Modeling of the Free-Radical Copolymerization Kinetics With

Revista de la Sociedad Química de México, Vol. 47, Núm. 1 (2003) 22-33

Investigación Modeling of the Free-Radical Copolymerization Kinetics with Crosslinking of Methyl Methacrylate / Ethylene Glycol Dimethacrylate Up to High Conversions and Considering Thermal Effects

Eduardo Vivaldo-Lima*, Rosalba García-Pérez, and Oswaldo J. Celedón-Briones

Departamento de Ingeniería Química, Facultad de Química, Conjunto E, Universidad Nacional Autónoma de México, Ciudad Universitaria, México D.F., CP 04510, México. Tel.: 5622-5256; Fax: 5622-5355; E-mail: [email protected]

Recibido el 13 de marzo del 2002; aceptado el 28 de enero del 2003

Resumen. Se usa un modelo matemático para copolimerización por Abstract. A mathematical model for the free radical copolymeriza- radicales libres con entrecruzamiento para modelar el comportamien- tion kinetics with crosslinking of vinyl / divinyl monomers is used to to del sistema de copolimerización metacrilato de metilo (MMA) / model the free-radical copolymerization of methyl methacrylate dimetacrilato de etilén glicol (EGDMA). Las predicciones del mode- (MMA)/ethylene glycol dimethacrylate (EGDMA). Good agreement lo se comparan contra datos experimentales de literatura, teniendo between model predictions and experimental data from the literature buena concordancia a bajas conversiones, y bajas concentraciones de is obtained at low conversions, and low crosslinker concentrations. agente de entrecruzamiento. Si se consideran en el modelo los efectos The agreement is also good at high conversions, and intermediate térmicos asociados a la polimerización con temperatura no uniforme crosslinker concentrations, if thermal effects in ampoule copolyme- en ampolletas de más de 0.5 mm de diámetro externo, la concordan- rization are incorporated into the model, through an energy balance. cia es aceptable aún a medias y altas conversiones, y a concentra- The thermal effects become important when ampoules greater than ciones medias de EGDMA. 0.5 mm in external diameter are used. Palabras clave: Entrecruzamiento, copolimerización, red polimérica, Keywords: Crosslinking, copolymerization, polymer network, MMA, EGDMA, modelos matemáticos. MMA, EGDMA, mathematical modeling.

Introduction (b) when the state of the art is sufficiently advanced to make a useful mechanistic model easily available. Mechanistic mo- Crosslinked polymers (polymer networks) are very important dels can contribute to scientific understanding, provide a basis in technology, medicine, biotechnology, and agriculture (as for extrapolation and even interpolation, and provide a repre- construction materials, polymer glasses with high mechanical sentation of the system’s response function that is more parsi- strength and high thermal stability, rubbers, ion-exchange monious than that obtained empirically. A mechanistic model resins and absorbents, insoluble polymer reagents, etc.). How- can suggest with greater certainty new sets of experimental ever, the treatment of polymer networks is difficult from any conditions that are worthy of investigation. perspective. Polymer gels are difficult to handle experimental- In the case of production of polymer networks via free-radi- ly, the characterization of their properties is a non-trivial sub- cal copolymerization of vinyl / divinyl monomers, the complex ject, with several conventional theories being inapplicable to and tedious experimental techniques can be reduced to the mini- those materials. The modelling of polymer network formation mum necessary by using mechanistic mathematical models of is also a formidable task. intermediate degree of complexity. Material and process design, As explained in Penlidis et al. [1], a mechanistic process and even process operator training, can be facilitated by the use of model is a mathematical form derived from consideration of a mathematical models and process simulators based upon them. supposed mechanism. Mechanistic models are typically for- There are several theories to explain gelation and polymer mulated in terms of differential equations, and are usually network formation. These theories are statistical or kinetic in non-linear in the parameters. Judgement is needed in deciding nature, and have been known for decades. Reviews in this when and when not to use mechanistic models. Mechanistic topic are available elsewhere [2,3]. However, the use of sound model development could be difficult and time consuming, mathematical models with predictive capabilities has been and might be improvident if all that was needed might be very limited. In the context of free-radical copolymerization achieved much more economically by empirical methods with crosslinking, quite complete and general kinetic models using factorials and response surface designs, i.e., empirical have been developed [4,5], but their application to actual sys- modeling. By contrast, a mechanistic approach is justified (a) tems require to make many assumptions and simplifications, whenever a basic understanding of the system is essential, or which makes them lose their appeal. Modeling of the Free-Radical Copolymerization Kinetics with Crosslinking of Methyl Methacrylate... 23

1 Table 1. Kinetic and Moment Equations. [Io]=0.00625 [Io]=0.0125 Initiation d(V[I]) [Io]=0.025 = -k d [I] 0.8 [Io]=0.05 Vdt [Io]=0.1 Non-SSH Overall conversion dx * = ( k p + k fm )(1 - x)[R ] 0.6 dt Moment equations for polymer radicals

Conversion 0.4 d( VY 0 ) = 2 fk [I] - ( k + k )Y 2 Vdt d tcn td 0

d( VY 1 ) * = 2 fk d [I] + ( k fm [M] + k fT [T]) + k p Y 0 Q 2 + k p [M]Y 0 0.2 Vdt

