Modeling of the Free Radical Copolymerization Kinetics of N-Butyl Acrylate, Methyl Methacrylate and 2-Ethylhexyl Acrylate Using PREDICI®

processes

Article Modeling of the Free Radical Copolymerization Kinetics of n-Butyl Acrylate, Methyl Methacrylate and 2-Ethylhexyl Acrylate Using PREDICI®

Javier A. Gómez-Reguera 1, Eduardo Vivaldo-Lima 1 , Vida A. Gabriel 2 and Marc A. Dubé 2,*

1 Departamento de Ingeniería Química, Facultad de Química, Universidad Nacional Autónoma de México, 04510 Ciudad de México, Mexico 2 Department of Chemical and Biological Engineering, Centre for Catalysis Research and Innovation, University of Ottawa, 161 Louis Pasteur Pvt., Ottawa, ON K1N 6N5, Canada * Correspondence: [email protected]; Tel.: +1-613-562-5915; Fax: +1-613-562-5174

Received: 30 April 2019; Accepted: 19 June 2019; Published: 26 June 2019

Abstract: Kinetic modeling of the bulk free radical copolymerizations of n-butyl acrylate (BA) and 2-ethylhexyl acrylate (EHA); methyl methacrylate (MMA) and EHA; as well as BA, MMA and EHA was performed using the software PREDICI®. Predicted results of conversion versus time, composition versus conversion, and molecular weight development are compared against experimental data at different feed compositions. Diffusion-controlled effects and backbiting for BA were incorporated into the model as they proved to be significant in these polymerizations. The set of estimated global parameters allows one to assess the performance of these copolymerization systems over a wide range of monomer compositions.

Keywords: modeling; polymerization kinetics; n-butyl acrylate; methyl methacrylate; 2-ethylhexyl acrylate

1. Introduction Copolymer synthesis is important in several areas due to the wide range of properties that can be achieved from the combination of different monomers. Potential applications for the materials that result from the virtually unlimited possible combinations have not been fully explored and studied [1]. Acrylate copolymers are of specific interest because of their widespread use in coatings, adhesives, resins, and many other products [1,2]. N-Butyl acrylate (BA), 2-ethyl hexyl acrylate (EHA), and methyl methacrylate (MMA) monomers are used in adhesive production and coating applications as binders in household paints [2,3] largely because their different glass transition temperatures allow for the control of adhesive and other mechanical properties [4,5]. The use of MMA and BA for polymerization processes has been studied abundantly. Their kinetic rate coefficients and the specific phenomena caused by their presence in polymerization recipes are well known and understood [6]. EHA, on the other hand, has not been studied to the same extent. Nonetheless, the propagation rate coefficient has been studied in bulk polymerizations at lower reaction temperatures (5–25 ◦C) [7]. Other studies in mini-emulsion [8] and micro-emulsion [9] polymerizations provide important comparative information on kinetic behavior. For example, the kinetic behavior of EHA was shown to be similar to that of BA in emulsion polymerizations at 75 ◦C[10]. The production of tertiary radicals resulting from transfer to polymer reactions, mostly due to backbiting reactions, was noted. The backbiting reactions led to a lower effective propagation rate coefficient. Extrapolation of the propagation rate coefficient from the bulk polymerization conditions (25 ◦C) of Beuermann et al. [7] to the emulsion polymerization conditions (75 ◦C) resulted in differing values from that estimated in the emulsion polymerization [10]. The backbiting reactions for EHA were shown to be higher than

Processes 2019, 7, 395; doi:10.3390/pr7070395 www.mdpi.com/journal/processes Processes 2019, 7, 395 2 of 33

that of BA in solution polymerizations at 70 ◦C[11]. Additional work in solution polymerization at low temperatures ( 25–10 C) showed a lower propagation rate coefficient than Beuermann et al. [7], − ◦ leading to speculation that a possible solvent effect was present [12]. The most recent pulsed laser polymerization study in bulk by Junkers et al. [13] provides propagation rate coefficients in the temperature range 20 to 80 ◦C. Thus, aside from the propagation rate coefficient, detailed kinetic studies of free radical polymerization of EHA in bulk or modeling studies of such polymerizations are rare. The present contribution uses some recent data published by some of the co-authors to begin a more comprehensive look at modeling EHA-based copolymers [14]. More recent studies involving EHA in living polymerizations underlie the need for this kinetic modeling [15–18]. Diffusion-controlled (DC) effects, particularly the auto-acceleration effect which manifests at 30–50% monomer conversions, have proven to be of great importance for reliable modeling results in free-radical polymerization (FRP) [19]. Monomers that polymerize by FRP have been classified into two types, depending on the effect that DC-termination has on the polymerization rate [20]. Class “A” monomers, of which BA is a typical example, show a notorious increase in polymerization rate during the full conversion range. On the other hand, Class “B” polymers, of which MMA is a representative case, present a “plateau region” at low conversions, followed by the previously mentioned acceleration up to high conversions [21]. EHA was not included in that classification since it became important in more recent years, and it is usually polymerized in emulsion and solution processes. Therefore, DC effects in bulk polymerization of EHA have not been reported nor fully explored. In this contribution, DC effects have been considered in the copolymerization of BA, MMA and EHA. Backbiting and the formation of tertiary radicals have been reported and measured in the polymerization of several acrylates [22,23], mainly BA, methyl acrylate (MA), EHA and dodecyl acrylate (DA). Backbiting for BA was included in our modeling approach because it has been reported to be important [24,25]. Although MMA is a widely studied monomer, there are no reports that we are aware of where backbiting is important for this monomer. In the case of EHA, there is evidence that backbiting may become important at temperatures in the range of 100 to 140 ◦C[23]. Since the experiments used in our study were conducted at 60 ◦C, it is well justified to neglect backbiting for EHA.

2. Experimental Section The polymerization conditions and experimental data used herein are from a previous experimental study of the addressed systems [14]. Even though monomers were purified to remove inhibitors by passing them through an inhibitor removal column, significant inhibition periods were observed in all polymerizations. This result may be attributed to residual oxygen which was already present in the glass ampoules where the polymerizations were carried out. Since our model did not include this phenomenon, the experimental data were slightly modified to remove the effect. The correction procedure consisted of fitting a predefined function to the conversion-time profiles, extrapolating to zero conversion and adjusting data to the new zero condition. An example of this procedure is shown in Figure1: Once the zero-conversion time was calculated with the empirical function (a logarithmic function in the example of Figure1), the times corresponding to the experimental data were corrected by subtracting the calculated induction times to the recorded experimental ones. Each experimental data has its own induction time since it is virtually impossible to determine the exact amount of residual oxygen for every ampoule. The fitting functions were logarithmic, polynomial or linear.

Adjusted time = Original time Estimated induction time (1) − The polymerizations were carried out using azobisisobutyronitrile (AIBN), at a concentration of 0.08 wt.% in glass ampoules submerged in a temperature-controlled oil bath set to 60 ◦C, so that isothermicity can be assumed. Near-isothermal conditions have been reported for ampoule reactors of high surface to volume ratios at even higher AIBN concentrations [26]. Experimental monomer feed compositions for the three systems studied in this contribution (BA/EHA, MMA/EHA, BA/EHA/MMA) are detailed in Tables1–3. The induction times ﬁtting functions are summarized in Table4. Processes 2019, 7, x FOR PEER REVIEW 3 of 36 Processes 2019, 7, 395 3 of 33

100 Original Experimental Data Adjusted Data 80 Logarithmic Fit

40 Conversion (wt. %)

0 0 2000 4000 6000 8000 10000 12000 14000 Time (s)

Figure 1. Comparison between original experimental data and adjusted data for BA-EHA copolymer Figurewith an1. initialbetween 50/50 original molar proportion. experimental data and adjusted data for BA-EHA copolymer with an

initial 50/50Table molar 1. Experimental proportion. conditions for the bulk copolymerization of BA and EHA at 60 ◦C.

Monomer Feed Monomer Feed BA EHA AIBN The polymerizations were carried out using azobisisobutyronitrile1 (AIBN),1 at a concentration1 of (BA Molar Fraction) (EHA Molar Fraction) (mol L− ) (mol L− ) (mol L− ) 0.08 wt.% in glass ampoules submerged in a temperature-controlled oil bath set to 60 °C, so that f BA = 0.3 f EHA = 0.7 1.596 3.699 0.00414 isothermicity canf BA be= 0.5assumed. Near-isothermalf EHA = 0.5 conditions 2.844 have been 2.836 reported 0.00414for ampoule reactors of high surface ftoBA volume= 0.7 ratios at evenf EHA = higher0.3 AIBN 4.284concentrations 1.837 [26]. Experimental 0.00414 monomer feed compositions for the three systems studied in this contribution (BA/EHA, MMA/EHA, BA/EHA/MMA)Table are 2. Experimental detailed in conditions Tables 1–3. for theThe bulk induction copolymerization times fitting of MMA functions/EHA at 60 are◦C. summarized in Table 4. Monomer Feed Monomer Feed MMA EHA AIBN 1 1 1 (MMA Molar Fraction) (EHA Molar Fraction) (mol L− ) (mol L− ) (mol L− )

f MMATable= 0.3 1. conditions for thef EHA bulk= 0.7 copolymerization 1.6847 of BA and 3.9407 EHA at 60 0.00414 °C. f MMA = 0.5 f EHA = 0.5 3.1794 3.1762 0.00438 Monomerf MMAFeed= 0.7 Monomerf EHA Feed= 0.3 5.1093 2.1897 0.00438 BA (mol L−1) EHA (mol L−1) (BA Molar Fraction) (EHA Molar Fraction) −1 Table 3. Experimental conditions for the bulk terpolymerization of BA, EHA and MMA atAIBN 60 ◦C. (mol L ) BA EHA Monomerf = Feed0.3 Monomer Feedf = 0.7 Monomer Feed 1.596 BA EHA 3.699 MMA 0.00414AIBN 1 1 1 1 (BA MolarfBA = Fraction) 0.5 (EHA Molar Fraction)fEHA = 0.5(MMA Molar Fraction) 2.844(mol L− ) (mol 2.836 L− ) (mol L− ) 0.00414(mol L− ) f BAfBA= =0.8 0.7 f EHA = 0.1fEHA = 0.3 f MMA = 0.1 4.284 5.456 0.683 1.837 0.677 0.00414 0.00414 f BA = 0.1 f EHA = 0.8 f MMA = 0.1 0.512 4.207 0.509 0.00419 f BA = 0.1 f EHA = 0.1 f MMA = 0.8 0.835 0.834 6.689 0.00450 Table 2. conditions for the bulk copolymerization of MMA/EHA at 60 °C. Table 4. Estimated induction times and extrapolation functions for each polymerization. Monomer Feed Monomer Feed MMA (mol L−1) EHA (mol L−1) AIBN (mol L−1) Fitting Equation Induction (MMA MolarSystem Fraction) (EHA Molarr2 Value Fraction) (x: Conversion; t: Time) Time (s) fMMA = 0.3 fEHA = 0.7 1.6847 3.9407 0.00414 fMMA =BA 0.5/50EHA fEHA 0.997 = 0.5 x = 38.457ln(t) 3.1794 288.17 3.1762 1796 0.00438 − 30BA/70EHA 0.993 x = 38.464ln(t) 278.76 1404 fMMA = 0.7 fEHA = 0.3 5.1093 − 2.1897 0.00438 70BA/30EHA 0.998 x = 40.656ln(t) 303.78 1758 − 50MMA/50EHA 0.996 x = 7E 08t2+ 0.002t 3.0858 1468 − − 30MMATable 3./70EHA conditions for the 0.996 bulk terpolymerix = 4Ezation08t2 + of0.0034t BA, EHA4.8554 and MMA at1405 60 °C. − − 70MMA/30EHA 0.996 x = 4E 08t2 + 0.0025t 4.1244 1608 Monomer Feed Monomer Feed Monomer −Feed BA− EHA MMA AIBN 10BA/10MMA/80EHA 0.969 x = 39.607ln(t) 293.78 1665 (BA Molar Fraction) (EHA Molar Fraction) (MMA Molar Fraction) −(mol L−1) (mol L−1) (mol L−1) (mol L−1) 10BA/80MMA/10EHA 0.987 x = 0.0038t 10.949 2881 fBA = 0.8 fEHA = 0.1 fMMA = 0.1 − 5.456 0.683 0.677 0.00414 80BA/10MMA/10EHA 0.899 x = 33.614ln(t) 251.67 1785 fBA = 0.1 fEHA = 0.8 fMMA = 0.1 − 0.512 4.207 0.509 0.00419 fBA = 0.1 fEHA = 0.1 fMMA = 0.8 0.835 0.834 6.689 0.00450

Processes 2019, 7, 395 4 of 33

3. Model Development Tables5–7 show the reaction steps used in PREDICI ® (version 11.Hilbert.4, Computing in Technology, CiT, Rastede, Lower Saxony, Germany), a software commonly used in polymer science and engineering [27–29]. The copolymerization schemes for the three studied systems (BA/EHA, EHA/MMA, BA/EHA/MMA) are based on conventional bulk free-radical copolymerization kinetics, but takeinto consideration the nature of each system, such as the occurrence of certain side reactions (e.g., backbiting for BA [25], the presence of additional cross reactions for the terpolymer, and building on previous knowledge on MMA and BA homo- and copolymerizations [19,20,25].

Table 5. Polymerization scheme and kinetic rate coeﬃcients used in PREDICI® for simulation of the

bulk copolymerization of BA and EHA at 60 ◦C (M1 = BA, M2 = EHA).

