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 “iso- thermal” 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

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