- { k fm [M] + k fT [T] + ( k tcn + k td )Y 0 + k fp ([Y 1 + Q1 ] - Y 1 )}Y 1

0 Moment equations for total polymer concentration 0 100 200 300 400 500 600 700 800 900 d(V[ +Q ]) Time (min) Y 0 0 =2 fk [I]+( k fm [M]+k fT [T])Y 0 Vdt d Fig. 1. Effect of initiator (AIBN) initial concentration on monomer 1 - k* Y ([ Y +Q ]-Y )- Y 2 conversion for the bulk free-radical homopolymerization of MMA, at p 0 1 1 1 2 ktcn 0 45 °C. Experimental data from Ito [20]. Molar concentrations as d(V[ Y1+Q1 ]) shown in the legend inside the plot. Solid lines are model predictions =2 fk [I]+( k fm [M]+k fT [T])Y 0 Vdt d using the SSH. Broken lines are model predictions without the SSH + [M] (transient model). k p Y 0 d(V[ Y 2+Q2 ]) =2 fk [I]+( k fm [M]+k fT [T])Y 0 Vdt d + k p [M]Y 0+2 k p [M]Y1 * 2 Hutchinson [6] developed a mathematical model of inter- +2 k pY1[ Y 2+Q2 ]+ktcwY1 mediate degree of complexity based upon the method of mo- Divinyl monomer ments. The model predicts the effect of branching on molecu- df  f − F  dx consumption 2 =  2 2  lar weight development (molecular weight averages), includ- dt  1 − x  dt ing internal cyclization and diffusion-controlled termination. The model was validated using experimental data from the li- Accumulated copolymer composition terature for the copolymerization of methyl methacrylate f 20 - f 2 (1- x) F 2 = (MMA) / ethylene glycol dimethacrylate (EGDMA). x A similar model was developed by Vivaldo-Lima et al. [2], but it was validated using experimental data for the Crosslink density copolymerization of styrene / divinylbenzene. The main di- *0 ρ d[xρ(x)] k p [2F 2(x)(1(1–- kcp)k cp )- (x)(1(1++kkcscs ))]x dx fferences with the model of Hutchinson [6] are related to the = dt (k1–p (1x)- x) dt calculation of cyclization, the equations used for diffusion- controlled reactions, and the fact of using different averages Transfer to small molecule of the termination kinetic rate constant. The predictive power d(V[T ]i ) * of the model of Vivaldo-Lima et al. [2] was demonstrated by = - k fTi [T i ][R ] Vdt using of the model to design a reaction recipe for a suspension copolymerization of styrene / divinylbenzene outside the ran- Temperature ∆ ges of variation of the operating variables used in the parame- dT (- H ) R p nw Cp UA = r - w dT w - (T - ) ter estimation stage, and then running the experiments at the T w dt nm Cp m nm Cp m dt nm Cpm designed conditions [7]. Zhu and Hamielec [8] experimentally proved that “isothermal” ampoule copolymerizations of MMA / EGDMA pre- / divinylbenzene, their maximum temperatures at the center of sent severe thermal effects, which manifest as the develop- the ampoules were in the same order of magnitude as the ones ment of temperature profiles with a maximum of up to 20 deg measured by Zhu and Hamielec for copolymerization of Celsius above the controlled (wall) temperature, at the center MMA / EGDMA, at similar concentrations of crosslinker of the ampoules. Vivaldo-Lima et al. [2] modeled these ther- (EGDMA) [8]. mal effects coupling an energy balance to their kinetic model, In this paper, the mathematical model of Vivaldo-Lima et and studied its effect on the copolymerization of styrene / al. [2] is used to model the copolymerization of MMA / divinylbenzene. Although there were no experimental data on EGDMA, using experimental data from the literature to vali- temperature profiles in ampoule copolymerizations of styrene date the model for this application. Thermal effects are consi- 24 Rev. Soc. Quím. Méx. Vol. 47, Núm. 1 (2003) Eduardo Vivaldo-Lima et al.

Table 2. Diffusion-Controlled Equations. · · R m,n,3 + M1 R m + 1,n,1 (k31) · · Initiator Efficiency R m,n,3 + M2 R m,n + 1,2 (k32) -D( 1 - 1 ) f = f 0 e V f V f 0 Crosslinking through pendant double bonds Propagation · * · * R m,n,1 + P r,s R m+r,s+n,3 (kp 13) · * · * o -B( 1 - 1 ) R m,n,2 + P r,s R m+r,s+n,3 (kp 23) V V k p = k e f f cr 2 ij R· + P* R· (kp* ) pij m,n,3 r,s m+r,s+n,3 33 Transfer to monomer Translational Termination · · R m,n,1 + M1 Pm,n + R 1,0,1 (kfm11) R· + M P + R· (kfm ) o -[A( 1 - 1 )] m,n,1 2 m,n 0,1,2 12 k tcn = k e V f V f 0 + k tcrd · · ij tcn R m,n,2 + M1 Pm,n + R 1,0,1 (kfm21) ij · · R m,n,2 + M2 Pm,n + R 0,1,2 (kfm22) x o 1 1 R· + M P + R· (kfm ) = k []P n 2 -[A( - )] + m,n,3 1 m,n 1,0,1 31 k tcwij e V f V f 0 k tcrd P w · · tcwij R m,n,3 + M2 Pm,n + R 0,1,2 (kfm32)

Reaction-Diffusion Termination Tranfer to small molecules o R· + T P + T· (kf ) = C (1 - x) m,n,1 m,n T1 k tcrd k p pse · · rd R m,n,2 + T Pm,n + T (kfT2) · · Fractional Free-Volume R m,n,3 + T Pm,n + T (kfT3) N V V = ∑ [0.025 +α (T - )] iVi] i = monomer 1, monomer 2, f i T g i VV Transfer to polymer i=1 t f · · R m,n,1 + Pr,s Pm,n + R r,s,1 (kfp11) polymer, solvent, CTA · · R m,n,1 + Pr,s Pm,n + R r,s,2 (kfp12) · · R m,n,2 + Pr,s Pm,n + R r,s,1 (kfp21) · · R m,n,2+ Pr,s Pm,n + R r,s,2 (kfp22) dered by coupling an energy balance to the kinetic and · · R m,n,3 + Pr,s Pm,n + R r,s,1 (kfp31) moment equations. · · R m,n,3 + Pr,s Pm,n + R r,s,2 (kfp32)