Value (L mol 1 s 1 Description Step in PREDICI® Parameter − − Reference Unless Otherwise Stated) Initiation Initiator decomposition I 2fI ki, f 7.4609E 6, 0.58 (s 1) [30] → • − − First Propagation for BA I + M1 P1(1) kp1 1333.8 [31] • → First Propagation for EHA I + M2 P2(1) kp2 33,395 [7] • → Propagation Self-propagation for BA P1(s) + M1 P1(s+1) kp1 1333.8 [31] • → • Self-propagation forEHA P2(s) + M2 P2(s+1) kp2 33,395 [7] • → • Cross-Propagation P1(s) + M2 P2(s+1) kp12 1333.8/r12 [14,31] • → • Cross-Propagation P2(s) + M1 P1(s+1) kp21 33,395/r21 [7,14] • → • Chain Transfer Chain Transfer to BA P1(s) + M1 P(s) + P1(1) kf1 0.1492 [31] • → • Chain Transfer to BA P2(s) + M1 P(s) + P1(1) kf21 3.5/r21 This work/[14] • → • Chain Transfer to EHA P1(s) + M2 P(s) + P2(1) kf12 0.1492/r12 [14,31] • → • Chain Transfer to EHA P2(s) + M2 P(s) + P2(1) kf2 3.5 This work • → • Intramolecular chain transfer to BA Backbitingfor BA P1(s) Q1(s) kbb 553 [32] •→ • Short-chainbranching BA Q1(s) + M1 P1(s+1) kp1tert 49.401 [33] • → • Short-chainbranching EHA Q1(s) + M2 P2(s+1) kp12tert 49.401/r12 [14,33] • → • Degradativechain transfer Q1(s) + M1 P(s) + P1(1) kf1tert 6 This work • → • Degradativechain transfer Q1(s) + M2 P(s) + P2(1) kf12tert 6/r12 This work/[14] • → • Termination Bycombination P1(s) + P1(r) P(s+r) ktc11 1.2259E6 [31] • •→ - P1(s) + P2(r) P(s+r) ktc21 √ktc11 ktc22 [31]/This work • •→ · - P2(s) + P2(r) P(s+r) ktc22 2.5E8 This work • •→ Bydisproportionation P1(s) + P1(r) P(s) + P(r) ktd11 8.5815E5 [31] • •→ - P1(s) + P2(r) P(s) + P(r) ktd21 √ktd11 ktd22 [31]/This work • •→ · - P2(s) + P2(r) P(s) + P(r) ktd22 2.5E8 This work • •→ Termination of BA tertiary radicals Bycombination Q1(s) + P1(r) P(s+r) ktc11 1.2259E6 [31] • •→ - Q1(s) + P2(r) P(s+r) ktc21 √ktc11 ktc22 [31]/This work • •→ · - Q1(s) + Q1(r) P(s+r) ktc11 1.2259E6 [31] • •→ Bydisproportionation Q1(s) + P1(r) P(s) + P(r) ktd11 8.5815E5 [31] • •→ - Q1(s) + P2(r) P(s) + P(r) ktd21 √ktd11 ktd22 [31]/This work • •→ · - Q1(s) + Q1(r) P(s) + P(r) ktd11 8.5815E5 [31] • •→ Processes 2019, 7, 395 5 of 33

Table 6. Polymerization scheme and kinetic rate coeﬃcients used in PREDICI® for simulation of the

bulk copolymerization of MMA and EHA at 60 ◦C (M1 = MMA, M2 = EHA).

Value (L mol 1 s 1, Description Step in PREDICI® Parameters − − Reference Unless Otherwise Stated) Initiation Initiator decomposition I 2fI ki, f 7.4609E 6, 5.8E 1 (s 1) [30] → • − − − First Propagation for MMA I + M1 P1(1) kp1 683.24 [31] • → First Propagation for EHA I + M2 P2(1) kp2 33,395 [7] • → Propagation Self-propagation for MMA P1(s) + M1 P1(s+1) kp1 683.24 [31] • → • Self-propagation for EHA P2(s) + M2 P2(s+1) kp2 33,395 [7] • → • Cross-Propagation P1(s) + M2 P2(s+1) kp12 683.24/r12 [14,31] • → • Cross-Propagation P2(s) + M1 P1(s+1) kp21 33,395/r21 [7,14] • → • Chain Transfer Chain Transfer to MMA P1(s) + M1 P(s) + P1(1) kf1 1.9321E 2 [31] • → • − Chain Transfer to MMA P2(s) + M1 P(s) + P1(1) kf21 3.5/r21 This work/[14] • → • Chain Transfer to EHA P1(s) + M2 P(s) + P2(1) kf12 1.9321E 2/r12 [14,31] • → • − Chain Transfer to EHA P2(s) + M2 P(s) + P2(1) kf2 3.5 This work • → • Termination By combination P1(s) + P1(r) P(s+r) ktc11 1.859E7 [31] • •→ - P1(s) + P2(r) P(s+r) ktc21 √ktc11 ktc22 [31]/This work • •→ · - P2(s) + P2(r) P(s+r) ktc22 2.5E8 This work • •→ By disproportionation P1(s) + P1(r) P(s) + P(r) ktd11 1.5382E7 This work • •→ - P1(s) + P2(r) P(s) + P(r) ktd21 √ktd11 ktd22 [31]/This work • •→ · - P2(s) + P2(r) P(s) + P(r) ktd22 2.5E8 This work • •→

Table 7. Polymerization scheme and kinetic rate coeﬃcients used in PREDICI® for simulation of the

bulk copolymerization of BA, EHA and MMA at 60 ◦C (M1 = BA, M2 = EHA, M3 = MMA).

Value (L mol 1 s 1, Description Step in PREDICI® Parameters − − Reference Unless Otherwise Stated) Initiation Initiator decomposition I 2fI ki, f 7.4609E 6, 0.58 (s 1) [30] → • − − First Propagation for BA I + M1 P1(1) kp1 1333.8 [31] • → First Propagation for EHA I + M2 P2(1) kp2 33,395 [7] • → First Propagation for MMA I + M3 P3(1) kp3 683.24 [31] • → Propagation Self-propagation for BA P1(s) + M1 P1(s+1) kp1 1333.8 [31] • → • Self-propagation for EHA P2(s) + M2 P2(s+1) kp2 33,395 [7] • → • Self-propagation forMMA P3(s) + M3 P3(s+1) kp3 683.24 [31] • → • Cross-Propagation P1(s) + M2 P2(s+1) kp12 1333.8/r12 [14,31] • → • Cross-Propagation P1(s) + M3 P3(s+1) kp13 1333.8/r13 [31,34] • → • Cross-Propagation P2(s) + M1 P1(s+1) kp21 33,395/r21 [7,14] • → • Cross-Propagation P2(s) + M3 P3(s+1) kp23 33,395/r23 [7,14] • → • Cross-Propagation P3(s) + M1 P1(s+1) kp31 683.24/r31 [31,34] • → • Cross-Propagation P3(s) + M2 P2(s+1) kp32 683.24/r32 [31,34] • → • Chain Transfer Chain Transfer to BA P1(s) + M1 P(s) + P1(1) kf1 0.1492 [31] • → • Chain Transfer to BA P2(s) + M1 P(s) + P1(1) kf21 3.5/r21 This work/[14] • → • Chain Transfer to BA P3(s) + M1 P(s) + P1(1) kf31 1.9321E 2/r31 [31,34] • → • − Chain Transfer to EHA P1(s) + M2 P(s) + P2(1) kf12 0.1492/r12 [14,31] • → • Chain Transfer to EHA P2(s) + M2 P(s) + P2(1) kf2 4.5 This work • → • Chain Transfer to EHA P3(s) + M2 P(s) + P2(1) kf32 1.9321E 2/r32 [14,31] • → • − Chain Transfer to MMA P1(s) + M3 P(s) + P3(1) kf13 0.1492/r13 [31,34] • → • Chain Transfer to MMA P2(s) + M3 P(s) + P3(1) kf23 3.5/r23 This work/[31] • → • Chain Transfer to MMA P3(s) + M3 P(s) + P3(1) kf3 1.9321E 2 [31] • → • − Intramolecular chain transfer of BA Backbiting for BA P1(s) Q1(s) kbb 553 [32] •→ • Short-chain branching BA Q1(s) + M1 P1(s+1) kp1tert 49.401 [33] • → • Short-chain branching EHA Q1(s) + M2 P2(s+1) kp12tert 49.401/r12 [14,33] • → • Short-chain branching EHA Q1(s) + M3 P3(s+1) kp13tert 49.401/r13 [33,34] • → • Degradative chain transfer Q1(s) + M1 P(s) + P1(1) kf1tert 6 This work • → • Degradative chain transfer Q1(s) + M2 P(s) + P2(1) kf12tert 6/r12 This work/[14] • → • Degradative chain transfer Q1(s) + M3 P(s) + P3(1) kf13tert 6/r13 This work/[34] • → • Processes 2019, 7, 395 6 of 33

Table 7. Cont.

Value (L mol 1 s 1, Description Step in PREDICI® Parameters − − Reference Unless Otherwise Stated) Termination By combination P1(s) + P1(r) P(s+r) ktc11 1.2259E6 [31] • •→ - P1(s) + P2(r) P(s+r) ktc21 √ktc11 ktc22 [31]/This work • •→ · - P1(s) + P3(r) P(s+r) ktc31 √ktc11 ktc33 [31] • •→ · - P2(s) + P2(r) P(s+r) ktc22 2.5E8 This work • •→ - P2(s) + P3(r) P(s+r) ktc32 √ktc22 ktc33 This work/[31] • •→ · - P3(s) + P3(r) P(s+r) ktc33 1.859E7 [31] • •→ By disproportionation P1(s) + P1(r) P(s) + P(r) ktd11 8.5815E5 [31] • •→ - P1(s) + P2(r) P(s) + P(r) ktd21 √ktd11 ktd22 [31]/This work • •→ · - P1(s) + P3(r) P(s) + P(r) ktd31 √ktd11 ktd33 [31] • •→ · - P2(s) + P2(r) P(s) + P(r) ktd22 2.5E8 This work • •→ - P2(s) + P3(r) P(s) + P(r) ktd32 √ktd22 ktd33 This work/[31] • •→ · - P3(s) + P3(r) P(s) + P(r) ktd33 1.5382E7 [31] • •→ Termination of BA tertiary radicals By combination Q1(s) + P1(r) P(s+r) ktc11 1.2259E6 [31] • •→ - Q1(s) + P2(r) P(s+r) ktc21 √ktc11 ktc22 This work • •→ · - Q1(s) + P3(r) P(s+r) ktc31 √ktc11 ktc33 [31] • •→ · - Q2(s) + Q2(r) P(s+r) ktc22 2.5E8 This work • •→ - Q2(s) + Q3(r) P(s+r) ktc32 √ktc22 ktc33 This work/[31] • •→ · - Q3(s) + Q3(r) P(s+r) ktc33 1.859E7 [31] • •→ By disproportionation Q1(s) + P1(r) P(s) + P(r) ktd11 8.5815E5 [31] • •→ - Q1(s) + P2(r) P(s) + P(r) ktd21 √ktd11 ktd22 [31]/This work • •→ · - Q1(s) + P3(r) P(s) + P(r) ktd31 √ktd11 ktd33 [31] • •→ · - Q2(s) + P2(r) P(s) + P(r) ktd22 2.5E8 This work • •→ - Q2(s) + P3(r) P(s) + P(r) ktd32 √ktd22 ktd33 This work/[31] • •→ · - Q3(s) + Q3(r) P(s) + P(r) ktd33 1.5382E7 [31] • •→

3.1. Initiation The initiation reaction involves two steps, as shown in Equations (2) and (3).

I 2fI• (2) →

I• + M P (1) (3) i→ i As with many other initiators, AIBN decomposes into two initiator primary free radicals (I•). The Arrhenius expression for the initiator decomposition kinetic rate coefficient is given by Equation (4) [6,31]. 1 4 k (s− ) = 1.07E14exp( 1.515 10 /T) (4) i − × The initiator primary free radicals react with either of the monomers producing polymer radicals of each type and size 1. Since not all initiator molecules produce primary free radicals due to side reactions, an efficiency factor (f ) for initiation is required. One explanation for the use of f is the so-called “cage effect”, which allows us to avoid considering side reactions such as termination between primary free radicals [35]. A value of f = 0.58 was used [30].

3.2. Propagation Since the polymerization scheme is based on the terminal model [1], four propagation and four-chain transfer to monomer reactions are involved. The self-propagation kinetic rate coeﬃcients for MMA and BA are shown in Equations (5) and (6) [6,31]. The cross-propagation kinetic rate coeﬃcients are expressed in terms of reactivity ratios, as detailed in Section 3.4.

1 1 3 kp (L mol− s− ) = 4.917E5exp( 4.353 10 /T) (5) MMA − × 1 1 3 kp (L mol− s− ) = 2.767E9exp( 9.630 10 /T) (6) BA − × For the EHA, the self-propagation kinetic rate coeﬃcient and pulsed laser polymerization (PLP) experimental data [7] were analyzed and adjusted to an Arrhenius relationship, leading to Equation (7). Processes 2019, 7, 395 7 of 33

The value for kpEHA used in our simulations (Table5) is consistent (order of magnitude agreement) with values reported in other studies for such parameters at the same temperature [13].

1 1 3 kp (L mol− s− ) = 9.00E6exp( 3.694 10 /T) (7) EHA − × 3.3. Chain Transfer Kinetic rate coeﬃcients for chain transfer to MMA and BA are given by Equations (8) and (9) [6,31].

1 1 3 kf (L mol− s− ) = 1.55E4exp( 7.475 10 /T) (8) MMA − × 1 1 3 kf (L mol− s− ) = 1.55E3exp( 7.475 10 /T) (9) BA − × Unfortunately, there was no reliable information in the literature for chain transfer to EHA kinetic 1 1 rate coeﬃcients in bulk polymerization. A value of kf BA = 3.5 L mol− s− at 60 ◦C was estimated in this contribution. The cross-chain transfer kinetic rate coeﬃcients were obtained using reactivity ratios as in the case of propagation.

3.4. Cross-Propagation and Cross-Chain Transfer Reactivity ratios, deﬁned by Equation (10), were used to calculate the cross-parameters.

kp1 k f 1 kp2 k f 2 r12 = = ; r21 = = (10) kp12 k f 12 kp21 k f 21

The reactivity ratios for BA/EHA, MMA/EHA and BA/EHA/MMA used in our calculations are summarized in Equations (11)–(15), and were taken from the composition results obtained from Gabriel and Dubé [14] (at 60 ◦C) using a computational package developed in MATLAB [36]. BA/EHA

kBA kp1 k f 1 kEHA kp2 k f 2 r12 = = = = 0.994; r21 = = = = 1.621 (11) kEHA kp12 k f 12 kBA kp21 k f 21

MMA/EHA

kMMA kp1 k f 1 kEHA kp2 k f 2 r12 = = = = 1.496; r21 = = = 0.315 (12) kEHA kp12 k f 12 kMMA kp21 k f 21

BA/EHA/MMA

kBA kp1 k f 1 kEHA kp2 k f 2 r12 = = = = 0.994; r21 = = = = 1.621 (13) kEHA kp12 k f 12 kBA kp21 k f 21

kBA kp1 k f 1 kMMA kp2 k f 2 r13 = = = = 2.022; r31 = = = = 0.343 (14) kMMA kp13 k f 13 kBA kp31 k f 31

kMMA kp1 k f 1 kEHA kp2 k f 2 r32 = = = = 1.496; r23 = = = = 0.315 (15) kEHA kp32 k f 32 kMMA kp31 k f 31 The reliability of reactivity ratios is fundamental for the accurate prediction of copolymer composition. As stated in the Mayo-Lewis equation, reactivity ratios are a measure of a monomer’s preference and tendency to bond [35]. In the case of cross chain transfer to monomer reactions, since the reacting species are the same as in cross propagation, it was assumed that the cross chain transfer reactions can be calculated using the same reactivity ratios. This assumption does not have a strong theoretical background, but it allowed us to reduce the number of estimated parameters. Processes 2019, 7, 395 8 of 33

3.5. Backbiting of BA and Reactions with Mid-Chain Tertiary Radicals Intramolecular chain transfer to polymer in BA polymerization is known to be important [37,38]. Mid-chain tertiary polymer radicals are formed from backbiting. They are quite stable and can propagate, transfer and terminate as any other polymer radicals. Backbiting and tertiary radical propagation, as well as their corresponding kinetic rate coeﬃcients, are given by Equations (16)–(19) [25]. Tertiary radical formation: kbb P1(s) Q1(s) (16) • → • 1 1 6 3 k (L mol− s− ) = 3.87 10 exp( 2.299 10 /T) (17) bb × − × Tertiary radical propagation: kp1tert Q1(s) + M1 P1(s + 1) (18) • → • 1 1 k (L mol− s− ) = 59.9exp( 64.2/T) (19) p1tert − Cross-propagation kinetic rate coeﬃcients were also obtained from BA/EHA and BA/EHA/MMA reactivity ratios, which are given by Equations (20) and (21), respectively.

kBA kp1 k f 1 kp1tert k f 1tert r12 = = = = 0.994; kp12tert = ; k f12tert = (20) kEHA kp12 k f 12 r12 r12

kBA kp1 k f 1 kp1tert k f 1tert r13 = = = = 2.022; kp13tert = ; k f13tert = (21) kMMA kp13 k f 13 r13 r13

There were no estimates of k f 1tert reported in the literature. It was therefore estimated in this study (k f 1tert = 6).

