Iterative robust numerical procedure for the determination of kinetic constants in Han’s model for NR cured with sulphur

Abstract

A closed form procedure to determine kinetic constants for NR vulcanized with sulphur is presented. The kinetic scheme originally proposed by Han et al. (Polymer (Korea) 22:223–230, 1998) and further modified by Milani et al. (Polym Test 32:1052–1063, 2013) is adopted as starting point to deduce a closed form expression for rubber curing degree. Rheometer experimental data collected at different temperatures are used to tune model parameters. After the normalization of the rheometer curves and the exclusion of induction period from calculations, the model requires the estimation of three kinetic constants, two of them describing incipient curing and stable crosslinks formations, the last reproducing reversion phenomenon. Whilst such constants are almost always determined by least-squares best fitting, here a numerical iterative procedure—much more stable and efficient from a computational standpoint—is proposed. The approach requires as input parameters only the degree of vulcanization at infinite (i.e. at the end of vulcanization), the instant where the maximum torque is reached and initial rate of vulcanization. The condition that the numerical curve reaches a maximum at a given time translates mathematically into a non-linear equation in two of the kinetic constants, which are determined iteratively in the paper. The numerical initial vulcanization rate is tuned in such a way to globally minimize the absolute error between numerical and experimental curves. The main capability of the procedure proposed stands in the very straightforward determination of reaction kinetic constants, avoiding demanding least-squares fittings on rheometer experimental data. A set of experimental data available, relying into rheometer curves of the same rubber blend conducted at five different temperatures are used to estimate the fitting capabilities of the mathematical model proposed. Very good agreement with experimental data is observed.