Transfer to initiator Highlights about the mathematical model · · R m,n,1 + I Pm,n + R in (kfi1) · · R m,n,2 + I Pm,n + R in (kfi2) The reaction mechanism for free-radical copolymerization R· + I P + R· (kf ) kinetics with crosslinking considered in this paper includes m,n,3 m,n in i3 the following reactions: chemical initiation, inhibition, propa- Termination gation, transfer to monomer, transfer to small molecules, · · R m,n,1 + R r,s,1 Pm,n + Pr,s (ktd11) transfer to polymer, transfer to initiator, crosslinking through P (k ) radical attack to pendant double bonds, and radical termina- m+r,n+s tc11 tion by combination and disproportionation. The reaction · · R m,n,1 + R r,s,2 Pm,n + Pr,s (ktd12) scheme is shown below. Symbols are explained in the nomen- P (k ) clature. m+r,n+s tc12 R· + R· P + P (k ) Inititation m,n,2 r,s,2 m,n r,s td22 · Pm+r,n+s (ktc22) I2 Rin (kd) · · R in + M1 R 1,0,1 (k1) · · · · R m,n,1 + R r,s,3 Pm,n + Pr,s (ktd13) R in + M2 R 0,1,2 (k2) Pm+r,n+s (ktc13) Inhibition · · · R m,n,2 + R r,s,3 Pm,n + Pr,s (ktd23) R m,n,1 + Z Pm,n (kz1) · Pm + Pr,n+s (ktc23) R m,n,2 + Z Pm,n (kz2) · R m,n,3 + Z Pm,n (kz3) · · R m,n,3 + R r,s,3 Pm,n + Pr,s (ktd33) P + P (k ) Propagation m m+r,n+s tc33 · · R m,n,1 + M1 R m + 1,n,1 (k11) · · The mathematical model developed by Vivaldo-Lima et R m,n,1 + M2 R m,n + 1,2 (k12) · · al. [2] is based on the Tobita-Hamielec [5] model for R m,n,2 + M1 R m + 1,n,1 (k21) · · crosslinking kinetics for the pre-gelation period, an improved R m,n,2 + M2 R m,n + 1,2 (k22) version of the Marten-Hamielec model for diffusion-contro- Modeling of the Free-Radical Copolymerization Kinetics with Crosslinking of Methyl Methacrylate... 25

0.8

0.6 0.6

[Io] = 0.01548 M [Io]=0.01548 M

Conversion 0.4 Conversion 0.4 [Io]=0.02018 M [Io]=0.0258 M

[Io]=0.05 M Non-SSH 0.2 0.2 [Io]=0.0258 M

0 0 0 100 200 300 400 500 600 0 20 40 60 80 100 120 140 160 180 Time (min) Time (min) Fig. 2. Effect of initiator (AIBN) initial concentration on monomer Fig. 4. Effect of initiator (AIBN) initial concentration on monomer conversion for the bulk free-radical homopolymerization of MMA, at conversion for the bulk free-radical homopolymerization of MMA, at 50 °C. Experimental data from Balke and Hamielec [21]. Molar con- 70 °C. Experimental data from Balke and Hamielec [21]. Molar concentrations of AIBN are shown in the legend inside the plot. Solid centrations of AIBN are shown in the legend inside the plot. Solid lines are model predictions using the SSH. Broken lines are model lines are model predictions using the SSH. Broken lines are model predictions without the SSH (transient model). predictions without the SSH (transient model).

0.9 10000000 0.8 Mn Mw 0.7 Non-SSH 0.6 1000000

0.5 [Io]=0.1 M 0.4 Conversion Mn, Mw 0.3 Non-SSH 100000

0.2

0.1

0 10000 0 20 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (min) Conversion

Fig. 3. Free-radical homopolymerization of MMA, at 60 °C, and Fig. 5. Molecular weight development on the bulk free-radical [AIBN]0 = 0.1 M. Experimental data of conversion versus time from homopolymerization of MMA, at 70 °C, and [AIBN]0 = 0.0258 M. Carswell et al. [22]. The solid line is the prediction of the model Experimental data from Balke and Hamielec [21]. Solid lines are using the SSH. The broken line is the prediction of the model without model predictions using the SSH. Broken lines are model predictions using the SSH (transient model). without the SSH (transient model). lled kinetics in free-radical polymerization [9] (which incor- The kinetic scheme can be treated as if it was a homopo- porates the recommendations of Zhu and Hamielec [10] on the lymerization by making use of the “pseudo-kinetic rate cons- use of different number and weight average termination con- tants method”, developed by Hamielec and MacGregor [11]. stants), and a simple phenomenological approach for the ter- The method of moments is used to follow the molecular mination kinetic constant during the post-gelation period weight development. Initiation, propagation and termination (although simple, this approach takes into account the unequal reactions are considered to be diffusion-controlled, and are reactivity of vinyl groups and cyclization reactions). Some modelled using a free-volume theory from the start of the features and characteristics of the kinetic model are described polymerization. Two averages, number- and weight-average below. termination rate constants, are used to model the mechanism The model consists of a set of ordinary differential and of bimolecular termination. The number average termination algebraic equations that describe the most important reactions kinetic rate constant, ktn, is used to calculate polymerization that take place during the copolymerization, according to the rate and number average molecular weight. The weight avera- reaction scheme shown above. These equations are listed in ge termination kinetic rate constant, ktw, is used to calculate Table 1. The most important equations dealing with diffusion- the weight average molecular weight. These averages depend controlled reactions, pseudo-homopolymer approach, and on polydispersity and conversion, and are defined in such a behaviour during the post-gelation period are listed in Tables way that no additional parameters are needed in the model. 2, 3, and 4, respectively. 26 Rev. Soc. Quím. Méx. Vol. 47, Núm. 1 (2003) Eduardo Vivaldo-Lima et al.