3.6. Termination Termination by combination and disproportionation kinetic rate coeﬃcients for BA and MMA copolymerizations were also taken from WATPOLY, as shown in Equations (22) and (23) [6,31].

1 1 8 2 ktc + k (L mol− min− ) = 4.68 10 exp( 8.73 10 /T) (22) MMA tdMMA × − × 1 1 9 2 ktc + k (L mol− min− ) = 5.88 10 exp( 7.01 10 /T) (23) BA tdBA × − × The ratio between both termination kinetic rate coeﬃcients for MMA and BA copolymerizations is given by Equation (24) [14].

k k tcMMA = 0.8275; tcBA = 0.7000 (24) ktdMMA ktdBA

Termination by combination and disproportionation kinetic rate coeﬃcients for EHA were estimated in this work, obtaining the values shown in Equations (25) and (26).

ktc + ktd = 5 108 (25) EHA EHA × ktc = ktd = 2.5 108 (26) EHA EHA × The missing “cross-termination” constants were calculated using the geometric average for every speciﬁc case, as shown in Equation (27). q ktij = ktiktj (27) Processes 2019, 7, 395 9 of 33

In the case of bimolecular radical termination including mid chain radicals, Q1(s) , it was assumed • that ktBA is independent of radical type [25,39]. More experimental or theoretical studies on termination of tertiary radicals to provide a sounder estimation of ktBA would be useful. Until then, the assumption of independence of radical type is needed to simplify our parameter estimation approach. Therefore, the termination kinetic rate coeﬃcients for such cases are the same as “cross-termination”.

3.7. Diffusion-Controlled Effects When implementing our model and trying to fit our experimental data, it was evident that diffusion-controlled (DC) effects were important. Since MMA was one of the monomers present in our reacting systems, this was not surprising [21,30]. All the reactions involving polymer molecules were assumed to be diffusion-controlled. DC effects were addressed using free-volume theory [25,40], as shown in Equation (28), where β is a free volume parameter andvf is fractional free volume, defined by Equation (29). Subscript “0” stands for initial conditions. T and Tgi are the reaction temperature and glass transition temperature for component i (monomer or polymer of each of the possible species, BA, EHA or MMA); αi is the expansion coefficient for species i, Vi is the volume of species i, and Vtis total volume.    o   1 1  k = k exp β  (28) i i   o  − υ f − υ f

# o f components X h i Vi υ f = 0.025 + αi T Tgi (29) − Vt i=1 The free volume parameters used in our calculations for the three copolymerization systems are summarized in Tables8 and9. As observed in Table9, di fferent β values were required, even for similar polymer molecules, in order to explain the behavior observed experimentally over a wide range of (binary) copolymer and terpolymer compositions. This is not surprising if one considers that the material properties of the reacting mixture, such as viscosity [41], depend on molecular weight development but also on copolymer content in the reactive mass and quite likely, copolymer composition. It is observed in Table9 that β for kinetic rate coefficients with the same name differs from system to system. This is a consequence of the nomenclature used for each system. For instance, kp1 refers to BA in the case of copolymerization of BA and EHA, whereas the same parameter is related to MMA in the copolymerization of MMA and EHA. Also, DC effects in both propagation and termination reactions were required for the copolymerization of BA and EHA whereas only diffusion-controlled termination was needed in the case of copolymerization of MMA and EHA. High values of β result in an earlier and more pronounced decrease of the corresponding kinetic rate coefficient, while kinetic rate coefficients where no DC effects are considered remain constant throughout the polymerization. In general, the presence of MMA causes higher polymerization rates. This behavior can be captured by the model by increasing β for termination or decreasing the corresponding β parameter for the propagation reaction. In other words, the βt/βp ratio for MMA must be greater (compared to the other monomers) for accurate predictions. The same trend applies for the “cross” kinetic rate coefficients when MMA is involved; βt/βp is also fundamental to tune the “onset” of the auto-acceleration effect. As discussed in the following section, it turned out that the only way to capture the behavior of the experimental data (particularly the conversion versus time profiles) was to include DC effects in most of the reactions and using different β values for similar polymer molecules.

Table 8. Free volume parameters used in our calculations.

Species TgM(K) TgP(K) αM αP BA 185 218 0.001 0.00048 EHA 167 223 0.001 0.00048 MMA 167 387 0.001 0.00048 Processes 2019, 7, 395 10 of 33

Table 9. β parameters used in our simulations; β = 0 implies that DC eﬀects were neglected in such cases.

Rate Coefficients Where DC Rate Coefficients Where DC System β Value Effects Were Considered Effects Were Neglected (β = 0) kp1 0.25 ki kp2 0.25 f kp12 0.25 kf1 kp21 0.5 kf21 kbb 5 kf12 BA-EHA ktc11 4 kf2 ktc21 0.5 kp1tert ktc22 2 kp12tert ktd11 4 kf1tert ktd21 0.5 kf12tert ktd22 2 —- kp2 0.5 ki kf1 1 f kf12 1 kp1, kp12 MMA-EHA kf2 1 kp2, kp21 kf21 1 ktc22, ktd22 ktc11 4 ktc21, ktd21 ktd11 4 —- kp1 1 ki kp12 1 f kp2 1.5 kf1 kp21 1 kf21 kp23 2 kf31 kp13 2 kf12 kp3 2 kf2 kp31 2 kf32 kp32 2 kf13 kbb 2.5 kf23 kf3 2 kbb BA-EHA-MMA kf13tert 2 kp1tert ktc11 6 kp12tert ktc21 2 kp13tert ktc22 2 kf1tert ktc31 10 kf12tert ktc32 10 —- ktc33 10 —- ktd11 6 —- ktd21 2 —- ktd22 2 —- ktd31 10 —- ktd32 10 —- ktd33 10 —-

3.8. Parameter Estimation Strategy Most of the kinetic rate coefficients used in our study came from reports available in the literature. The kinetic rate coefficients estimated by us are summarized in Table 10. The other estimated parameters are the βs indicated in Table9. As explained earlier, all the parameters estimated in this work are related to EHA. The estimation procedure consisted of a combination between a careful literature review, detailed parameter sensitivity analyses and the use of educated guesses (trial and error guided from the information gained from the literature review and the parameter sensitivity analyses). An example of the parameter estimation strategy is shown in AppendixA. In the case of bimolecular radical termination between polymer radicals terminated in EHA monomer units, it was assumed that ktcEHA = ktdEHA. This is of course an approximation, but it is sufficient since, in many instances, in our model it is the total termination kinetic rate coefficient (kt = ktd + ktc) which is needed. Processes 2019, 7, 395 11 of 33

Table 10. Estimated kinetic rate coeﬃcients in this work.

Parameter Value Description ktc 2.5 108 EHA termination by combination EHA × ktd 2.5 108 EHA termination by disproportionation EHA × kfEHA 45 Chain transfer to EHA kftertEHA 60 EHA intramolecular chain transfer to BA tertiary radicals kftertMMA 35 MMA intramolecular chain transfer to BA tertiary radicals

Regarding chain transfer to a monomer, our parameter sensitivity analyses showed that kfEHA, kftertEHA and kftertMMA had a negligible effect on the polymerization rate, but that reaction is important for molecular weight development. Thus, initial guesses for these parameters were assigned order of magnitude estimates, based on values of that parameter for other monomers, and they were fine-tuned using molecular weight development experimental data. Our parameter estimation strategy included addressing the BA/EHA and MMA/EHA copolymer systems first, and then proceeding with the terpolymer case. Although we could get fairly good (order of magnitude) overall predictions of the polymerization rate, copolymer composition and molecular weight development for most of the cases studied by adjusting the intrinsic kinetic rate coefficients, the reproduction of unusual behavior (e.g., opposite trends in the conversion-time profiles when changing feed composition, overlap of profiles in certain instances, for system MMA/EHA) was possible only if strong DC effects (large β values) were used in some cases.

4. Results and Discussion

4.1. Conversion versus Time Proﬁles

ProcessesAs observed2019, 7, x FOR in PEER Figures REVIEW2–4, good agreement between experimental data and calculated12 proﬁlesof 36 of conversion versus time was obtained for all the conditions studied of the three copolymerization systemssystems analyzed analyzed inin thisthis contribution.contribution. It It is is also also observed observed from from the the experimental experimental profiles proﬁles for forthe thethree three reacting systems that the polymerization rate is higher, and higher conversions are reached when reacting systems that the polymerization rate is higher, and higher conversions are reached when EHA EHA is present in a major proportion. This is consistent with the fact that kp for EHA turned out to is present in a major proportion. This is consistent with the fact that kp for EHA turned out to be the be the highest for the three monomers considered in this study. highest for the three monomers considered in this study.

100

40 Model predictions

Conversion (wt. Conversion%) (wt. BA/EHA = 30/70 20 BA/EHA = 50/50 BA/EHA = 70/30

0 0 2000 4000 6000 8000 10000 12000 14000 Time (s)

FigureFigure 2. 2.Comparison of experimental of experimentaland calculated and profiles calculated of total proﬁles monomer of totalconversion monomer versus conversion time at 60 versus°C timefor copolymerization at 60 ◦C for copolymerization of BA and EHA of at BA different and EHA initial at di concentrationsﬀerent initial concentrationsof BA. of BA.

100

Model Predictions

80 MMA/EHA = 30/70 MMA/EHA = 50/50 MMA/EHA = 70/30

40 Conversion %) (wt.

0 0 5000 10000 15000 20000 25000 Time (s)

Figure 3. of experimental and calculated profiles of total monomer conversion versus time at 60 °C for copolymerization of MMA and EHA at different initial concentrations of MMA.

Processes 2019, 7, x FOR PEER REVIEW 12 of 36

systems analyzed in this contribution. It is also observed from the experimental profiles for the three reacting systems that the polymerization rate is higher, and higher conversions are reached when EHA is present in a major proportion. This is consistent with the fact that kp for EHA turned out to be the highest for the three monomers considered in this study.

100

40 Model predictions

Conversion (wt. Conversion%) (wt. BA/EHA = 30/70 20 BA/EHA = 50/50 BA/EHA = 70/30

0 0 2000 4000 6000 8000 10000 12000 14000 Time (s)

Figure 2. of experimental and calculated profiles of total monomer conversion versus time at 60 °C Processesfor2019 copolymerization, 7, 395 of BA and EHA at different initial concentrations of BA. 12 of 33

100

Model Predictions

80 MMA/EHA = 30/70 MMA/EHA = 50/50 MMA/EHA = 70/30

40 Conversion %) (wt.

0 0 5000 10000 15000 20000 25000 Time (s)

ProcessesFigureFigure 2019 3. ,3. 7Comparison, ofx FOR experimental PEER REVIEW of experimental and calculated and profiles calculated of tota proﬁlesl monomer of total conversion monomer versus conversion time at 60 versus °C13 of 36 timefor atcopolymerization 60 ◦C for copolymerization of MMA andof EHA MMA at different and EHA initial at di concentrationsﬀerent initial concentrations of MMA. of MMA.

100

Conversion (wt. %) (wt. Conversion Model Predictions

20 BA/EHA/MMA = 10/10/80 BA/EHA/MMA = 80/10/10 BA/EHA/MMA = 10/80/10 0 0 5000 10000 15000 20000 25000 Time (s)

FigureFigure 4. 4.Comparison of experimental of experimental and calculated and profiles calculated of tota proﬁlesl monomer of total conversion monomer versus conversion time at 60 versus °C timefor atcopolymerization 60 ◦C for copolymerization of BA, EHA and of BA, MMA EHA at anddifferent MMA initial at di concentrations.ﬀerent initial concentrations.

