A Two‐Step Interval and Modified Rosenbrock Method for Kinetic Parameter Estimation in Copolymerization Systems: A Robust and Reliable Approach

Abstract

Summary: The kinetics of solution free radical copolymerization of isobutyl methacrylate (i‐BMA) and lauryl methacrylate (LMA) in benzene, initiated with 2,2‐azoisobutyronitrile (AIBN) were studied at different monomer feed compositions at low conversion levels. In order to avoid the complications of copolymerization kinetics, the pseudo‐kinetic rate constant method was applied in constant and variable volume polymerization systems. A two‐step procedure based on interval analysis and the modified Rosenbrock method was used to estimate the kinetic parameters of copolymerization. In the first step, initiation, coupled propagation‐termination and transfer rate parameters were determined from steady state kinetic equations using interval analysis. Since the objective function is non‐linear, non‐convex and has multiple local optima, a robust computational technique, based on the Interval Newton/Generalized Bisection (IN/GB) algorithm, was developed to solve this set of non‐linear algebraic equations. This method was used with mathematical and computational guarantees of certainty to find the global optimum. In the second step, the system of mole balance, population balance and moment equations, which are highly stiff ordinary differential equations, were discritized and solved by the modified Rosenbrock method. The results of the first computational step were inserted as an active or equality constraint in the second step to calculate the individual elementary rate parameters of the reaction. Statistical analysis indicated that the copolymer composition is well described by the terminal unit model (TUM), but the implicit penultimate unit effect (IPUE) model of Fukuda and coworkers is more suitable for describing the rate data. In contrast to most previously studied systems, it was found that propagation and coupled rate parameters are greater than those predicted by the TUM.Variation of number average molecular weight of the copolymer in polymerization system for various initial monomer feed compositions at different reaction times (solid lines are computed results).imageVariation of number average molecular weight of the copolymer in polymerization system for various initial monomer feed compositions at different reaction times (solid lines are computed results).

Kinetic study of methacrylate copolymerization systems by thermoanalysis methods

“…Also, the values of the coupled rate parameter, k p / k t 0.5 , at different monomer feed compositions were estimated and are shown in Figure 7. The solid line in this figure shows the prediction of the implicit penultimate unit model (IPUM) of Fukuda and coworkers44, 45 with the reactivity ratios previously reported by Habibi and Vasheghani‐Farahani 43…”

“…Because the induction period represents the required time for the consumption of primary radicals, its measurement provides a direct calculation of the in situ initiation rates. f was previously determined43 by the mean kinetic chain length approach and was equal to 0.62. It was assumed that the efficiency of initiation was constant and was not influenced by the presence of the inhibitor.…”

Kinetic study of an in situ inhibited polymerization using robust M-statistics

“…6a. The solid line in this figure is the prediction of Implicit Penultimate Unit Model (IPUM) of Fukuda et al (1985Fukuda et al ( , 1992 using the previously reported reactivity ratios by Habibi and Vasheghani-Farahani (2004). Also the estimated values of 2fk d at different monomer feed compositions are shown in Fig.…”

“…The initiator efficiency, f which was previously determined by the "Mean kinetic chain length" method (Habibi and Vasheghani-Farahani, 2004) is set to 0.62. It is assumed that the efficiency of initiation is constant and not influenced by the presence of inhibitor.…”

Bayesian Modeling and Markov Chain Monte Carlo Simulations for a Kinetic Study of Homo‐ and Co‐ Polymerization Systems

“…Based on the values of elementary rate constants, [67] a couple of MCMC simulations were performed. All elementary reactions (i.e., initiation, propagation, termination by combination, termination by disproportionation and transfer) were considered in the simulation.…”