All diffusion-controlled reactions are modelled using a

Polimerizacion en masa de MMA iniciada por AIBN “series” structure for the effective kinetic rate constants, as

1 opposed to the rather common “parallel” approach. The diffe- Balke y Hamielec,1973 rences between these two modelling approaches are explained T= 90 °C 0.8 in detail in Vivaldo-Lima et al. [9]. The model equations can be solved using the steady-state-hypothesis (SSH) for polymer 0.6 radicals, but this is reliable only during the pre-gelation period. [Io]=0.01548 mol AIBN/ L 0.4 Conversión Cyclization reactions are modelled using the equations [Io]=0.0258 mol AIBN/ L proposed by Tobita and Hamielec [5], although only average 0.2 Transiente cyclization densities are calculated, instead of the full density Tiem po m in distributions. Likewise, only the average crosslinking density 0 0 5 10 15 20 25 30 35 40 as function of time is calculated. Tobita and Hamielec [5] ge- Fig. 6. Effect of initiator (AIBN) initial concentration on monomer neralized Flory’s theory for the post-gelation period by using conversion for the bulk free-radical homopolymerization of MMA, at a crosslinking density distribution. Instead, in this paper the 90 °C. Experimental data from Balke and Hamielec [21]. Molar con- original Flory-Stockmayer equation for calculation of the sol centrations of AIBN are shown in the legend inside the plot. Solid fraction is used, but his simplifying assumptions regarding lines are model predictions using the SSH. Broken lines are model equal reactivity of double bonds, absence of cyclization and predictions without the SSH (transient model). independence of double bonds were removed.

Table 3. Pseudo-Kinetic Rate Constants (Pseudo-Homopolymer Results and discussion Approach). Propagation 3 2 Parameter Estimation Strategy φ * k p = ∑∑ k ij i f j Whenever possible, kinetic rate constants were obtained from i=1 j =1 references where accepted experimental techniques were used Crosslinking (e.g., the “Pulsed Laser Polymerization”, PLP, to measure 3 * * * propagation kinetic rate constants). In some cases, values esti- φ 2 ρ ρ k p = ∑ k pi3 i( F - a - c ) i =1 mated within the frame of a given model were used, provided Inhibition that a clear description of the estimation procedure was avai- 3 φ * lable. For the situations where no experimental or reliable k z = ∑ k zi i i =1 estimates were available, an accepted estimation procedure was used: the “error in variables method”, EVM, which is a Transfer to Monomer 3 2 weighted non-linear multivariable regression procedure. More φ * k fm = ∑∑ k fmij i f j details on the use of this method are provided elsewhere i =1 j =1 [2,12]. Tables 5 and 6 provide a summary of the kinetic rate constants and other physical parameters, respectively. Transfer to a Small Molecule 3 φ * k fT = ∑ k fT i i Homopolymerization of MMA i =1 The performance of the model was first tested using experi- Transfer to Polymer mental data from the literature for the free-radical homopoly- 3 2 φ * merization of MMA. An extensive study using 2,2’-azobis k fp = ∑∑k fpij i F j i =1 j =1 (isobutyronitrile) (AIBN) and benzoyl peroxide (BPO) as ini- tiators (concentrations ranging from 0.006 to 0.042 mole / L), Transfer to Initiator several temperatures: 45, 50, 60, 60, 70, and 90 °C, and bulk 3 φ * and solution (in benzene) processes, was carried out. Experi- k fI = ∑k fI i i i =1 mental data from different laboratories [16-24] were reproduced with the kinetic model described above, using zero as Termination by Disproportionation the initial concentration of crosslinker. The agreement bet- 3 3 φ *φ * ween model predictions and experimental data is very good. ktd = ∑∑ktdij i j i=1 j=1 Fig. 1 shows a comparison of experimental data of Ito Termination by Combination [20] versus predicted results for monomer conversion of 3 3 MMA at 45 °C. As observed, the agreement is very good over φ *φ * k tc = ∑∑k tc ij i j all the range of AIBN initial concentrations. Similar results i=1 j =1 are observed in Figs. 2, 3, and 4, but at temperatures of 50, 60 and 70 °C, respectively. Experimental data of Figs. 2 and 4 Modeling of the Free-Radical Copolymerization Kinetics with Crosslinking of Methyl Methacrylate... 27

Copolimerización de MMA/EGDMA a 70°C 0.3%peso de EGDMA iniciada por Pesos moleculares Mw y Mn 1.00E+07 AIBN Polimerización en masa de MMA a 90°C 1 [Io]=0.0258 mol/L AIBN

1.00E+06 0.8

QSSH 0.6 Transiente 1.00E+05 x Tobita 1990 Mn Mw Mn 0.4

Mn 1.00E+04 0.2 Mw Transiente 0 1.00E+03 0 20 40 60 80t (min) 100 120 140 0 0.2 0.4x 0 .6 0.8 1

Fig. 7. Molecular weight development on the bulk free-radical Fig. 10. Total monomer conversion of the free-radical copolymeriza- homopolymerization of MMA, at 90 °C, and [AIBN]0 = 0.0258 M. tion of MMA / EGDMA, at 70 °C, [AIBN]0 = 0.01548 M, and Experimental data from Balke and Hamielec [21]. Solid lines are [EGDMA]0 = 0.3 wt. %. Experimental data from Tobita [25]. The model predictions using the SSH. Broken lines are model predictions solid line is the model prediction using the SSH. The broken line was without the SSH (transient model). obtained without using the SSH (transient model).

Concentración de Radicales en la homopolimerizac ión de MMA a 70°C iniciada por AIBN 0.3%w t 1.0E-02 Copolimerización de MMA/EGDMA a 70°C 0.5%peso de EGDMA 1

1.0E-03 0.8 1.0E-04

0.6 1.0E-05 SSH QSSH x

[R*] mol/l [R*] Transiente 1.0E-06 Transiente 0.4 Tobita 1990 Zhu et al. 1990a (exp) 1.0E-07 0.2

1.0E-08 0 0 20 40 60 80 100 120 140 t (min) 0 20 40 60 80t (min) 100 120 140

Fig. 8. Evolution of total radical concentration on the bulk free-radi- Fig. 11. Total monomer conversion of the free-radical copolymeriza- cal homopolymerization of MMA, at 70 °C, and [AIBN]0 = 0.3 wt. tion of MMA/EGDMA, at 70 °C, [AIBN]0 = 0.01548 M, and %. Experimental data from Zhu et al. [23]. Solid lines are model pre- [EGDMA]0 = 0.5 wt. %. Experimental data from Tobita [25]. The dictions using the SSH. Broken lines are model predictions without solid line is the model prediction using the SSH. The broken line was the SSH (transient model). obtained without using the SSH (transient model).