If oneIf one compares compares the the BA BA/EHA/EHAand and MMAMMA/EHA/EHA copolymerization systems, systems, as as shown shown in inFigures Figures 2 2 andand3, it3, isit is observed observed that that BABA polymerizespolymerizes faster faster than than MMA MMA and and EHA EHA polymerizes polymerizes faster faster than thanBA. As BA. Asobserved observed in in the the previous previous section, section, the the kp kpvalues values are areconsistent consistent with with these these observations. observations. InIn the the case case of copolymerization of copolymerization of BA of and BA EHA, and it EHA, is observed it is inobserved Figure2 inthat Figure higher 2 polymerization that higher ratespolymerization are obtained rates when are the obtained amount when of EHA the is amount increased. of EHA However, is increased. that increase However, in polymerization that increase in rate is onlypolymerization marginal since rate theis only profiles marginal are close since to the one prof another.iles are This close result to one can another. be explained This result by comparing can be kpexplained and kt for by both comparing monomers. kp and Although kt for both kp formonome EHA isrs. almost Although one kp order for EHA of magnitude is almost higherone order than of kp formagnitude BA, kt for higher EHA isthan two kp orders for BA, of magnitudekt for EHA higheris two thanorders kt of for magnitude BA. Since higher the polymerization than kt for BA. rate is proportionalSince the polymerization to kp and inversely rate is proportional proportional to tokp ktand0.5, inversely their effects proportional almost cancel to kt out.0.5, their However, effects it is strangealmost cancel to observe out. fromHowever, the experimental it is strange datato observe that the from final the conversion experimental for case data 50BA that/50EHA the final was higherconversion than in for case case 70BA 50BA/50EHA/30EHA, whenwas higher the trend than wasin case exactly 70BA/30EHA, the opposite when at the very trend low was conversions. exactly the opposite at very low conversions. This effect could not be captured by the model without DC effects. Even when single β parameters for similar polymer molecules were used, the observed trend could not be replicated. The only way to capture this effect was to use different β and kt values for EHA (see summary of β parameters in Table 9). Additional experimental data (i.e., extending the experimental study to include other monomer compositions for each co- and terpolymer system) as well as EHA homopolymerization studies would be required to more accurately estimate the model parameters. Another possible explanation for the extra fine tuning of EHA parameters is the fact that we estimated induction times and corrected the experimental sampling times accordingly. Similar behavior as the one described above is observed in Figure 3 for the copolymerization of MMA and EHA. Although more noticeable differences in the polymerization rate are observed in this case, these differences are still moderate. As observed for the BA/EHA system, in the case of copolymerization of MMA and EHA, the experimental data show higher conversion values in the very low conversion region and lower final conversions in the medium to high conversion regions for system 70MMA/30EHA, compared to system 50MMA/50EHA. The same explanation as in BA/EHA applies for MMA/EHA. Another important aspect observed in Figure 3 for the copolymerization of MMA and EHA is the presence of strong DCEs, particularly the auto-acceleration effect (typical “S”-shaped conversion versus time profiles, both experimental and calculated). This auto-acceleration effect (diffusion-controlled termination) is manifested in all the copolymerizations where MMA is present, even the case with the lowest content of MMA (30MMA/70EHA). As a matter of fact, this case with the lowest MMA content is the one wherethe

Processes 2019, 7, 395 13 of 33

This effect could not be captured by the model without DC effects. Even when single β parameters for similar polymer molecules were used, the observed trend could not be replicated. The only way to capture this effect was to use different β and kt values for EHA (see summary of β parameters in Table9). Additional experimental data (i.e., extending the experimental study to include other monomer compositions for each co- and terpolymer system) as well as EHA homopolymerization studies would be required to more accurately estimate the model parameters. Another possible explanation for the extra fine tuning of EHA parameters is the fact that we estimated induction times and corrected the experimental sampling times accordingly. Similar behavior as the one described above is observed in Figure3 for the copolymerization of MMA and EHA. Although more noticeable differences in the polymerization rate are observed in this case, these differences are still moderate. As observed for the BA/EHA system, in the case of copolymerization of MMA and EHA, the experimental data show higher conversion values in the very low conversion region and lower final conversions in the medium to high conversion regions for system 70MMA/30EHA, compared to system 50MMA/50EHA. The same explanation as in BA/EHA applies for MMA/EHA. Another important aspect observed in Figure3 for the copolymerization of MMA and EHA is the presence of strong DCEs, particularly the auto-acceleration effect (typical “S”-shaped conversion versus time profiles, both experimental and calculated). This auto-acceleration effect (diffusion-controlled termination) is manifested in all the copolymerizations where MMA is present, even the case with the lowest content of MMA (30MMA/70EHA). As a matter of fact, this case with the lowest MMA content is the one wherethe strongest diffusion-controlled termination is observed (see Figure3). This behavior can be explained by the large difference in propagation kinetic rate coefficients between the two monomers (kp for EHA is 2 orders in magnitude higher than kp for MMA), so that a decrease in termination will have a more noticeable effect in the species with the faster propagation. Accurate estimation of ktin our MMA-containing copolymerizations is quite important, given their lower propagation rates and the nature of the experimental results obtained, which show very similar conversions reached at quite different initial monomer concentrations. As shown in Figure4 for the terpolymerization of BA, EHA and MMA, more noticeable differences in the polymerization rate expressed as conversion versus time at different comonomer initial compositions are observed in this case, compared to the binary copolymerization systems analyzed earlier. These differences can again be explained in terms of different propagation rates among the three monomers (with EHA being the fastest monomer), but also by the monomer initial concentrations, where one monomer dominates the process. A close look at the β values reported in Table9 shows again that DC effects and specifically diffusion-controlled termination are relevant for MMA, since higher β values are obtained for kinetic rate coefficients involving MMA and MMA cross interactions. As a matter of fact, the best overall fit to experimental data was obtained when system 10BA/10EHA/80MMA was used to estimate free volume parameters for MMA (estimation of β values for MMA related reactions).

4.2. Evolution of Kinetic Rate Coeﬃcients Due to DC Eﬀects It was stated above that DC effects were needed to reproduce the observed experimental phenomena, particularly the effect of feed comonomer composition on the polymerization rate and molecular weight development. In order to understand such behavior, it is useful to visualize how the kinetic rate coefficients that are DC evolve over time. The evolution with conversion of kp and kt for system 30BA/70EHA, kp and kt for system 70EHA/30MMA, kp for system 10BA/10EHA/80MMA, kp for system 80BA/10EHA/10MMA, kp for system 10BA/80EHA/10MMA, kt for system 10BA/10EHA/80MMA, kt for system 80BA/10EHA/10MMA, and kt for system 10BA/80EHA/10MMA, are shown in Figures5–12, respectively. Processes 2019, 7, x FOR PEER REVIEW 14 of 36

strongest diffusion-controlled termination is observed (see Figure 3). This behavior can be explained by the large difference in propagation kinetic rate coefficients between the two monomers (kp for EHA is 2 orders in magnitude higher than kp for MMA), so that a decrease in termination will have a more noticeable effect in the species with the faster propagation. Accurate estimation of ktin our MMA-containing copolymerizations is quite important, given their lower propagation rates and the nature of the experimental results obtained, which show very similar conversions reached at quite different initial monomer concentrations. As shown in Figure 4 for the terpolymerization of BA, EHA and MMA, more noticeable differences in the polymerization rate expressed as conversion versus time at different comonomer initial compositions are observed in this case, compared to the binary copolymerization systems analyzed earlier. These differences can again be explained in terms of different propagation rates among the three monomers (with EHA being the fastest monomer), but also by the monomer initial concentrations, where one monomer dominates the process. A close look at the β values reported in Table 9 shows again that DC effects and specifically diffusion-controlled termination are relevant for MMA, since higher β values are obtained for kinetic rate coefficients involving MMA and MMA cross interactions. As a matter of fact, the best overall fit to experimental data was obtained when system 10BA/10EHA/80MMA was used to estimate free volume parameters for MMA (estimation of β values for MMA related reactions).

4.2. Evolution of Kinetic Rate Coefficients Due to DC Effects It was stated above that DC effects were needed to reproduce the observed experimental phenomena, particularly the effect of feed comonomer composition on the polymerization rate and molecular weight development. In order to understand such behavior, it is useful to visualize how the kinetic rate coefficients that are DC evolve over time. The evolution with conversion of kp and kt for system 30BA/70EHA, kp and kt for system 70EHA/30MMA, kp for system 10BA/10EHA/80MMA, kp for system 80BA/10EHA/10MMA, kp for system 10BA/80EHA/10MMA, kt for system 10BA/10EHA/80MMA, kt for system 80BA/10EHA/10MMA, and kt for system Processes 2019, 7, 395 14 of 33 10BA/80EHA/10MMA, are shown in Figures 5–12, respectively.

109

108

kp1 107 kp2 kp12 kp21

6 ktc11 10 ktc22 ktc21 ktd11 105 ktd22 ktd21 30BA/70EHA (L*mol/s)

104

103

0 20406080100 Conversion (wt. %)

Processes 2019, 7, x FOR PEER REVIEW 15 of 36 Figure 5. Evolution with conversion of propagation and termination kinetic rate coeﬃcients for system ProcessesFigure 2019, 7,5. x FORwith PEER conversion REVIEW of propagation and termination kinetic rate coefficients for system15 of 36 30BA/70EHA due to DCeﬀects. 30BA/70EHA due to DCeffects.8 10 108

107 107 kp2 ktc11 kp2 ktd11 ktc11 106 ktd11 106

105 105 70EHA/30MMA (L*mol/s) 70EHA/30MMA (L*mol/s)

104 104

0 20406080100 0 20406080100Conversion (wt. %) Conversion (wt. %) Figure 6. with conversion of propagation and termination kinetic rate coefficients for system Figure 6. Evolution with conversion of propagation and termination kinetic rate coeﬃcients for system Figure70EHA/30MMA 6. with conversion due to DCeffects. of propagation and termination kinetic rate coefficients for system 70EHA70EHA/30MMA/30MMA due due to to DCe DCeffects.ﬀects.

106 106

105 kp1 5 10 kp2 kp1 kp12 4 kp2 10 kp21 kp12 4 kp3 10 kp21 kp31 kp3 3 kp32 10 kp31 kp13 3 kp32 10 kp23 kp13 102 kp23 102

10BA/10EHA/80MMA (L*mol/s) 10BA/10EHA/80MMA 101

100 0 0 20406080100 10 0 20406080100Conversion (wt. %) Conversion (wt. %) Figure 7. Evolution with conversion of propagation kinetic rate coeﬃcient for system 10BA/10EHA/ Figure 7. with conversion of propagation kinetic rate coefficient for system 10BA/10EHA/80MMA 80MMAFiguredue to due DCeffects.7. with to DCe conversion ﬀects. of propagation kinetic rate coefficient for system 10BA/10EHA/80MMA due to DCeffects.

Processes 2019, 7, x FOR PEER REVIEW 16 of 36 Processes 2019, 7, 395 15 of 33 Processes 2019, 7, x FOR PEER REVIEW 16 of 36

106 106

105 105 kp1 kp2 kp1 4 kp12 10 kp2 kp21 104 kp12 kp3 kp21 kp31 3 kp3 10 kp32 kp31 3 kp13 10 kp32 kp23 kp13 2 10 kp23 102 80BA/10EHA/10MMA (L*mol/s) 101 80BA/10EHA/10MMA (L*mol/s) 101

100 0 0 20406080100 10 0 20406080100Conversion (wt. %) Conversion (wt. %) Figure 8. with conversion of propagation kinetic rate coefficient for system 80BA/10EHA/10MMA Figure 8. Evolution with conversion of propagation kinetic rate coeﬃcient for system 80BA/10EHA/ dueFigure to DCeffects.8. with conversion of propagation kinetic rate coefficient for system 80BA/10EHA/10MMA 10MMA due to DCeﬀects. due to DCeffects.

106 106

105 105 kp1 kp2 kp1 4 kp12 10 kp2 kp21 4 kp12 10 kp3 kp21 kp31 kp3 3 kp32 10 kp31 kp13 103 kp32 kp23 kp13 102 kp23 102 10BA/80EHA/10MMA (L*mol/s) 101 10BA/80EHA/10MMA (L*mol/s) 101

100 0 0 20 40 60 80 100 10 0 20Conversion 40 (wt. 60 %) 80 100 Conversion (wt. %) Figure 9. Evolution with conversion of propagation kinetic rate coeﬃcient for system 10BA/80EHA/ ProcessesFigure 2019 9., 7 ,with x FOR conversion PEER REVIEW of propagation kinetic rate coefficient for system 10BA/80EHA/10MMA17 of 36 10MMAdueFigure to due DCeffects.9. with to DCe conversion ﬀects. of propagation kinetic rate coefficient for system 10BA/80EHA/10MMA

due to DCeffects. 109

108

107 ktc11 ktc22 6 ktc21 10 ktd11

5 ktd22 10 ktd21 ktc33 104 ktc31 ktc32 103 ktd33 ktd31 ktd32 102 10BA/10EHA/80MMA (L*mol/s) 10BA/10EHA/80MMA

101

100 020406080100 Conversion (wt. %)

FigureFigure 10. Evolution10. with conversion with conversion of termination of termination kinetic rate kinetic coefficient rate coe forﬃ systemcient for 10BA/10EHA/80MMA system 10BA/10EHA / 80MMAdue to due DCeffects. to DCe ﬀects.

109

108 ktc11 7 10 ktc22 ktc21 106 ktd11 ktd22

5 ktd21 10 ktc33 ktc31 104 ktc32 ktd33 3 ktd31 10 ktd32

102 80BA/10EHA/10MMA (L*mol/s)

101

100 0 20 40 60 80 100 Conversion (wt. %)

Figure 11. with conversion of termination kinetic rate coefficient for system80BA/10EHA/10MMA due to DCeffects.

Processes 2019, 7, x FOR PEER REVIEW 17 of 36

109

108

107 ktc11 ktc22 6 ktc21 10 ktd11

5 ktd22 10 ktd21 ktc33 104 ktc31 ktc32 103 ktd33 ktd31 ktd32 102 10BA/10EHA/80MMA (L*mol/s) 10BA/10EHA/80MMA

101

100 020406080100 Conversion (wt. %)

Figure 10. with conversion of termination kinetic rate coefficient for system 10BA/10EHA/80MMA Processes 2019, 7, 395 16 of 33 due to DCeffects.

109

108 ktc11 7 10 ktc22 ktc21 106 ktd11 ktd22

5 ktd21 10 ktc33 ktc31 104 ktc32 ktd33 3 ktd31 10 ktd32

102 80BA/10EHA/10MMA (L*mol/s)

101

100 0 20 40 60 80 100 Conversion (wt. %)

Figure 11. ProcessesFigure 2019, 711.Evolution, x FORwith PEER conversion with REVIEW conversion of termination of termination kinetic rate kinetic coefficient rate coe forﬃ system80BA/10EHA/10MMAcient for system80BA/10EHA18 of 36/ 10MMAdue to due DCeffects. to DCe ﬀects.

