from the aqueous-phase copolymerization of the cationic monomer MAPTAC with nonionized AA was conducted, to broaden and generalize the knowledge of copolymerization kinetics of cationic monomers. The rates of polymerization and relative consumption rates of AA and MAPTAC as a function of conversion were measured across varying initial monomer contents (w mon,0 = 0.05 − 0.35) and initial comonomer compositions, with some experiments conducted with added salt to manipulate counterion concentration. Functional representations of MAPTAC propagation and termination rate coefficients were formulated to capture the effect of charge screening on the rate coefficients, extended to also represent the influence of C Cl − on comonomer composition drifts. As found for the TMAEMC system, an increase in w mon,0 resulted in increased relative incorporation of MAPTAC into the copolymer, indicative of reduced electrostatic repulsion. The correctness of this interpretation is verified by the ability of the representation to also capture the influence of adding NaCl on the comonomer composition drift. Compared to TMAEMC, the influence of C Cl − on MAPTAC copolymer composition was not as strong, in agreement to the PLP-SEC study of homopropagation kinetics. A comparison to the AA-TMAEMC system demonstrates that the same generalized model structure can be used to represent copolymerizations of AA with both cationic monomers, despite the amide vs ester functionality. The terminal model of propagation was combined with a treatment of AA intramolecular chain transfer to develop a complete mechanistic representation of AA-MAPTAC copolymerization. Monomer conversions profiles were well represented by combining homopolymerization models combined with the geometric mean treatment of cross-termination. However, further experiments are required to test the ability of the model to represent conversion and copolymer composition profiles, as well as copolymer molecular weights, over a range of operating temperature.