Polim erización en solución de MMA en benceno T=70°C Copolimerización de MMA/EGDMA a 70°C 1.0% peso de EGDMA 1 1

0.8 0.8

0.6 0.6 QSSH Conversión x 0% vol Benceno Transiente 0.4 0.4 20%vol Benceno Tobita 1990 40%vol Benceno 60%vol Benceno 0.2 0.2 80%vol Benceno 90%vol Benceno Tiem po m in 0 0 0 100 200 300 400 0 20 40 60t (min) 80 100 120 140

Fig. 9. Effect of benzene concentration of the solution free-radical Fig. 12. Total monomer conversion of the free-radical copolymeriza- homopolymerization of MMA, at 70 °C, and [AIBN]0 = 0.0413 M. tion of MMA / EGDMA, at 70 °C, [AIBN]0 = 0.01548 M, and Experimental data from Schulz and Harborth [24]. Solid lines are [EGDMA]0 = 1.0 wt. %. Experimental data from Tobita [25]. The model predictions using the SSH. solid line is the model prediction using the SSH. The broken line was obtained without using the SSH (transient model). 28 Rev. Soc. Quím. Méx. Vol. 47, Núm. 1 (2003) Eduardo Vivaldo-Lima et al.

no fine tuning of the parameters was needed to reproduce the 1 experimental data. So far, all the cases shown here have been polymeriza- 0.8 tions carried out in bulk. Fig. 9 shows the performance of the model for polymerizations carried out in solution. The solvent

0.6 used was benzene. The experimental data are from Schulz and 1%EGDMMA (Tobita) Harborth [24]. The agreement is very good, and once again, 5%EDMMA (Tobita) no fine-tuning of parameters was needed. 0.4 15%EGDMMA (Tobita) From Figs. 1 to 9 it is observed that our model is very Overall Conversion 1%EGDMMA (non-isothermal) good for free-radical homopolymerizations, under a large 0.2 5%EGDMMA (Non-isothermal) range of temperatures, initiator initial concentrations, and sol-

15%EGDMMA (Non-isothermal) vent concentrations. This agreement provided confidence on

0 the estimates of the kinetic rate constants for the homopoly- 0 20406080100merization situation. The only parameters that were estimated Time (min) using the error in variables method (EVM) for the homopoly- Fig. 13. Effect of crosslinker initial concentration (intermediate merization case were A and B (free-volume parameters for range) on total monomer conversion for the free-radical copolyme- diffusion-controlled reactions), and ktn, the overall number rization of MMA / EGDMA, at 70 °C, and [AIBN]0 = 0.01548 M. average termination kinetic rate constant. Experimental data from Tobita [25]. Solid lines are model predictions using the SSH. Broken lines are model predictions without the SSH (transient model). Copolymerization of MMA / EGDMA

With exception of the parameters mentioned before for homopolymerization of MMA, the only estimated parameters are form Balke and Hamielec [21], and those of Fig. 3 from for the copolymerization situation were the crosslinking kine- Carswell et al. [22]. tic rate constants (k*ij). The main assumptions regarding the Fig. 4 shows a comparison of experimental data from values of some of the kinetic parameters are summarized in Balke and Hamielec [21] and model predictions for number Table 5. and weight average molecular weights at 70 °C, and [AIBN]0 = 0.0258 M. The solid lines are model predictions using the Table 4. Key Equation During the Post-Gelation Period. SSH, whereas the broken ones are model predictions without Sol Fraction the SSH. The agreement is very good, except at very high (τ+β )(1-ρ ) -( τ +β )  wg e β τ β ρ τ β ρ conversions where model predictions of M drop more rapidly ws (x) = 3 ( + )[ ln(1 - wg )-( + - 1] + [ ln(1 - wg ) n [ ln(1-ρ )-(τ+β )]  than the experimental values. Similar results were obtained at wg  2 β the other temperatures and initiator initial concentrations of - (τ + β )]τ - [ ln(1 - ρ w )-( τ + β )] [τ + (τ + β )] g 2  Figs. 1 to 4. The number average molecular weight in a free-radical Number Average Chain Length of Primary Molecules polymerization is determined by the ratio of rate of propaga- sol 2P (x) tion to rate of radical termination. Therefore, this ratio will sol np depend on the ratio of k [M] / (sqrt(k *2*f*k *[I])). The slow P (x)= p t d n sol increase in M is caused by the auto-acceleration effect (the 2 – ρ sol (x)P (x) n np decrease in kt), but at high conversions the propagation reaction also becomes diffusion-controlled (kp decreases). The Weight Average Chain Length of Primary Molecules sharp decrease in M at very high conversions, shown in Fig. n sol 5, may be caused by an exaggerated diffusion-controlled P (x) sol wp effect on kp, or a weak diffusion-controlled effect on the initia- P (x)= w sol tion reaction (a not enough decrease of f). 1 – ρ sol (x)P (x) Figs. 6 and 7 show a comparison of experimental data by wp Balke and Hamielec [21] and model predictions, at 90 °C. The agreement is again very good, as in the previous cases at Radical Termination (assumed dominated lower temperatures. Fig. 8 shows a comparison of experimen- by reaction-diffusion termination) tal data and model predictions of total radical concentration  gp 0 x  ≈ C 3 x (x )C 3 versus time for the bulk free-radical homopolymerization of k t k tc rd = C (x w sol ) + C w gel k p pse (1 - x)  rd rd  MMA, at 70 °C, and [AIBN]0 = 0.3 wt. %. The experimental data were obtained by Zhu et al. [23] using electro spin reso- gp ktw | nance (ESR). The agreement is very good, which shows some C = pse xgp rd kp | (1- xgp) of the predictive capabilities of the model, given the fact that pse xgp Modeling of the Free-Radical Copolymerization Kinetics with Crosslinking of Methyl Methacrylate... 29