109

108 ktc11 ktc22 7 10 ktc21 ktd11 106 ktd22 ktd21 ktc33 105 ktc31 ktc32 104 ktd33 ktd31 103 ktd32

102 10BA/80EHA/80MMA (L*mol/s)

101

100 0 20406080100 Conversion (wt. %)

Figure 12. Evolution with conversion of termination kinetic rate coefficient for system10BA/80EHA/ Figure 12. with conversion of termination kinetic rate coefficient for system10BA/80EHA/10MMA 10MMA due to DCeffects. due to DCeffects. It is observed in Figure5 that kt and kp for system 30BA /70EHA decrease very slowly as conversion It is observed in Figure 5 that kt and kp for system 30BA/70EHA decrease very slowly as increases, remaining almost constant during the polymerization. This behavior is typical of systems conversion increases, remaining almost constant during the polymerization. This behavior is typical withof verysystems weak with DC very eff ectsweak and DC agree effects well and withagree the we conversionll with the conversion versus time versus profile time shown profile in shown Figure 2 forin the Figure same 2 copolymerizationfor the same copolymerization system. Figure system.6, onFigure the 6, other on the hand, other shows hand, shows the behavior the behavior of system of 70EHAsystem/30MMA, 70EHA/30MMA, which displays which displays strong DCe strongffects. DCeffects. DCtermination DCtermination is particularly is particularly strong strong for systemfor 70EHAsystem/30MMA, 70EHA/30MMA, since kp decreases since kp moderately, decreases moderately, but kt decreases but kt by decreases several orders by several of magnitude. orders of This behaviormagnitude. agrees This well behavior with the agrees conversion well with versus the timeconversion profile versus shown time in Figure profile3. shown Similar in behavior Figure 3. was obtainedSimilar for behavior systems was 50BA obtained/50EHA, for 70BA system/30EHA,s 50BA/50EHA, 50EHA/50MMA 70BA/30EHA, and 30EHA 50EHA/50MMA/70MMA, although and results30EHA/70MMA, for those systems although were results not included for those here syst dueems to space were considerations. not included here due to space considerations.It is also observed in Figures5 and6, and in Table9, that system BA /EHA required estimating more βItparameters is also observed than in system Figures MMA 5 and/EHA. 6, and Thisin Table is related 9, that tosystem the natureBA/EHA of required each system. estimating System MMAmore/EHA β parameters is governed than by system the strong MMA/EHA. auto-acceleration This is related eff ectto the (DC nature termination) of each system. of MMA System whereas MMA/EHA is governed by the strong auto-acceleration effect (DC termination) of MMA whereas system BA/EHA required to estimate β parameters for several reactions involving polymer molecules system BA/EHA required to estimate β parameters for several reactions involving polymer in order to reproduce the experimental behavior of that system. molecules in order to reproduce the experimental behavior of that system. The terpolymer systems showed more complex behavior than the binary copolymerization ones. The terpolymer systems showed more complex behavior than the binary copolymerization As observedones. As observed in Figures in7 Figures–12 for the7–12 terpolymer for the terpolymer systems, systems, each case each shows case remarkablyshows remarkably di fferent different behavior. behavior. For instance, as observed in Figure 7, kp3, kp31, kp32, kp13 and kp23 for system 10BA/10EHA/80MMA decay much faster than in the other terpolymer systems shown in Figures 8 and 9. The common feature in those cases is that the involved kinetic rate coefficients are related to MMA and cross propagation reactions with MMA, a monomer with strong DC effects. Similar decaying behavior for the termination kinetic rate coefficients, particularly kt33, kt32 and kt31 (both combination and disproportionation), all of them related to MMA, once more, can be observed in Figures 10–12. This DC dependency of kp and kt for the terpolymer systems agrees well with the conversion versus time profiles observed in Figure 4.

4.3. Copolymer Composition versus Conversion Profiles As expected, accurate predictions of copolymer composition versus conversion strongly depend on the reactivity ratios used in the calculations. Given the excellent agreement between experimental data and calculated profiles of copolymer composition versus conversion at several initial compositions for the copolymerizations of BA and EHA, as well as MMA/EHA (see Figures 13 and 14), it is fair to say that the estimates of reactivity ratios for such systems [14] were very good.

Processes 2019, 7, x FOR PEER REVIEW 19 of 36

Processes 2019, 7, x FOR PEER REVIEW 19 of 36 However, as observed in Figures 15–17, the same does not hold entirely true for the Processes 2019, 7, 395 17 of 33 terpolymerizationHowever, as cases. observed The agreementin Figures between15–17, theexperimental same does data not andhold model entirely predictions true for ofthe copolymerterpolymerization composition cases. versus The conversion agreement for between terpolymerizations experimental 80BA/10EHA/10MMA data and model predictions (see Figure of For17)copolymer instance,and 10BA/80EHA/10MMA ascomposition observed in versus Figure conversion 7,(see kp3, kp31,Figure for kp32, terpolymerizations16) kp13is andvery kp23 good, 80BA/10EHA/10MMA for systembut it 10BA is poor/10EHA (seefor/80MMA Figurecase decay10BA/10EHA/80MMA,17) muchand faster10BA/80EHA/10MMA than in as the observed other terpolymer (seein Figure Figure systems 15. 16)On shownais firstvery inthought, Figuresgood, 8one andbut may9.it The isargue commonpoor that for featurebetter case reactivity ratio estimates may be needed, particularly for MMA-BA interactions. It is observed in in10BA/10EHA/80MMA, those cases is that the involved as observed kinetic in rateFigure coe ffi15.cients On a are first related thought, to MMA one andmay crossargue propagation that better Figure 15 that the largest deviation between experimental data and model predictions occurs when reactionsreactivity with ratio MMA, estimates a monomer may withbe needed, strong DCparticular effects.ly Similar for MMA-BA decaying interactions. behavior for It the is terminationobserved in initial MMA content is highest. Given the lower kp value for MMA, compared to the corresponding kineticFigure rate 15 coethatffi thecients, largest particularly deviation kt33, between kt32 and experi kt31mental (both data combination and model and predictions disproportionation), occurs when all values for BA and EHA, it may be argued that MMA incorporates slower into the copolymer and its ofinitial them relatedMMA content to MMA, is oncehighest. more, Given can bethe observed lower kp invalue Figures for MMA,10–12. compared This DC dependency to the corresponding of kp and composition could be slightly lower, as observed in the experimental profile (solid squares) of Figure kt valuesfor the for terpolymer BA and EHA, systems it may agrees be wellargued with that the MMA conversion incorporates versus timeslower profiles into the observed copolymer in Figure and its4. 15.composition The adjustments could carriedbe slightly out lower,to eliminate as observed inductio inn the times experimental from experimental profile (solid data squares) may be an of issueFigure 4.3.here.15. Copolymer The However, adjustments Composition on a closer carried versuslook, out it Conversionto is eliminate inferred Profiles frominductio Tablen times 9, that from although experimental the reactivity data may ratios be atan zero issue conversion coincide with the available ones, the fact that different free volume parameters (β values) here.As However, expected, on accurate a closer predictions look, it is inferred of copolymer from Ta compositionble 9, that although versus conversion the reactivity strongly ratios depend at zero forconversion the cross-propagation coincide with terms the available were used, ones, that the caus fact esthat the different ratios to free change volume (according parameters to Equation (β values) on the reactivity ratios0 used in the calculations. Given the excellent agreement between experimental (29);for ratiosthe cross-propagation of kpi remain unchanged, terms were but used, since that the caus βs arees the different, ratios tothe change actual (according ratios do not to Equationremain dataconstant). and calculated profiles of copolymer composition versus conversion at several initial compositions (29); ratios of kpi0 remain unchanged, but since the βs are different, the actual ratios do not remain for the copolymerizations of BA and EHA, as well as MMA/EHA (see Figures 13 and 14), it is fair to constant). say that the estimates of reactivity ratios for such systems [14] were very good. Exp Data BA=30% Exp Data BA=50% 1.0 Exp Exp Data Data BA=70% BA=30% Model BA=30% 0.9 Exp Data BA=50% 1.0 Model Exp BA=50%Data BA=70% Model BA=70% 0.8 Model BA=30% 0.9 Model BA=50% 0.7 Model BA=70% 0.8

0.6 0.7

0.5 0.6

0.4 0.5

0.3 0.4

0.2

BA CompositionBA (mol fraction) 0.3

0.1 0.2 BA CompositionBA (mol fraction) 0.0 0.1 0 20406080100 0.0 Conversion (wt. %) 0 20406080100 Conversion (wt. %) Figure 13. of experimental data and calculated profiles of BA composition versus conversion in the Figure 13. Comparison of experimental data and calculated proﬁles of BA composition versus copolymerization of BA and EHA at different initial compositions of BA. conversionFigure 13. in of the experimental copolymerization data and of BAcalculated and EHA profiles at diﬀ oferent BA initialcomposition compositions versus ofconversion BA. in the copolymerization of BA and EHA at different initial compositions of BA. 1.0

0.91.0

0.80.9

0.70.8

0.60.7

0.50.6

0.40.5

0.30.4 Exp Data MMA=30% Exp Data MMA=50% 0.20.3 Exp Exp Data Data MMA=70% MMA=30%

MMA Composition (mol fraction) (mol Composition MMA Model Exp MMA=30%Data MMA=50% Model MMA=50% 0.10.2 Exp Data MMA=70% Model MMA=70%

MMA Composition (mol fraction) (mol Composition MMA Model MMA=30% 0.00.1 Model MMA=50% 0 Model 20 MMA=70% 40 60 80 100 0.0 Conversion (wt. %) 0 20 40 60 80 100 Figure 14. Comparison of experimental dataConversion and calculated (wt. %) profiles of MMA composition versus Figure 14. of experimental data and calculated profiles of MMA composition versus conversion in conversionthe copolymerization in the copolymerization of MMA and EHA of MMA at different and EHA initial at di compositionsfferent initial of compositions MMA. of MMA. Figure 14. of experimental data and calculated profiles of MMA composition versus conversion in However,the copolymerization as observed of MMA in Figures and EHA 15 at– different17, the initial same compositions does not of hold MMA. entirely true for the terpolymerization cases. The agreement between experimental data and model predictions of copolymer Processes 2019, 7, 395 18 of 33 composition versus conversion for terpolymerizations 80BA/10EHA/10MMA (see Figure 17) and 10BA/80EHA/10MMA (see Figure 16) is very good, but it is poor for case 10BA/10EHA/80MMA, as observed in Figure 15. On a first thought, one may argue that better reactivity ratio estimates may be needed, particularly for MMA-BA interactions. It is observed in Figure 15 that the largest deviation between experimental data and model predictions occurs when initial MMA content is highest. Given the lower kp value for MMA, compared to the corresponding values for BA and EHA, it may be argued that MMA incorporates slower into the copolymer and its composition could be slightly lower, as observed in the experimental profile (solid squares) of Figure 15. The adjustments carried out to eliminate induction times from experimental data may be an issue here. However, on a closer look, it is inferred from Table9, that although the reactivity ratios at zero conversion coincide with the available ones, the fact that different free volume parameters (β values) for the cross-propagation terms were Processes 2019, 7, x FOR PEER REVIEW 0 20 of 36 used, that causes the ratios to change (according to Equation (29); ratios of kpi remain unchanged, but Processes 2019, 7, x FOR PEER REVIEW 20 of 36 since the βs are different, the actual ratios do not remain constant). 1.0 1.0

0.8 0.8

0.6 Model fMMA 0.6 Model Model fBA fMMA Model Model fEHA fBA Exp Model Data fEHA fMMA 0.4 Exp Exp Data Data fBA fMMA 0.4 Exp Exp Data Data fEHA fBA Exp Data fEHA

0.2 0.2 Composition (10BA-10EHA-80MMA) Composition Composition (10BA-10EHA-80MMA) Composition 0.0 0.0 0 20406080100 0 20406080100Conversion (wt. %) Conversion (wt. %) Figure 15. of experimental data and calculated profiles of copolymer composition for each monomer Figure 15. Comparison of experimental data and calculated proﬁles of copolymer composition for each speciesFigure versus15. of experimental conversion for data a system and calculated with 10BA/10EHA/80MMA profiles of copolymer initial composition concentrations. for each monomer monomerspecies species versus versusconversion conversion for a system for awith system 10BA/10EHA/80MMA with 10BA/10EHA initial/80MMA concentrations. initial concentrations.

1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 Model fMMA 0.6 Model Model fBA fMMA 0.6 Model Model fEHA fBA 0.5 Exp Model Data fEHA fMMA 0.5 Exp Exp Data Data fBA fMMA 0.4 Exp Exp Data Data fEHA fBA 0.4 Exp Data fEHA 0.3 0.3 0.2 0.2

Composition (10BA-80EHA-10MMA) 0.1

Composition (10BA-80EHA-10MMA) 0.1 0.0 0.0 0 20 40 60 80 100 0 20Conversion 40 (wt. 60 %) 80 100 Conversion (wt. %) FigureFigure 16. 16.Comparison of experimental of experimental data and calculated data and profiles calculated of copolymer proﬁles of composition copolymer for composition each monomer for each monomerspeciesFigure species versus16. of experimental conversion versus conversion for data a system and for calculated awith system 10BA/80EHA/10MMA profiles with 10BA of copolymer/80EHA initial/ 10MMAcomposition concentrations. initial for concentrations.each monomer species versus conversion for a system with 10BA/80EHA/10MMA initial concentrations.

Processes 2019, 7, 395 19 of 33 Processes 2019, 7, x FOR PEER REVIEW 21 of 36

1.0

0.9

0.8

0.7 Exp Data fMMA 0.6 Exp Data fBA Exp Data fEHA 0.5 Model fMMA Model fBA 0.4 Model fEHA

0.3

0.2

Composition (80BA-10EHA-10MMA) Composition 0.1

0.0 Processes 2019, 7, x FOR PEER0 REVIEW 102030405060708090100 22 of 36 Conversion (wt. %) It is observed in Figures 24 and 25 that our model is able to produce order-of-magnitude Figure 17. Comparison of experimental data and calculated profiles of copolymer composition for each estimationsFigure of17. final of experimental molecular data weights and calculated (Mn and profilesMw at highof copolymer conversions), composition but their for each time monomer (conversion) monomer species versus conversion for a system with 80BA/10EHA/10MMA initial concentrations. evolutionspecies is versuscaptured conversion very poorly.for a system As witha matter 80BA/10EHA/10MMA of fact, the observed initial concentrations. trends are opposite to the calculated profiles: The observed trend is that molecular weight increases within the same order of 4.4. Molecular4.4. Molecular Weight Weight versus versus Conversion Conversion Profiles Profiles magnitude values, and the calculated profiles show huge decreasing trends (of about two orders of magnitude).As observedAs observed Although in Figures in thisFigures 18is –not23 18–23,, uncommon the agreementthe agreemen in polymerization betweent between experimental experimentalmodeling data studies, data and these calculatedand resultscalculated profiles may of molarbeprofiles attributed mass of molar (M ton ,Mthe massw estimatedand (M Ð)n, M versus wk andEHA chain Ð) conversion versus transfer conversion for to themonomer copolymerizations for the kinetic copolymerizations rate constant of BA and of coefficients. BA EHA and as EHA well As as MMAstatedas andwell in EHAasSection MMA is reasonably3.8 and of EHA this iscontribution, good. reasonably Order good.k ofEHA magnitude was Order used of magnitudeto agreement fit BA/EHA agreement was and obtained EHA/MMA was obtained in both copolymers in cases. both It is observedmolarcases. masses. fromIt is observed the Since experimental crosslinkingfrom the experimental data and of Mgelationn,M dataw occurredand of M Ðn,that M inw changesandthe ÐBA/E that onHA changes initial system,monomer on ourinitial estimation monomer composition of dok notEHAcomposition havewas likely significant do overpredicted. not ehaveffects significant on More the molecular effectscomplete on weights experithe molecularmental and dispersity studiesweights withand achieved. dispersitya focus The on achieved. samecapturing trend The the was predictedearlysame stages trend by the ofwas model the predicted polymerization at low by conversions, the model and gathering at but low significant conv moreersions, data deviations but on significantmolecular are observed deviationsweight atdevelopment high are observed conversions are at high conversions (calculated Mw-and consequently Ð– being much higher than the corresponding (calculatedneeded toM improvew-and consequently our model. Ð– being much higher than the corresponding experimental data). experimental data).

