Simplified closed form numerical approaches to predict natural rubber behavior under sulfur curing in standard rheometer tests

In this paper, a complex numerical and a simplified mathematical closed form approach with robust kinetic base are proposed to interpret NR sulfur vulcanization. A preliminary phase of preparation of meta-rheometer curves from few experimental data may be necessary to have at disposal the whole curve to fit with the model when experimental data at disposal are a few. Then, on such data (either experimental or artificially generated) partial reaction kinetic constants characterizing the vulcanization process within the models proposed are derived. When needed, meta-data are obtained using a direct C2 natural cubic spline interpolation of the rheometer curve, which proved to fit the experimental data well. Both the presence and absence of reversion are discussed and how they are reflected in the model calculations. The chemical schemes, translated mathematically into differential equations systems, are suitably re-arranged to derive single analytical equations, which represents the crosslinking degree evolution vs time. The parameters of the single equations may be determined setting the kinetic constants of the chemical model by means of best fitting in the first model (more complex) and with the direct solution of a non linear system of equations in the second (simplified) approach. The major improvement of the second procedure here proposed is to utilize some ad hoc values for the kinetic constants that do not necessarily require an optimization algorithm, thus by-passing the usage of a least squares minimization routine.