Fracción de gel a través de la copolimerización de MMA /EGDMA a 70°C 1.0% Conversión en el punt o de gelac ión (Xg) y su peso de EGDMA

depende ncia de los niveles de EGDMA 1 0.3

0.8 Exp. Li et al. 1989 0.2 0.6 Xg Wg

0.4 0.1 Wg QSSH Wg Transiente Wg Tobita, 1990 0.2 0 0 0.2 0.4 0.6 0.8 1 0 % peso EGDMA 0 50 100 150 tiempo (min)

Fig. 14. Effect of crosslinker initial concentration on the gelation Fig. 17. Gel formation in the free-radical copolymerization of MMA / point for the free-radical copolymerization of MMA / EGDMA, at 70 EGDMA, at 70 °C, [EGDMA]0 = 1.0 wt. %, and [AIBN]0 = 0.01548 °C, and [AIBN]0 = 0.01548 M. Comparison of experimental data (Li M. Comparison of experimental data (Tobita [25]), and model predic- et al. [26]), and model predictions (solid line). tions (solid line).

Fracción de gel a través de la copolimerización de MMA /EGDMA a 70°C 0.3% peso de EGDMA

1 Concentración de Radicales en la copolimerización de MMA/EGDMA a 70°C 0.3 % peso de EGDMA 1.0E-02 0.8 1.0E-03

0.6 1.0E-04 Wg 0.4 Wg QSSH 1.0E-05 Wg Transiente [R*] m ol/l Wg Tobita, 1990 1.0E-06 0.2 QSSH 1.0E-07 Transiente 0 [R*] Zhu et al. 1990a (exp) 1.0E-08 0 50 100 150 t (m in) 0 20 40 t (m in) 60 80 100

Fig. 15. Gel formation in the free-radical copolymerization of MMA / Fig. 18. Evolution of total radical concentration on the bulk free-radi- EGDMA, at 70 °C, [EGDMA]0 = 0.3 wt. %, and [AIBN]0 = 0.01548 cal copolymerization of MMA / EGDMA, at 70 °C, [EGDMA]0 = 0.3 M. Comparison of experimental data (Tobita [25]), and model predic- wt. %, and [AIBN]0 = 0.01548 M. Experimental data from Zhu et al. tions (solid line). [23]. Solid lines are model predictions using the SSH. Broken lines are model predictions without the SSH (transient model).

Fracción de gel a través de la copolimerización de MMA /EGDMA a 70°C 0.5% Concentración de Radicales en la copolimerización de MMA/EGDMA a 70°C peso de EGDMA 1.0 % peso de EGDMA 1.0E-02 1

1.0E-03 0.8 1.0E-04 0.6 1.0E-05 Wg

0.4 Wg QSSH [R*] m ol/l 1.0E-06 Wg Transiente QSSH Wg Tobita, 1990 0.2 1.0E-07 Transiente Zhu et al.(1990a) 0 1.0E-08 0 50 t(min) 100 150 0 20 40 t (m in) 60 80 100 Fig. 16. Gel formation in the free-radical copolymerization of MMA / Fig. 19. Evolution of total radical concentration on the bulk free-radi- EGDMA, at 70 °C, [EGDMA]0 = 0.5 wt. %, and [AIBN]0 = 0.01548 cal copolymerization of MMA / EGDMA, at 70 °C, [EGDMA]0 = 1.0 M. Comparison of experimental data (Tobita [25]), and model predic- wt. %, and [AIBN]0 = 0.01548 M. Experimental data from Zhu et al. tions (solid line). [23]. Solid lines are model predictions using the SSH. Broken lines are model predictions without the SSH (transient model). 30 Rev. Soc. Quím. Méx. Vol. 47, Núm. 1 (2003) Eduardo Vivaldo-Lima et al.

Table 5. Estimates of the kinetic rate constants. Parameter Value or functionality Remarks

–1 × 15 – 15460 kd, s 1.0533 10 exp[ T (K) ] For AIBN [13] 1.25 × 10–5 (@ 70 °C) For BPO [13] –1 –1 × 5 –2189.23 k11, L mol s 4.9167 10 exp[ T (° C) ] [14] –1 –1 k22, L mol s 2 k11 [6] r12 (= k11 / k12) 0.674 ± 0.045 [15] r21 (= k22 / k21) 1.34 ± 0.18 [15] –1 –1 k*13 = k*23, L mol s 415 (@ f =[0.3-0.5], T = 70 °C) Best fit, this paper 380 (@ f = 1.0, T = 70 °C) –1 –1 k*33, L mol s 0.0 Neglected –1 –1 k31 = k32, L mol s 200 (@ 70 °C) Set to a reasonable value, this paper –1 –1 kz1 = kz2 = kz3, L mol s 0 Not used (no inhibitor present) –1 –1 kfm11, L mol s kfm11 –2853.5 [6] = 0.0741 exp[ T(K) ] k11 –1 –1 kfm22, L mol s kfm11 [6] –1 –1 kfmij, L mol s kfmii / rij [6] (crossed terms) –1 –1 kfT, L mol s 0 For Benzene –1 –1 kfp, L mol s 0 Neglected –1 –1 kfI, L mol s 0 For AIBN (neglected for BPO) –1 –1 × 10 – 15460 ktn11 (= ktcn11 + ktdn11), L mol s 2.8926 10 exp[ T (K) ] EVM for ktdn11, (approximate 95 % confidence × 3 – 15460 2.43 10 exp[ T (K) ] limits around 5 % of the base values), this paper ktdn11 / ktcn11 [6], [16] –1 –1 ktn22 (= ktcn22 + ktdn22), L mol s ktn11 [6], [2] ktdn22 / ktcn22 ktdn11 / ktcn11 [6] –1 –1 ktnij = ktnji, L mol s [2], [6] ktnij ktnji