As observed in Figures 21 and 22 for copolymerization of MMA BA/EHA= and 0.3/0.7 EHA, the model predicts 6 BA/EHA= 0.5/0.5 a slightly faster increase3.5x10 of molar mass as the polymerization proceeds, namely, the calculated BA/EHA= 0.7/0.3 profiles of both Mn and Mw slightly overpredict the molar masses Model achieved. BA/EHA= It0.3/0.7 is also observed in 6 these figures that although3x10 the experimental profiles seem to overlap, Model BA/EHA= the effect 0.5/0.5 of increasing the Model BA/EHA= 0.7/0.3 amount of MMA in the initial monomer mixture is captured in the opposite direction by the model. 6 The experimental 2.5x10profiles show that molecular weight increases as the amount of MMA is decreased, whereas the calculated profiles show the opposite trend. To solve this issue, a more complete experimental2x10 6study for the homopolymerization of EHA would be useful, since the chain transfer to the EHA kinetic rate coefficient had to be estimated in this study. However, the increasing trend of molar mass as time elapses is nicely captured by the model. This increasing

trend was possible1.5x10 to reproduce6 by including diffusion-controlled dependence to the chain transfer to monomer reaction. Otherwise, flat profiles were obtained. In the case of copolymerization of BA and EHA, although order of magnitude predictions of

Mn and Mw wereNumber AverageMolar(g/mol) Mass obtained, which was partly possible by the inclusion of DC effects in the chain transfer to monomer and backbiting reactions; the evolution with conversion of the profiles was 106 captured very poorly. This020406080100 may be explained by the fact that crosslinking and gelation occurred in the experimental system and these phenomenaConversion were (wt. not%) included in the model. Furthermore, as noted in the original data reference [14], because the samples were filtered prior to analysis, only FigureFigure 18. 18.Comparison of experimental of experimental data and datacalculated and calculated profiles of proﬁles number of average number molar average mass molar versus mass the soluble part of the polymer could be analyzed. This situation also explains the large discrepancy versusconversion conversion in the in copolymerization the copolymerization of BA ofand BA EHA and at EHA different at di ﬀinitialerent compositions initial compositions of BA. of BA. between experimental and calculated values of Ð observed in Figure 20.

6x106

5x106

4x106

3x106

2x106

BA/EHA= 0.3/0.7 BA/EHA= 0.5/0.5 BA/EHA= 0.7/0.3 Model BA/EHA= 0.3/0.7 Mass Average Molar Mass (g/mol) Model BA/EHA= 0.5/0.5 Model BA/EHA= 0.7/0.3 106 0 20406080100 Conversion (wt. %)

Figure 19. of experimental data and calculated profiles of weight average molar mass versus conversion in the copolymerization of BA and EHA at different initial compositions of BA.

Processes 2019, 7, x FOR PEER REVIEW 22 of 36

It is observed in Figures 24 and 25 that our model is able to produce order-of-magnitude estimations of final molecular weights (Mn and Mw at high conversions), but their time (conversion) evolution is captured very poorly. As a matter of fact, the observed trends are opposite to the calculated profiles: The observed trend is that molecular weight increases within the same order of magnitude values, and the calculated profiles show huge decreasing trends (of about two orders of magnitude). Although this is not uncommon in polymerization modeling studies, these results may be attributed to the estimated kEHA chain transfer to monomer kinetic rate constant coefficients. As stated in Section 3.8 of this contribution, kEHA was used to fit BA/EHA and EHA/MMA copolymers molar masses. Since crosslinking and gelation occurred in the BA/EHA system, our estimation of kEHA was likely overpredicted. More complete experimental studies with a focus on capturing the early stages of the polymerization and gathering more data on molecular weight development are needed to improve our model.

BA/EHA= 0.3/0.7 6 BA/EHA= 0.5/0.5 3.5x10 BA/EHA= 0.7/0.3 Model BA/EHA= 0.3/0.7 3x106 Model BA/EHA= 0.5/0.5 Model BA/EHA= 0.7/0.3

2.5x106

2x106

1.5x106 Number AverageMolar(g/mol) Mass

106 020406080100 Conversion (wt. %)

Processes 2019Figure, 7, 39518. of experimental data and calculated profiles of number average molar mass versus 20 of 33 conversion in the copolymerization of BA and EHA at different initial compositions of BA.

6x106

5x106

4x106

3x106

2x106

BA/EHA= 0.3/0.7 BA/EHA= 0.5/0.5 BA/EHA= 0.7/0.3 Model BA/EHA= 0.3/0.7 Mass Average Molar Mass (g/mol) Model BA/EHA= 0.5/0.5 Model BA/EHA= 0.7/0.3 106 0 20406080100 Conversion (wt. %)

Figure 19. Comparison of experimental data and calculated proﬁles of weight average molar mass ProcessesFigure 2019, 719., x FORof experimentalPEER REVIEW data and calculated profiles of weight average molar mass versus23 of 36 versusconversion conversion in the in copolymerization the copolymerization of BA ofand BA EHA and at EHA different at di ﬀinitialerent compositions initial compositions of BA. of BA.

2.5 BA/EHA= 0.5/0.5 BA/EHA= 0.7/0.3 BA/EHA= 0.3/0.7 Model BA/EHA= 0.5/0.5 Model BA/EHA= 0.7/0.3 Model BA/EHA= 0.3/0.7

2.0 Dispersity 1.5

1.0 0 20406080100 Conversion (wt. %)

FigureFigure 20. 20.Comparison of experimental of experimental data and data calculated and calculated profiles proﬁles of Ð ofversus Ð versus conversion conversion in inthe the copolymerizationcopolymerization of of BA BA and and EHA EHA at at di differentﬀerent initialinitial compositionscompositions of of BA. BA.

As observed in Figures 21 and 22 for copolymerization of MMA and EHA, the model predicts a slightly faster increase of molar mass as the polymerization proceeds, namely, the calculated profiles of both Mn and Mw slightly overpredict the molar masses achieved. It is also observed in these figures that although the experimental profiles seem to overlap, the effect of increasing the amount of MMA in the initial monomer mixture is captured in the opposite direction by the model. The experimental profiles show that molecular weight increases as the amount of MMA is decreased, whereas the calculated profiles show106 the opposite trend. To solve this issue, a more complete experimental study for the homopolymerization of EHA would be useful, since the chain transfer to the EHA kinetic rate coefficient had to be estimated in this study. However, the increasing trend of molar mass as time elapses is nicely captured by the model. This increasing trend was possible to reproduce by including diffusion-controlled dependence to the chain transfer to MMA/EHA= monomer 0.3/0.7 reaction. Otherwise, flat profiles MMA/EHA= 0.5/0.5 were obtained. MMA/EHA= 0.7/0.3 Model MMA/EHA= 0.3/0.7 Model MMA/EHA= 0.5/0.5 Number Average Molar Mass (g/mol) Mass Molar Average Number Model MMA/EHA= 0.7/0.3

105 0 102030405060708090100 Conversion (wt. %)

Figure 21. of experimental data and calculated profiles of number average molar mass versus conversion in the copolymerization of MMA and EHA at different initial compositions of MMA.

Processes 2019, 7, x FOR PEER REVIEW 23 of 36

2.5 BA/EHA= 0.5/0.5 BA/EHA= 0.7/0.3 BA/EHA= 0.3/0.7 Model BA/EHA= 0.5/0.5 Model BA/EHA= 0.7/0.3 Model BA/EHA= 0.3/0.7

2.0

Processes 2019, 7, 395 Dispersity 21 of 33 1.5

In the case of copolymerization of BA and EHA, although order of magnitude predictions of Mn and Mw were obtained, which was partly possible by the inclusion of DC effects in the chain transfer to monomer and backbiting reactions; the evolution with conversion of the profiles was captured very 1.0 poorly. This may be explained by0 the fact 20406080100 that crosslinking and gelation occurred in the experimental system and these phenomena were not includedConversion in the (wt. model. %) Furthermore, as noted in the original data reference [14], because the samples were filtered prior to analysis, only the soluble part of the polymerFigure could 20. be analyzed.of experimental This situationdata and alsocalculated explains profiles the large of discrepancyÐ versus conversion between experimentalin the and calculatedcopolymerization values ofof ÐBA observed and EHA at in different Figure 20initial. compositions of BA.

106

MMA/EHA= 0.3/0.7 MMA/EHA= 0.5/0.5 MMA/EHA= 0.7/0.3 Model MMA/EHA= 0.3/0.7 Model MMA/EHA= 0.5/0.5 Number Average Molar Mass (g/mol) Mass Molar Average Number Model MMA/EHA= 0.7/0.3

105 0 102030405060708090100 Conversion (wt. %)

Figure 21. Comparison of experimental data and calculated proﬁles of number average molar mass ProcessesFigure 2019 , 21.7, x ofFOR experimental PEER REVIEW data and calculated profiles of number average molar mass versus24 of 36 versus conversion in the copolymerization of MMA and EHA at diﬀerent initial compositions of MMA. conversion in the copolymerization of MMA and EHA at different initial compositions of MMA.

107

106

MMA/EHA= 0.3/0.7 MMA/EHA= 0.5/0.5

MMA/EHA= 0.7/0.3 Model MMA/EHA= 0.3/0.7

Mass AverageMolar Mass (g/mol) Model MMA/EHA= 0.5/0.5 Model MMA/EHA= 0.7/0.3

105 0 102030405060708090100 Conversion (wt. %)

FigureFigure 22. 22.Comparison of experimental of experimental data and calculated data and calculatedprofiles of proﬁles weight ofaverage weight molar average mas molar versus mas versusconversion conversion in the in copolymerization the copolymerization of MMA of MMA and EHA and EHAat different at diﬀ erentinitial initialcompositions compositions of MMA. of MMA.

MMA/EHA= 0.5/0.5 5 MMA/EHA= 0.7/0.3 MMA/EHA= 0.3/0.7 Model MMA/EHA= 0.5/0.5 Model MMA/EHA= 0.7/0.3 4 Model MMA/EHA= 0.3/0.7

Dispersity 2

0 0 102030405060708090100 Conversion (wt. %)

Figure 23. of experimental data and calculated profiles of Ð conversion in the copolymerization of MMA and EHA at different initial compositions of MMA.

Processes 2019, 7, x FOR PEER REVIEW 24 of 36

107

106

MMA/EHA= 0.3/0.7 MMA/EHA= 0.5/0.5 MMA/EHA= 0.7/0.3 Model MMA/EHA= 0.3/0.7

Mass AverageMolar Mass (g/mol) Model MMA/EHA= 0.5/0.5 Model MMA/EHA= 0.7/0.3

105 0 102030405060708090100 Conversion (wt. %)

Figure 22. of experimental data and calculated profiles of weight average molar mas versus Processes 2019, 7, 395 22 of 33 conversion in the copolymerization of MMA and EHA at different initial compositions of MMA.

MMA/EHA= 0.5/0.5 5 MMA/EHA= 0.7/0.3 MMA/EHA= 0.3/0.7 Model MMA/EHA= 0.5/0.5 Model MMA/EHA= 0.7/0.3 4 Model MMA/EHA= 0.3/0.7

Dispersity 2

0 0 102030405060708090100 Conversion (wt. %)

FigureFigure 23. 23. ofComparison experimental of data experimental and calculated data profiles and calculatedof Ð conversion proﬁles in the of copolymerization Ð conversion in of the copolymerizationMMA and EHA ofat different MMA and initial EHA compositions at diﬀerent of initial MMA. compositions of MMA.

It is observed in Figures 24 and 25 that our model is able to produce order-of-magnitude estimations of final molecular weights (Mn and Mw at high conversions), but their time (conversion) evolution is captured very poorly. As a matter of fact, the observed trends are opposite to the calculated profiles: The observed trend is that molecular weight increases within the same order of magnitude values, and the calculated profiles show huge decreasing trends (of about two orders of magnitude). Although this is not uncommon in polymerization modeling studies, these results may be attributed to the estimated kEHA chain transfer to monomer kinetic rate constant coefficients. As stated in Section 3.8 of this contribution, kEHA was used to fit BA/EHA and EHA/MMA copolymers molar masses. Since crosslinking and gelation occurred in the BA/EHA system, our estimation of kEHA was likely overpredicted. More complete experimental studies with a focus on capturing the early stages of the polymerization and gatheringProcesses more2019, 7, datax FOR on PEER molecular REVIEW weight development are needed to improve our model. 25 of 36

107

106

Model BA/EHA/MMA= 0.1/0.1/0.8 Model BA/EHA/MMA= 0.8/0.1/0.1 Model BA/EHA/MMA= 0.1/0.8/0.1 BA/EHA/MMA= 0.1/0.1/0.8 Number Average Molar Mass (g/mol) Mass Molar Average Number BA/EHA/MMA= 0.8/0.1/0.1 BA/EHA/MMA= 0.1/0.8/0.1 105 020406080100 Conversion (wt. %) . Figure 24. Comparison of experimental data and calculated proﬁles of number average molar mass versusFigure conversion 24. of experimental for terpolymerizations data and ofcalculated BA, EHA prof andiles MMA of number at the initial average concentrations molar mass indicated versus conversion for terpolymerizations of BA, EHA and MMA at the initial concentrations indicated in the in the legend. legend.

107

106 Model BA/EHA/MMA= 0.1/0.1/0.8 Model BA/EHA/MMA= 0.8/0.1/0.1 Model BA/EHA/MMA= 0.1/0.8/0.1 BA/EHA/MMA= 0.1/0.1/0.8 Mass Average Molar Mass (g/mol) Mass Molar Average Mass BA/EHA/MMA= 0.8/0.1/0.1 BA/EHA/MMA= 0.1/0.8/0.1 105 0 20406080100 Conversion (wt. %)

Figure 25. of experimental data and calculated profiles of weight average molar mass versus conversion for terpolymerizations of BA, EHA and MMA at the initial concentrations indicated in the legend.

