Effective GA approach for a direct evaluation of reaction kinetic within EPDM accelerated sulphur crosslinking

A direct genetic algorithm (GA) approach with kinetic base, to provide effective numerical estimates of vulcanization level for EPDM cross-linked with accelerated sulphur is presented. The model requires a preliminary characterization of rubber through standard rheometer tests. A recently presented kinetic exponential model is used as starting point to develop the algorithm proposed. In such a model, three kinetic constants have to be determined by means of a non-linear least-squares curve fitting. The approach proposed circumvents a sometimes inefficient and not convergent non-linear data fitting, disregarding at a first attempt reversion and finding the local minimum of a suitable two-variable error function, to have an estimate of the first two kinetic constants. A comparison between present GA approach and traditional gradient based algorithms is discussed. The last constant, representing reversion is again evaluated through a minimization performed on a single variable error function. The applicability of the approach is immediate and makes the model extremely appealing when fast and reliable estimates of crosslinking density of cured EPDM are required. To show the capabilities of the approach proposed, a comprehensive comparison with both available experimental data and results obtained numerically with a least square exponential model for a real compound at different temperatures is provided.     

EPDM accelerated sulfur vulcanization: a kinetic model based on a genetic algorithm

A simple closed form equation for the prediction of crosslinking of EPDM during accelerated sulfur vulcanization is presented. Such a closed form solution is derived from a second order non homogeneous differential equation, deduced from a kinetic model. The kinetic model is based on the assumption that, during vulcanization, a number of partial reactions occurs, both in series and in parallel, which determine the formation of intermediate compounds, including activated and matured polymer. Once written standard first order differential equations for each partial reaction, the differential equation system so obtained is rearranged and, after few considerations, a single second order non homogeneous differential equation with constant coefficients is derived, for which a solution may be found in closed form, provided that the non-homogeneous term is approximated with an exponential function. To estimate numerically the degree of crosslinking, kinetic model constants are evaluated through a simple data fitting, performed on experimental rheometer cure curves. The fitting procedure is a new one, and is achieved using an ad-hoc genetic algorithm, provided that a few points, strictly necessary to estimate model unknown constants with sufficient accuracy, are selected from the whole experimental cure curve. To assess the results obtained with the model proposed, a number of different compounds are analyzed, for which experimental or numerical data are available from the literature. The important cases of moderate and strong reversions are also considered, experiencing a convincing convergence of the analytical model proposed. For the single cases analyzed, partial reaction kinetic constants are also provided.     

Closed form numerical approach for a kinetic interpretation of high-cis polybutadiene rubber vulcanization with sulphur

A novel mathematical approach to predict the vulcanization degree of high-cis polybutadiene rubber vulcanized with sulphur is presented. The model has kinetic base, it is constituted by four reactions occurring in series and parallel and takes contemporarily into consideration, within a simplified but reliable scheme, the actual reactions occurring during polybutadiene sulphur curing, namely primary crosslinking and possible de-vulcanization. The first order differential equation system obtained is suitably rearranged and a closed form expression for the vulcanization degree is derived, depending the four kinetic constants characterizing the chemistry describing reactions. Instead of using classic least-squares optimization routines to characterize kinetic constants on experimental data, a simplified but reliable approach is proposed, where a system of four non-linear equations is solved with a recursive strategy, allowing estimating kinetic constants that proved to fit well normalized experimental data. The procedure is fast and its reliability is tested on a number of experimental data available, relying into a high-cis polybutadiene rubber cured under different temperatures and accelerators concentrations. Very good approximations of experimental data are obtained, also in comparison with a heuristic numerical approach where optimization is obtained interactively.     

A new simple numerical model based on experimental scorch curve data fitting for the interpretation of sulphur vulcanization

Sulphur was the first agent used to vulcanize commercial elastomers (e.g. natural rubber) and allows meaningful cost reductions during the industrial process (production cost ratio between peroxides and accelerated sulphur is around 5). Therefore, accelerated sulphur vulcanization is the most popular technique for the production of polydiene and EPDM elastomers items. At present, crosslinking mechanisms are not analytically known in detail, therefore reticulation kinetic has to be deduced from mechanical properties obtained during standardized tests, as for instance the oscillating disc rheometer. In the present paper, we propose a numerical model to fit experimental rheometer data based on a simple composite three functions curve, able to describe the increase of the viscosity at successive curing times at different controlled temperature to use during the production of thick items vulcanized with sulphur. It is believed that rheometer curve is able to give an indirect information on the rubber reticulation kinetic at different temperatures, to use in a successive step to establish simplified analytical kinetic formulas to adopt in the accelerated sulphur vulcanization of polydiene and EPDM elastomers. In the model, it is necessary to collect rheometer curves at different specimen temperatures, because vulcanization in industrial practice occurs at variable temperatures during curing, with considerable differences from the core to boundary of the item. Once that rheometer curves are suitably collected in a database, they are used to predict the optimal vulcanization of real items industrially produced. Finally, a so called alternating tangent approach (AT) is implemented to determine optimal input parameters (curing external temperature T _n and rubber exposition time t) to use in the production process. Output mechanical property (objective function) to optimize is represented by the average tensile strength of the item. A meaningful example of engineering interest, consisting of a thick 2D EPDM cylinder is illustrated to validate the model proposed.     