Figures 10 to 12 show a comparison of experimental data OD) is not constant. From the three solid lines, the one show- of Tobita [25] and model predictions for the bulk free-radical ing the slower polymerization rate at conversions higher than copolymerization of MMA / EGDMA at low concentrations 0.5, corresponds to 1 wt. % of EGDMA. The line at an inter- of EGDMA: 0.3, 0.5 and 1 wt. %, respectively. The solid lines mediate polymerization rate corresponds to 5 wt. % of are model predictions using the SSH, and the broken lines are EGDMA, an the one showing the fastest polymerization rate without the SSH. At 0.3 and 0.5 there is not significant diffe- corresponds to 15 wt. % of EGDMA. rence between using or not the SSH, although the non-SSH The qualitative behavior observed with the isothermal simulations lie closer to the experimental data. At 1 wt. % of simulations is adequate; namely, polymerization rate increas- EGDMA, however, there is a larger difference between the es with crosslinker initial concentration. The agreement simulations with and without using the SSH, and this time the between experimental data and isothermal model predictions SSH simulations lie closer to the experimental data. In the is very good at low and intermediate conversions, and accept- three cases the SSH simulations lie above the non-SSH ones. able at higher conversions, for the case of 1 wt. % of This behavior is explained by the fact that the model parame- EGDMA. However, in the cases of 5 and 15 wt. % of ters estimated for MMA homopolymerization were obtained EGDMA, severe discrepancies are observed. Zhu and Hamie- using a regression program with the SSH model implemented lec [8] experimentally demonstrated that homopolymeriza- on it. tions of MMA and copolymerizations of MMA / EGDMA Fig. 13 shows a comparison of experimental data of carried out in glass ampoules from 0.5 to 1 mm OD in con- Tobita [25] versus non-SSH model predictions for overall trolled temperature bath circulators present severe thermal monomer conversion. The isolated circles, squares and trian- effects, manifested by increases in temperature along the cen- gles are experimental data at 1, 5 and 15 wt. % of EGDMA, tral lines of the ampoules of about 20 °C higher than the con- respectively. Also shown in the plot are two sets of three lines. trolled temperature at the walls of the ampoules. In the actual The three lines of one set are solid lines, and the other three system, temperature profiles along the radial direction of the lines of the second set are small symbols along solid lines. ampoules are developed, with a maximum at the center of the The solid lines are model predictions assuming that the ampoule. This time and space-dependent, non-isothermal process is isothermal. The lines with small symbols along behavior causes the rate of polymerization to increase, and them are model predictions assuming that temperature along gelation to occur sooner. the central line of the ampoules used by Tobita [25] (0.5 mm Modeling of the Free-Radical Copolymerization Kinetics with Crosslinking of Methyl Methacrylate... 31

Table 6. Cyclization and free-volume parameters. Longitud de cadena promedio en número y peso a través de la Parameter Value or Remarks copolimerización de MMA/EGDMA a 70°C y 0.3% peso de EGDMA functionality 1.0E+06 Pn QSSH kcp, dimensionless 0.04 [17] 1.0E+05 Pn Transiente kcs, dimensionless 0.5 [6] Pw QSSH A, dimensionless 1.45 ± 0.407 EVM, this paper Pw transiente B, dimensionless 0.7 ± 0.22 EVM, this paper 1.0E+04 D, dimensionless 0.001 [2] Pn Pw ln (V ) = –0.839 – 639 Vfcr2, dimensionless fcr2 T(K) [18] 1.0E+03 –1 Crd, L mol 1.117 [19] f, dimensionless 0.6 For AIBN, [18] Tiempo (min) 1.0E+02 0.7 For BPO, [2] 0 20 40 60 80 Tg, 0K 167.15 For MMA, [19] Tg, 0K 387.15 For PMMA, [13] Fig. 20. Molecular weight development on the bulk free-radical Tg, 0K 167.15 For EGDMA, [6] copolymerization of MMA / EGDMA, at 70 °C, [EGDMA]0 = 0.3 wt. %, and [AIBN] = 0.01548 M. Solid lines are model predictions Tg, 0K 414.15 For PEGDMA, [6] 0 using the SSH. Broken lines are model predictions without the SSH Tg, 0K 171.15 For Benzene, [13] (transient model). Tg, 0K 173.15 For BPO, [2] α, 0K–1 0.001 For MMA, [6] α, 0K–1 4.8 × 10–4 For PMMA, [6] Longitud de cadena promedio en número y peso a través de la copolimerización de MMA/EGDMA a 70°C y 1.0% peso de EGDMA α, 0K–1 0.001 For EGDMA, [6] 1.0E+07 α 0 –1 × –4 , K 4.8 10 For PEGDMA, [6] Pn QSSH α 0 –1 1.0E+06 , K 0.001 For Benzene, [13] Pn Transiente Pw QSSH 1.0E+05 Pw Transiente

To model that non-isothermal behaviour, an energy ba- Pn Pw 1.0E+04 lance was coupled to the mass conservation equations (last row of Table 1). That energy balance assumes that the reac- 1.0E+03 tion takes place in a stirred tank reactor, with a jacket cooled with water at 20 °C. Temperature and concentrations are 1.0E+02 assumed homogeneous inside the tank reactor. It is also 0 20406080x assumed that the maximum heat removal capacity of the cooling system is given by the maximum heat generation in a free- Fig. 21. Molecular weight development on the bulk free-radical copolymerization of MMA / EGDMA, at 70 °C, [EGDMA] = 1.0 radical homopolymerization of MMA. When this maximum 0 wt. %, and [AIBN]0 = 0.01548 M. Solid lines are model predictions heat removal capacity is reached in the copolymerization case, using the SSH. Broken lines are model predictions without the SSH temperature starts increasing. (transient model). The lines with small symbols are simulations with the previously explained energy balance incorporated into the mathematical model. These lines were obtained using a value Figures 15 to 17 show predicted calculations versus expe- of UA = 0; namely, adiabatic polymerization. Although there rimental data of weight fraction of gel, as function of time, at is a significant improvement over the isothermal simulations, initial values of 0.3, 0.5, and 1 wt. % of crosslinker, respecti- the agreement with experimental data is not good enough. In vely. The agreement is good, although the model predicts a the cases of 5 and 15 wt. % of EGDMA, the non-isothermal faster gel formation than those experimentally observed. The simulations show a quite faster polymerization rate, moving non-SSH simulations show a better agreement than the SSH towards the experimental data, but not reaching them. For 1 ones, although both can be considered good approximations to wt. % of EGDMA, and for smaller concentrations of the experimental data. EGDMA, the isothermal simulations are good enough. The evolution of total radical concentration with time for Fig. 14 shows a good agreement between predicted and copolymerizations started with 0.3 and 1 wt. % of EGDMA, is experimental data for gelation point, as a function of crosslin- shown if Figs. 18 and 19. It is observed that the total radical ker initial concentration. It is observed that the higher the concentration increases several orders of magnitude from the crosslinker concentration, the sooner gelation takes place. value at the onset of gelation until the end of the polymeriza- This is reasonable, since more pendant double bonds accessi- tion. The model captures quite nicely this behavior, which is ble to crosslinking are present when the amount of EGDMA is manifested by the good agreement between model simulations higher. and experimental data. 32 Rev. Soc. Quím. Méx. Vol. 47, Núm. 1 (2003) Eduardo Vivaldo-Lima et al.