5. Conclusions MMA and BA are well-studied monomers whose polymerization kinetic rates coefficients are well known. However, this is not the case with EHA. Several important EHA kinetic rate coefficients (e.g., termination and transfer to monomer) had to be estimated in this study from experimental data not designed for such purposes. The overall agreement between experimental data and model calculations for the three copolymerization systems was fairly good. Copolymer composition was captured very well with

Processes 2019, 7, x FOR PEER REVIEW 25 of 36

107

106

Figure 24. of experimental data and calculated profiles of number average molar mass versus conversion for terpolymerizations of BA, EHA and MMA at the initial concentrations indicated in the Processeslegend.2019, 7, 395 23 of 33

107

Figure 25. Comparison of experimental data and calculated profiles of weight average molar mass Figure 25. of experimental data and calculated profiles of weight average molar mass versus versusconversion conversion for terpolymerizations for terpolymerizations of BA, ofEHA BA, and EHA MMA and at MMA the initial at the concentrations initial concentrations indicated indicated in the inlegend. the legend. 5. Conclusions 5. Conclusions MMA and BA are well-studied monomers whose polymerization kinetic rates coefficients are MMA and BA are well-studied monomers whose polymerization kinetic rates coefficients are well known. However, this is not the case with EHA. Several important EHA kinetic rate coefficients well known. However, this is not the case with EHA. Several important EHA kinetic rate (e.g.,coefficients termination (e.g., and termination transfer to and monomer) transfer hadto mo tonomer) be estimated had to in be this estimated study from in experimentalthis study from data notexperimental designed for data such not purposes. designed for such purposes. TheThe overall overall agreement agreement betweenbetween experimentalexperimental data data and and model model calculations calculations for for the the three three copolymerizationcopolymerization systems systems was was fairly fairly good. good. Copolymer Copolymer composition composition was was captured captured very very well well with with the model for systems BA/EHA and MMA/EHA. Some deviations between calculated and experimental data were observed in some cases for conversion versus time. Predictions of molecular weight development were rather poor in some cases, particularly for the terpolymer cases. Further experimental and modeling studies are needed, particularly for the estimation of chain transfer in EHA in bulk copolymerizations. From the experimental side, it would be useful to put more emphasis on reducing further or eliminating induction times (more through purification procedures) and gathering more experimental data for molecular weight development. Regarding the theoretical side, it would be useful to carry out simulations where BA is present with inclusion of crosslinking and gelation phenomena. Finally, it is important to point out that our simulations were carried out using a global set of parameters. Namely, the parameters were not changed in individual cases to improve data fitting. If we had done that, our simulated profiles would have fit the experimental data much better, but that was not the point of our study.

Author Contributions: J.A.G.-R. performed all the simulations and parameter estimation procedure, guided by E.V.-L. M.A.D. and V.A.G. conceived the experimental research. V.A.G. conducted all experiments and characterization. M.A.D. and E.V.-L. provided technical direction to the project. J.A.G.-R. wrote the paper with feedback from V.A.G., while M.A.D. and E.V.-L. revised the final document. Funding: This research was funded by: DGAPA-UNAM, grant numbers PAPIIT IG100718 and IV100119; Facultad de Química-UNAM, grant number PAIP 5000-9078. Acknowledgments: Additional financial support from the following sources is gratefully acknowledged: (a) Consejo Nacional de Ciencia y Tecnología (CONACYT, México), MEng scholarship granted to J.A.G.-R.; and (b) Natural Sciences and Engineering Research Council (NSERC) of Canada. Conflicts of Interest: The authors declare no conflict of interest. Processes 2019, 7, 395 24 of 33

Nomenclature

BA n-Butyl Acrylate EHA 2-Ethylhexyl Acrylate MMA Methyl Methacrylate DC Diffusion-Controlled (Effects) PSA Pressure Sensitive Adhesive AIBN Azobisisobutyronitrile f i Monomer Fraction for species i I Initiator I Primary Radical • Mi Monomer for species i Pi(s) Secondary Radical for species i • Qi(s) Tertiary Radical for species i • Pi Polymer for species i Pi(s) Secondary Radical for species i • ki Initiator Decomposition Kinetic Rate Coefficient f Initiator Efficiency Factor kpi Propagation Kinetic Rate Coefficient for species i kpij Cross-Propagation Kinetic Rate Coefficient for species iand j kfi Chain Transfer Kinetic Rate Coefficient for species i kfij Cross-Chain Transfer Kinetic Rate Coefficient for species i kbb Backbiting Kinetic Rate Coefficient for n-Butyl Acrylate kti Global Termination Kinetic Rate Coefficient for species i ktci Termination by Combination Kinetic Rate Coefficient for species i ktdi Termination by Disproportionation Kinetic Rate Coefficient for species i kfitert Tertiary Radical Chain Transfer Kinetic Rate Coefficient for species i kpitert Tertiary Radical Propagation Kinetic Rate Coefficient for species i ktcij Cross-Termination by Combination Kinetic Rate Coefficient for species i ktdij Cross-Termination by Disproportionation Kinetic Rate Coefficient for species i rij Reactivity Ratio beween species iand j β Attraction/Separation Overlap Factor in free-volume theory αM Expansion Coefficient for Monomer αP Expansion Coefficient for Polymer vf Free Volume at a given calculated time vfo Free volume at initial conditions Vmi Monomer Volume for species i Vpi Polymer Volume for species i TgP Polymer Glass Temperature TgM Monomer Glass Temperature T System Temperature

Appendix A As stated in Section 3.8, the parameter estimation procedure consisted of a combination between careful literature review, detailed parameter sensitivity analyses and the use of educated guesses (trial and error guided from the information gained from the literature review and the parameter sensitivity analyses). The strategy is illustrated in this appendix with a few examples related to the estimation of ktEHA, kfEHA and the β-parameters for DC effects. As shown in Figure A1 for system 30BA/70EHA, ktEHA has a strong effect on polymerization rate and because of that it was used to fit conversion versus time data at low conversions (up to 20–30 wt. % monomer conversion). From the results shown in Figure A1, it seemed that kt = 1 108 L mol 1 s 1 EHA × − − was the best fit. However, as shown in Figures A2 and A3 for systems 50BA/50EHA and 70BA/30EHA, respectively, that value was not the best for such systems. Therefore, the value of ktEHA was set by using a single value (kt = 5 108 L mol 1 s 1) that could reproduce reasonably well the experimental data EHA × − − Processes 2019, 7, x FOR PEER REVIEW 27 of 36

ktdij Cross-Termination by Disproportionation Kinetic Rate Coefficient for species i rij Reactivity Ratio beween species iand j β Attraction/Separation Overlap Factor in free-volume theory

𝛼 Expansion Coefficient for Monomer 𝛼 Expansion Coefficient for Polymer vf Free Volume at a given calculated time

vfo Free volume at initial conditions Vmi Monomer Volume for species i Vpi Polymer Volume for species i

TgP Polymer Glass Temperature TgM Monomer Glass Temperature T System Temperature

Appendix A As stated in Section 3.8, the parameter estimation procedure consisted of a combination between careful literature review, detailed parameter sensitivity analyses and the use of educated guesses (trial and error guided from the information gained from the literature review and the parameter sensitivity analyses). The strategy is illustrated in this appendix with a few examples related to the estimation of ktEHA, kfEHA and the β-parameters for DC effects. As shown in Figure A1 for system 30BA/70EHA, ktEHA has a strong effect on polymerization rate and because of that it was used to fit conversion versus time data at low conversions (up to 20–30 wt. % monomer conversion). From the results shown in Figure A1, it seemed that ktEHA = 1 × 108 L mol−1 Processess−12019 was, 7 ,the 395 best fit. However, as shown in Figures A2 and A3 for systems 50BA/50EHA and25 of 33 70BA/30EHA, respectively, that value was not the best for such systems. Therefore, the value of ktEHA was set by using a single value (ktEHA = 5 × 108 L mol−1 s−1) that could reproduce reasonably well the for theexperimental three compositions data for the and three the experimentalcompositions dataand the (at experimental diﬀerent compositions) data (at different for system compositions) MMA/EHA (proﬁlesfor system not shown MMA/EHA for those (profiles cases). not shown for those cases).

kt = 1E5 EHA kt = 1E6 EHA kt = 1E7 EHA kt = 1E8 100 EHA kt = 2.5E8 (Model Value) EHA kt = 1E9 EHA

ktEHA= 1E10 Exp Data BA/EHA = 0.3/0.7 80

40 Conversion (wt. %) (wt. Conversion

0 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s) Processes 2019, 7, x FOR PEER REVIEW 28 of 36 Figure A1. Eﬀect of ktEHA on conversion versus time proﬁles for system 30BA/70EHA.

Figure A1. Effect of ktEHA on conversion versus time profiles for system 30BA/70EHA. ktEHA= 1E5

ktEHA= 1E6 100 ktEHA= 1E7

ktEHA= 1E8

ktEHA= 2.5E8 (Model Value)

ktEHA= 1E9 80 ktEHA= 1E10 Exp Data BA/EHA = 0.5/0.5

40 Conversion (wt. %) (wt. Conversion

0 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s)

Figure A2. Eﬀect of ktEHA on conversion versus time proﬁles for system 50BA/50EHA.

Figure A2. Effect of ktEHA on conversion versus time profiles for system 50BA/50EHA.

kt = 1E5 EHA kt = 1E6 EHA kt = 1E7 100 EHA

ktEHA= 1E8

ktEHA= 2.5E8 (Model Value) kt = 1E9 EHA kt = 1E10 80 EHA Exp Data BA/EHA = 0.7/0.3

40 Conversion (wt. %) (wt. Conversion

0 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s)

Figure A3. Effect of ktEHA on conversion versus time profiles for system 70BA/30EHA.

The calculated conversion versus time profiles agreed well with experimental data at low conversions but deviated at the high conversion region. The agreement at high conversions was improved by considering DC effects, namely, by adjusting βp and βt. When carrying out parameter sensitivity analyses for those parameters, it was found that it was the ratio of them, which was useful for data fitting purposes. Table A1 and Figures A4 to A6 show some of the data sets used for estimation of βp and βt (Table A1 and Section 3.7) and the profiles obtained for the BA/EHA system. As shown in Figure A7 DC effects also affect the composition versus conversion profiles, a fact that was also considered in the estimation of the DC β-parameters.

Processes 2019, 7, x FOR PEER REVIEW 28 of 36

ktEHA= 1E5

ktEHA= 1E6 100 ktEHA= 1E7

ktEHA= 1E8

ktEHA= 2.5E8 (Model Value)

ktEHA= 1E9 80 ktEHA= 1E10 Exp Data BA/EHA = 0.5/0.5

40 Conversion (wt. %) (wt. Conversion

0 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s)

Processes 2019, 7, 395 26 of 33 Figure A2. Effect of ktEHA on conversion versus time profiles for system 50BA/50EHA.

kt = 1E5 EHA kt = 1E6 EHA kt = 1E7 100 EHA

ktEHA= 1E8

ktEHA= 2.5E8 (Model Value) kt = 1E9 EHA kt = 1E10 80 EHA Exp Data BA/EHA = 0.7/0.3

40 Conversion (wt. %) (wt. Conversion

0 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s)

Figure A3. Effect of ktEHA on conversion versus time profiles for system 70BA/30EHA. Figure A3. Effect of ktEHA on conversion versus time profiles for system 70BA/30EHA. The calculated conversion versus time profiles agreed well with experimental data at low conversionsThe butcalculated deviated conversion at the high versus conversion time profiles region. agreed The well agreement with experimental at high conversions data at low was improvedconversions by considering but deviated DC at e fftheects, high namely, conversion by adjusting region. βThep and agreementβt. When at carryinghigh conversions out parameter was sensitivityimproved analyses by considering for those DC parameters, effects, namely, it was by found adjusting that it β wasp and the βt. ratio When of carrying them, which out parameter was useful for datasensitivity fitting analyses purposes. for Tablethose A1parameters, and Figures it was A4 foun–A6d show that it some was the of theratio data of them, sets usedwhich for was estimation useful of βforpProcesses and dataβ t 2019fitting (Table, 7, x purposes.FORA1 PEERand REVIEW Section Table 3.7A1 )and and Figures the profiles A4 to obtained A6 show for some the BA of/ EHAthe data system. sets 29used As ofshown 36 for in Figureestimation A7 DC of β epff andects β alsot (Table affect A1 theand composition Section 3.7) and versus the profiles conversion obtained profiles, for the a BA/EHA fact that system. was also Table A1. Tested βt/βp ratios for BA/EHA system. consideredAs shown in in the Figure estimation A7 DC of effects the DC alsoβ-parameters. affect the composition versus conversion profiles, a fact that was also considered in the estimation of the DC β-parameters. BA βt/βp EHA βt/βp BA/EHA βt/βp Figure

SetTable 1 A1. Tested 32 βt/βp ratios 16 for BA/EHA 4 system. A4 Set 2 (Model) 16 8 2 A5 (2) BA β /β EHA β /β BA/EHA β /β Figure Set 3t p 8 4t p 1 t A6p Set 1 32 16 4 A4

Set 2 (Model) 16 8 2 A5 (2) Set 3 8 4 1 A6

100

40 BA/EHA= 0.3/0.7 BA/EHA= 0.5/0.5 Conversion (wt. %) (wt. Conversion BA/EHA= 0.7/0.3 20 Model Predictions (DCE set 1)

0 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s)

Figure A4. Effect of the ratio βt/βp (DC effects) on conversion versus time profiles for system BA/EHA (Set 1 ofFigure Table A4. A1 Effect). of the ratio βt/βp (DC effects) on conversion versus time profiles for system BA/EHA (Set 1 of Table A1).

100

40 Model predictions

Conversion %) (wt. BA/EHA = 30/70 20 BA/EHA = 50/50 BA/EHA = 70/30

0 0 2000 4000 6000 8000 10000 12000 14000 Time (s) .

Figure A5. (Figure2) Effect of the ratio βt/βp (DC effects) on conversion versus time profiles for system BA/EHA (Set 2 of Table A1).

Processes 2019, 7, x FOR PEER REVIEW 29 of 36

Table A1. Tested βt/βp ratios for BA/EHA system.

BA βt/βp EHA βt/βp BA/EHA βt/βp Figure Set 1 32 16 4 A4 Set 2 (Model) 16 8 2 A5 (2) Set 3 8 4 1 A6

100

40 BA/EHA= 0.3/0.7 BA/EHA= 0.5/0.5 Conversion (wt. %) (wt. Conversion BA/EHA= 0.7/0.3 20 Model Predictions (DCE set 1)

0 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s)

Processes 2019, 7, 395 27 of 33 Figure A4. Effect of the ratio βt/βp (DC effects) on conversion versus time profiles for system BA/EHA (Set 1 of Table A1).

