# Effective closed form mathematical approach to determine kinetic constants of NR vulcanized with sulphur and accelerators at different concentrations

## Abstract

The basic reaction scheme due to Han and co-workers for NR vulcanized with sulphur is adopted and modified taking into account the single contributions of the different accelerators, focusing in particular on some experimental data ad hoc obtained at Pirelli’s laboratories, where NR was vulcanized at different temperatures (from 150 to 180 °C) and concentrations of sulphur, using TBBS and DPG in the mixture as co-agents. Typically, the chain reactions are initiated by the formation of macro-compounds that are responsible of the formation of the unmatured crosslinked polymer. This first reaction depends on the reciprocal concentrations of all components and their chemical nature. In presence of two accelerators, it was considered that the reactions between each single accelerator and the NR raw material occur in parallel, making the reasonable assumption that there are no mutual reactions between the two accelerators. From the kinetic scheme adopted, a closed form solution was found for the crosslink density, with the only limitation that the induction period is excluded from computations. Even kinetic constants are evaluated in closed form, avoiding a numerically demanding least-squares best fitting on rheometer experimental data. Two series of experiments available, relying into rheometer curves at different temperatures and differentmore »

- Authors:

- Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan (Italy)
- Pirelli Tyre, Via Alberto e Piero Pirelli 25, 20126 Milan (Italy)
- CHEMCO Consultant, Via J.F. Kennedy 2, 45030 Occhiobello (Italy)

- Publication Date:

- OSTI Identifier:
- 22391163

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: AIP Conference Proceedings; Journal Volume: 1648; Journal Issue: 1; Conference: ICNAAM-2014: International Conference on Numerical Analysis and Applied Mathematics 2014, Rhodes (Greece), 22-28 Sep 2014; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 36 MATERIALS SCIENCE; CHAIN REACTIONS; CONCENTRATION RATIO; CROSS-LINKING; DENSITY; DIAGRAMS; LEAST SQUARE FIT; MATHEMATICAL MODELS; RAW MATERIALS; RUBBERS; SULFUR; VULCANIZATION

### Citation Formats

```
Milani, Gabriele, E-mail: milani@stru.polimi.it, E-mail: gabriele.milani@polimi.it, Hanel, Thomas, Donetti, Raffaella, and Milani, Federico.
```*Effective closed form mathematical approach to determine kinetic constants of NR vulcanized with sulphur and accelerators at different concentrations*. United States: N. p., 2015.
Web. doi:10.1063/1.4913073.

```
Milani, Gabriele, E-mail: milani@stru.polimi.it, E-mail: gabriele.milani@polimi.it, Hanel, Thomas, Donetti, Raffaella, & Milani, Federico.
```*Effective closed form mathematical approach to determine kinetic constants of NR vulcanized with sulphur and accelerators at different concentrations*. United States. doi:10.1063/1.4913073.

```
Milani, Gabriele, E-mail: milani@stru.polimi.it, E-mail: gabriele.milani@polimi.it, Hanel, Thomas, Donetti, Raffaella, and Milani, Federico. Tue .
"Effective closed form mathematical approach to determine kinetic constants of NR vulcanized with sulphur and accelerators at different concentrations". United States.
doi:10.1063/1.4913073.
```

```
@article{osti_22391163,
```

title = {Effective closed form mathematical approach to determine kinetic constants of NR vulcanized with sulphur and accelerators at different concentrations},

author = {Milani, Gabriele, E-mail: milani@stru.polimi.it, E-mail: gabriele.milani@polimi.it and Hanel, Thomas and Donetti, Raffaella and Milani, Federico},

abstractNote = {The basic reaction scheme due to Han and co-workers for NR vulcanized with sulphur is adopted and modified taking into account the single contributions of the different accelerators, focusing in particular on some experimental data ad hoc obtained at Pirelli’s laboratories, where NR was vulcanized at different temperatures (from 150 to 180 °C) and concentrations of sulphur, using TBBS and DPG in the mixture as co-agents. Typically, the chain reactions are initiated by the formation of macro-compounds that are responsible of the formation of the unmatured crosslinked polymer. This first reaction depends on the reciprocal concentrations of all components and their chemical nature. In presence of two accelerators, it was considered that the reactions between each single accelerator and the NR raw material occur in parallel, making the reasonable assumption that there are no mutual reactions between the two accelerators. From the kinetic scheme adopted, a closed form solution was found for the crosslink density, with the only limitation that the induction period is excluded from computations. Even kinetic constants are evaluated in closed form, avoiding a numerically demanding least-squares best fitting on rheometer experimental data. Two series of experiments available, relying into rheometer curves at different temperatures and different concentrations of sulphur and accelerator, are utilized to evaluate the fitting capabilities of the mathematical model. Very good agreement between numerical output and experimental data is experienced in all cases analysed.},

doi = {10.1063/1.4913073},

journal = {AIP Conference Proceedings},

number = 1,

volume = 1648,

place = {United States},

year = {Tue Mar 10 00:00:00 EDT 2015},

month = {Tue Mar 10 00:00:00 EDT 2015}

}