Finally, Figs. 20 and 21 show model predictions of num- C1 Empirical parameter to account for initiator efficiency ber and weight average chain length versus time for the decrease during the post-gelation period. copolimerizations started with 0.3 and 1 wt. % of EGDMA, C2 Empirical parameter to account for propagation kinetic respectively. Unfortunately there were no experimental data constant decrease during the post-gelation period. available for molecular weight development for this system, C3 Empirical parameter to account for termination kinetic but the profiles show the typical features of a crosslinking sys- constant decrease during the post-gelation period. tem. Weight average chain length increases exponentially up Cp Heat capacity, J / K. to the gelation point, and afterwards, the weight average chain C°rd Parameter for reaction-diffusion termination. length of the sol fraction decreases progressively due to sol D Effectiveness factor to account for overlap of free-volume consumption by the gel phase. and separation of fragment-radical molecules. f, f0 Initiator efficiency (superscript “0” accounts for initial conditions). Concluding remarks f2, f20 Divinyl monomer molar fraction (also shown as f 0DVB). The mathematical model developed by Vivaldo-Lima et al. F2 Instantaneous relative composition of monomer 2 in the [2], which was originally validated for copolymerization of polymer (accumulated composition if shown with an over- styrene / divinylbenzene, was proved to effectively explain line). ∆ and reproduce experimental data for the copolymerization of (– H)r Heat of reaction, J / mol. MMA / EGDMA. Model predictions at low crosslinker con- [I] Initiator concentration, mol / L. centrations are reliable, although its behavior at high cross- kij Effective (diffusion-controlled) propagation kinetic con- linker concentrations, despite still being qualitatively correct, stant for radical type i (i = 1, 2, or 3) and adding monomer –1 –1 becomes less reliable. unit j (j = 1, 2), L mol s . Also represented as kpij. It was demonstrated that heat effects are important in k*i3 Effective (diffusion-controlled) propagation kinetic con- copolymerizations of MMA/ EGDMA, even if the experi- stant for addition of a pendant double bond (macromono- ments are carried out in thin ampoules immersed in controlled mer) into a radical with end unit of monomer i, L mol–1 –1 temperature baths. A first approach to model these effects by s . Also represented as kp*i3. use of a simplified energy balance was presented in this paper. kcp Proportionality constant between primary cyclization den- Although there was significant improvement over the isother- sity and mole fraction of divinyl monomer bound in the mal model, the idealized situation is not yet a good enough polymer chains. representation of the actual reaction system in a glass ampoule kcs Proportionality constant between the average number of with time-dependent temperature profiles in the radial direc- secondary cycles per crosslink and the fraction of “free” tion. pendant double bonds in the primary polymer molecule. –1 Different experimental responses were simultaneously kd Initiator decomposition kinetic constant, s . and effectively reproduced with the mathematical model, kfI Pseudo-kinetic rate constant for chain transfer to initiator, L without having to perform extensive data fitting. Most of the mol–1 s–1. model parameters were taken from the literature, and extra kfm Pseudo-kinetic rate constant for chain transfer to care was taken to obtain the most reliable estimates of those monomer, L mol–1 s–1. parameters. Whenever possible reliable experimentally mea- kfp Pseudo-kinetic rate constant for chain transfer to polymer, sured values were used, and when not, an accepted statistical L mol–1 s–1. procedure was used (the “error in all variables model”, EVM). kfT Pseudo-kinetic rate constant for chain transfer to a small molecule T, L mol–1 s–1. kfTi Kinetic constant for chain transfer of radical type i (i = 1,2 Acknowledgements or 3) to a small molecule T, L mol–1 s–1. –1 –1 kp Pseudo-kinetic propagation rate constant, L mol s . Financial support from Consejo Nacional de Ciencia y kp* Pseudo-kinetic rate constant for crosslinking reaction, L Tecnología (CONACYT), through research Projects I 30027A mol–1 s–1. 0 and 31170-U, and Dirección General de Asuntos del Personal k tc Intrinsic chemical kinetic constant for termination by Académico (DGAPA) of UNAM, through Project PAPIIT combination, L mol–1 s–1. 0 IN120599, is gratefully acknowledged. k td Intrinsic chemical kinetic constant for termination by disproportionation, L mol–1 s–1. ktn Number average pseudo-kinetic rate constant for termina- Nomenclature tion, L mol–1 s–1. ktw Weight average pseudo-kinetic rate constant for termina- A Effectiveness factor to account for overlap of free-volume tion, L mol–1 s–1. and separation of reactive radicals (or heat transfer area, m2, m- Accounts for meta isomer. in Table 1). [M] Total monomer concentration, mol/L. Modeling of the Free-Radical Copolymerization Kinetics with Crosslinking of Methyl Methacrylate... 33

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