100

40 Model predictions

Conversion %) (wt. BA/EHA = 30/70 20 BA/EHA = 50/50 BA/EHA = 70/30 Processes 2019, 7, x FOR PEER REVIEW 30 of 36

0 0 2000 4000 6000 8000 10000 12000 14000 Time (s) Processes 2019, 7, x FOR PEER100 REVIEW . 30 of 36 Figure A5. (Figure2)E ffect of the ratio βt/βp (DC effects) on conversion versus time profiles for system BA/EHA (Set 2 ofFigure Table A5. A1 (Figure2)). Effect of the ratio βt/βp (DC effects) on conversion versus time profiles for system BA/EHA80 (Set 2 of Table A1). 100

60 80

BA/EHA= 0.3/0.7 40 60 BA/EHA= 0.5/0.5 BA/EHA= 0.7/0.3

Conversion (wt. %) Conversion (wt. Model Prediction (DCE set 3) 20 BA/EHA= 0.3/0.7 40 BA/EHA= 0.5/0.5 BA/EHA= 0.7/0.3

Conversion (wt. %) Conversion (wt. Model Prediction (DCE set 3) 200 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s) 0 0 2000 4000 6000 8000 10000 12000 14000 16000 Figure A6. Effect of the ratio βt/βp (DC effects)Time on (s) conversion versus time profiles for system BA/EHA (Set 3 of Table A1).

Figure A6. Effect of the ratio βt/βp (DC effects) on conversion versus time profiles for system BA/EHA (Set 3 of TableFigure A1A6.). Effect of the ratio βt/βp (DC effects) on conversion versus time profiles for system BA/EHA (Set 3 of Table A1). 0.7

0.6 0.7

0.5 0.6

0.4 0.5

0.3 0.4 Model without DCE's 0.2 Model with DCE's 0.3 Exp Data MMA/EHA= 0.3/0.7

Model without DCE's

MMA Composition (mol fraction) (mol Composition MMA 0.1 0.2 Model with DCE's Exp Data MMA/EHA= 0.3/0.7

0.0

MMA Composition (mol fraction) (mol Composition MMA 0.1 0 102030405060708090100 Conversion (wt. %) 0.0 Figure A7. Comparison of0 modeling 102030405060708090100 proﬁles of copolymer composition (MMA content) versus Conversion (wt. %) conversion,Figure considering A7. Comparison or neglecting of modeling DC eprofilesﬀects, forof copolymer system 30MMA composition/70EHA. (MMA content) versus conversion, considering or neglecting DC effects, for system 30MMA/70EHA. Figure A7. Comparison of modeling profiles of copolymer composition (MMA content) versus Itconversion, is common considering knowledge or neglecting in free radical DC effects, polymerization for system 30MMA/70EHA. that chain transfer reactions to small molecules may significantly affect molar mass development without affecting polymerization rate. This wasIt is thecommon case for knowledge chain transfer in free to radical EHA. polymerization As observed in that Figure chain A8, transfer changing reactions kfEHA to by small three ordersmolecules of magnitude may significantly resulted affect on minimal molar mass effect development on polymerization without rate, affecting but significant polymerization changes rate. on molecularThis was weightthe case development for chain transfer (see Figure to EHA. A9). As observed in Figure A8, changing kfEHA by three orderskfEHA of wasmagnitude therefore resulted used toon fit minimal molecular effect weight on poly developmentmerization rate,data butfor thesignificant studied changes systems. on As observedmolecular in weight Figures development A10 to A12, (seethe bestFigure agreement A9). with experimental data of Mn versus conversion EHA was obtainedkf was with therefore kfEHA =used 2 L molto fit−1 moleculars−1, but the weight best agreement development with data experimental for the studied data systems.of Mw versus As observed in Figures A10 to A12, the best agreement with experimental data of Mn versus conversion was obtained with kfEHA = 2 L mol−1 s−1, but the best agreement with experimental data of Mw versus

Processes 2019, 7, x FOR PEER REVIEW 31 of 36

conversion was obtained with kfEHA = 5 L mol−1 s−1 (see Figures A13 to A15). A compromise was establishedProcesses by 2019 using, 7, x FOR a finalPEER REVIEWvalue of kfEHA = 3.5 L mol−1 s−1. As explained in the body of the31 of 36 Processes 2019 7 manuscript,, , 395gelation was an issue for systems containing BA, so that explains the deviation.28 Chain of 33 − −1 transferconversion to tertiary was radicals obtained for with EHA kfEHA was = 5 adjuL molsted1 ssimilarly, (see Figures but A13its effectto A15). on Amolar compromise mass was established by using a final value of kfEHA = 3.5 L mol−1 s−1. As explained in the body of the developmentIt is common had knowledgeless impact inon freemolar radical mass polymerizationresults. that chain transfer reactions to small manuscript, gelation was an issue for systems containing BA, so that explains the deviation. Chain molecules may significantly affect molar mass development without affecting polymerization rate. transfer to tertiary radicals for EHA was adjusted similarly, but its effect on molar mass This was the case for chain transfer to EHA. As observed in Figure A8, changing kfEHA by three orders development had less impact on molar mass results. of magnitude resulted100 on minimal effect on polymerization rate, but significant changes on molecular weight development (see Figure A9).

80 100

60 80

60 kf = 3.5 40 EHA kf = 35 EHA kf = 0.35 EHA Exp Data kf = 3.5 20 40 EHA kf = 35 EHA BA/EHA 50/50 Conversion (wt.%) 50/50 Conversion BA/EHA kf = 0.35 EHA Exp Data 0 20 0 2000 4000 6000 8000 10000 12000 14000 16000 BA/EHA 50/50 Conversion (wt.%) 50/50 Conversion BA/EHA Time (s) 0 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s) Figure A8. Effect of kfEHA on conversion versus time profiles for system 50BA/50EHA.

Figure A8. Eﬀect of kfEHA on conversion versus time proﬁles for system 50BA/50EHA.

Figure A8. EffectBA/EHA= of kf EHA0.5/0.5 on conversion versus time profiles for system 50BA/50EHA. 107

BA/EHA= 0.5/0.5

7 6 10 10

6 5 10 10 Exp Data kf = 3.5 EHA 5 4 10 10 kf = 35 EHA Exp Data kf = 0.35 EHA kf = 3.5 EHA 4 3 10 kf = 35 10 EHA kf = 0.35 EHA Number Average Molar Mass (g/mol)

3 2 10 10 0 20406080100 Number Average Molar Mass (g/mol) Conversion (wt. %) 102 Figure A9. Eﬀect of0 kfEHA on Mn 20406080100versus conversion proﬁles for system 50BA/50EHA.

Figure A9. Effect of kfEHA on Mn versus conversionConversion profiles (wt. %) for system 50BA/50EHA. kfEHA was therefore used to ﬁt molecular weight development data for the studied systems. As observed in Figures A10–A12, the best agreement with experimental data of Mn versus conversion was obtained with kf = 2 L mol 1 s 1, but the best agreement with experimental data of M FigureEHA A9. Effect of− kfEHA− on Mn versus conversion profiles for system 50BA/50EHA. w 1 versus conversion was obtained with kfEHA = 5 L mol−1 s− (see Figures A13–A15). A compromise

Processes 2019, 7, 395 29 of 33

Processes 2019, 7, x FOR PEER REVIEW 32 of 36 1 1 was established by using a ﬁnal value of kfEHA = 3.5 L mol− s− . As explained in the body of the manuscript, gelation was an issue for systems containing BA, so that explains the deviation. Chain transfer to tertiary radicals for EHA was adjustedkf = 2 L mol similarly,-1s-1 but its eﬀect on molar mass development Processes 2019, 7, x FOR PEER REVIEW6 EHA 32 of 36 had less impact on molar3.5x10 mass results.

3.0x106 kf = 2 L mol-1s-1 3.5x106 EHA 2.5x106 3.0x106 2.0x106 2.5x106 1.5x106 2.0x106 6 1.0x10 Model Predictions 1.5x106 5 BA/EHA= 0.3/0.7 5.0x10 BA/EHA= 0.5/0.5

Number Average Molar Mass (g/mol) 6 BA/EHA= 0.7/0.3 1.0x10 Model Predictions 0.0

5 0 20406080100 BA/EHA= 0.3/0.7 5.0x10 BA/EHA= 0.5/0.5

Number Average Molar Mass (g/mol) Conversion BA/EHA= 0.7/0.3(wt. %) 0.0 0 20406080100 Figure A10. Comparison of experimental and calculated profiles of Mnversus conversion for system Conversion (wt. %) BA/EHA using kfEHA = 2 L mol−1 s−1. Figure A10. Comparison of experimental and calculated proﬁles of Mnversus conversion for system 1 1 BAFigure/EHA using A10. kfEHAComparison= 2 L of mol experimental− s− . and calculated profiles of Mnversus conversion for system −1 −1 BA/EHA using kfEHA = 2 L mol s . -1 -1 6 kf = 3.5 L mol s 3.5x10 EHA

3.0x106 -1 -1 6 kf = 3.5 L mol s 3.5x10 EHA 2.5x106 3.0x106

2.0x106 2.5x106

1.5x106 2.0x106

6 1.0x10 Model Predictions 6 1.5x10 5 BA/EHA= 0.3/0.7 5.0x10 BA/EHA= 0.5/0.5 6 Number Average Molar Mass (g/mol) Mass Molar Average Number 1.0x10 ModelBA/EHA= Predictions 0.7/0.3 0.0 5 BA/EHA= 0.3/0.7 5.0x10 0 20406080100 BA/EHA= 0.5/0.5 Number Average Molar Mass (g/mol) Mass Molar Average Number Conversion BA/EHA= (wt. %)0.7/0.3 0.0 Figure A11. Comparison of experimental and calculated proﬁles of Mn versus conversion for system 01 204060801001 BAFigure/EHA using A11. kfEHAComparison= 3.5 Lof molexperimental− s− . and calculated profiles of Mn versus conversion for system Conversion (wt. %) BA/EHA using kfEHA = 3.5 L mol−1 s−1.

Figure A11. Comparison of experimental and calculated profiles of Mn versus conversion for system BA/EHA using kfEHA = 3.5 L mol−1 s−1.

Processes 2019, 7, x FOR PEER REVIEW 33 of 36

kf = 5 L mol-1s-1 EHA ProcessesProcesses2019 2019, 7, 395, 7, x FOR3.5x10 PEER6 REVIEW 3033 of 33of 36

3.0x106 kf = 5 L mol-1s-1 EHA 3.5x106 2.5x106

3.0x106 2.0x106

2.5x106 1.5x106

2.0x106 6 1.0x10 Model Predictions 1.5x106 5.0x105 BA/EHA= 0.3/0.7

Number Average Molar Mass (g/mol) 6 BA/EHA= 0.5/0.5 1.0x10 Model Predictions 0.0 BA/EHA= 0.7/0.3

0 20406080100 5.0x105 BA/EHA= 0.3/0.7 Conversion (wt. %) Number Average Molar Mass (g/mol) BA/EHA= 0.5/0.5 0.0 BA/EHA= 0.7/0.3 0 20406080100 Figure A12. Comparison of experimental and calculated profiles of Mn versus conversion for system Conversion (wt. %) BA/EHA using kfEHA = 5 L mol−1 s−1.

Figure A12. Comparison of experimental and calculated proﬁles of Mn versus conversion for system 1 1 BA/FigureEHA using A12. kfEHAComparison= 5 L of mol experimental− s− . and calculated profiles of Mn versus conversion for system −1 −1 BA/EHA using kfEHA6 = 5 L mol s . 7x10 kf = 2 L mol-1s-1 EHA

6x106

6 7x10 -1 -1 6 kf = 2 L mol s 5x10 EHA

6x106 4x106

5x106 3x106

4x106 6 2x10 Model Predictions 6 3x10 6 BA/EHA= 0.3/0.7 1x10 BA/EHA= 0.5/0.5

Weight Average Molar Mass (g/mol) Mass Molar Average Weight 6 BA/EHA= 0.7/0.3 2x10 Model Predictions 0

6 0 20406080100 BA/EHA= 0.3/0.7 1x10 BA/EHA= 0.5/0.5

Weight Average Molar Mass (g/mol) Mass Molar Average Weight Conversion (wt. %) BA/EHA= 0.7/0.3

Figure A13. Comparison0 of experimental and calculated proﬁles of Mw versus conversion for system 01 204060801001 BAFigure/EHA usingA13. Comparison kfEHA = 2 L of mol experimental− s− . and calculated profiles of Mw versus conversion for system Conversion (wt. %) BA/EHA using kfEHA = 2 L mol−1 s−1.

Figure A13. Comparison of experimental and calculated profiles of Mw versus conversion for system BA/EHA using kfEHA = 2 L mol−1 s−1.

Processes 2019, 7, x FOR PEER REVIEW 34 of 36

Processes 2019, 7, x FOR PEER6 REVIEW kf = 3.5 L mol-1s-1 34 of 36 7x10 EHA Processes 2019, 7, 395 31 of 33

6x106 6 kf = 3.5 L mol-1s-1 7x10 EHA 5x106

6x106 4x106

5x106 3x106

6 4x10 2x106 Model Predictions

6 3x10 BA/EHA= 0.3/0.7 1x106 BA/EHA= 0.5/0.5

Weight Average Molar Mass (g/mol) Molar Mass Average Weight BA/EHA= 0.7/0.3 2x106 Model Predictions 0 020406080100 BA/EHA= 0.3/0.7 1x106 BA/EHA= 0.5/0.5

Weight Average Molar Mass (g/mol) Molar Mass Average Weight Conversion BA/EHA= (wt.0.7/0.3 %)

0 020406080100 Figure A14. Comparison of experimental and calculated profiles of Mw versus conversion for system BA/EHA using kfEHA = 3.5 L mol−1 s−1. Conversion (wt. %)

Figure A14. Comparison of experimental and calculated proﬁles of Mw versus conversion for system 1 1 BAFigure/EHA usingA14. Comparison kfEHA = 3.5 of L molexperimental− s− . and calculated profiles of Mw versus conversion for system −1 −1 BA/EHA using kf7x10EHA6 = 3.5 L mol s . kf = 5 L mol-1s-1 EHA 6x106

7x106 5x106 kf = 5 L mol-1s-1 EHA 6x106 4x106

5x106 3x106

4x106 2x106 Model Predictions 3x106 1x106 BA/EHA= 0.3/0.7

Weight Average Molar Mass (g/mol) Mass Molar Average Weight BA/EHA= 0.5/0.5 2x106 BA/EHA= 0.7/0.3 0 Model Predictions 0 20406080100 1x106 BA/EHA= 0.3/0.7

Weight Average Molar Mass (g/mol) Mass Molar Average Weight Conversion BA/EHA= (wt. 0.5/0.5 %) BA/EHA= 0.7/0.3 Figure A15. Comparison0 of experimental and calculated proﬁles of Mw versus conversion for system BA/EHA using kfEHA =05 L mol 1 s 204060801001. Figure A15. Comparison of experimental− − and calculated profiles of Mw versus conversion for system BA/EHA using kfEHA = 5 L mol−1 s−1. Conversion (wt. %) References

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