Parabola-Hyperbola P-H kinetic model for NR sulphur vulcanization

A Parabola-Hyperbola (P-H) kinetic model for NR sulphur vulcanization is presented. The idea originates from the fitting composite Parabola-Parabola-Hyperbola (P-P-H) function used by the authors in [1,2] to approximate experimental rheometer curves with the knowledge of a few key parameters of vulcanization, such as the scorch point, initial vulcanization rate, 90% of vulcanization, maximum point and reversion percentage. After proper normalization of experimental data (i.e. excluding induction and normalizing against maximum torque), the P-P-H model reduces to the discussed P-H composite function, which is linked to the kinetic scheme originally proposed by Han and co-workers [3]. Typically, it is characterized by three kinetic constants, where classically the first two describe incipient curing and stable/instable crosslinks and the last reproduces reversion.The powerfulness of the proposed approach stands into the very reduced number of input parameters required to accurately fit normalized experimental data (i.e. rate of vulcanization at scorch, vulcanization at 90%, maximum point and reversion percentage), and the translation of a mere geometric data-fitting into a kinetic model. Kinetic constants knowledge from simple geometric fitting allows characterizing rubber curing also at temperature different from those experimentally tested.The P-H model can be applied also in the so-called backward direction, i.e. assuming Han's kinetic constants known from other models and deriving the geometric fitting parameters as result.Some existing experimental data available, relying into rheometer curves conducted at 5 different temperatures on the same rubber blend are used to benchmark the P-H kinetic approach proposed, in both backward and forward direction. Very good agreement with previously presented kinetic approaches and experimental data is observed.     

Closed form analytical approach for a second order non-linear ODE interpreting EPDM vulcanization with peroxides

In this paper, the recently presented kinetic model proposed in Milani and Milani (J Math Chem 51(3):1116–1133, 2013) to interpret EPDM peroxide vulcanization is extensively revised and the resultant second order ODE is solved by means of an approximate but effective closed form analytical approach. The model has kinetic base and it is aimed at predicting, by means of a very refined approach, the vulcanization degree of rubber vulcanized with peroxides. Such a procedure takes contemporarily into consideration, albeit within a simplified scheme, the actual reactions occurring during peroxidic curing, namely initiation, H-abstraction, combination and addition, and supersedes the simplified approach used in practice, which assumes for peroxidic curing a single first order reaction. The main drawback of the overall procedure proposed in Milani and Milani (J Math Chem 51(3):1116–1133, 2013) is that the single second order non-linear differential equation obtained mathematically and representing the crosslink evolution with respect to time, was solved numerically by means of a Runge–Kutta approach. Such a limitation is here superseded and a major improvement is proposed allowing the utilization of an approximate but still effective closed form solution. After some simplifications applied on some parts of the solving function not allowing direct closed form integration, an analytical function is proposed. Kinetic parameters within the analytical model are evaluated through least squares where target data are represented by few experimental normalized rheometer curve values. In order to have an insight into the reliability of the numerical approach proposed, a case of technical interest of an EPDM with low unsaturation and crosslinked with three different peroxides at three increasing temperatures is critically discussed.     

Differential model accounting for reversion for EPDM vulcanized with peroxides

One of the main drawbacks of EPM/EPDM rubber vulcanization by peroxides is the lack of selectivity, which leads to a number of side reactions. The reaction mechanisms at the base of peroxides crosslinking are generally known and include the formation of alkyl and allyl (in the EPDM case) macro-radicals through H-abstraction from the macromolecular chains and the combination of these macro-radicals, which macroscopically is known with the term “vulcanization”. In the paper, a simple but effective mathematical model having kinetic base, to predict the vulcanization degree of rubber vulcanized with peroxides, is presented. The approach takes contemporarily into consideration, albeit within a simplified scheme, the actual reactions occurring during peroxidic curing, namely initiation, H-abstraction, combination and addition, and supersedes the simplified approach used in practice, which assumes for peroxidic curing a single first order reaction. After a suitable re-arrangement of the first order system of differential equations obtained from the actual kinetic system adopted, a single second order non-linear differential equation is obtained and numerically solved by means of a Runge–Kutta approach. Kinetic parameters to set are evaluated by means of a standard least squares procedure where target data are represented by experimental values available, i.e. normalized rheometer curves or percentage crosslink density experimentally evaluated by means of more sophisticated procedures. In order to have an insight into the reliability of the numerical approach proposed, two cases of technical interest are investigated in detail: the first is an EPDM crosslinked with two different peroxides, whereas the second is a compound with high level of unsaturation, showing reversion at medium-high vulcanization temperature ( $175^\circ \text{ C}$ ).     

Double Exponential Evaluation under Non‐Pseudo–First‐Order Conditions: A Mixed Second‐Order Process Followed by a First‐Order Reaction

In this paper, a double exponential approximating approach is described for a quite common kinetic model (mixed second‐order formation of an intermediate followed by its first‐order decay) under non‐pseudo–first‐order conditions (i.e., when the initial ratio of the two reactants is between 1 and 10). For the evaluation, first the exact kinetic curves predicted by the two‐step model were calculated and then fitted to a double exponential function. The goodness of the fits and the estimated parameters of the double exponential function for both I and P concentrations were determined as a function of the rate constants and initial concentrations in the two‐step model. It was found that the fit of the double exponential function is acceptable or very good under these conditions despite the fact that none of the reagents is in a large excess. Since UV–vis absorption spectroscopy is probably the most common technique to follow kinetic traces, we also made efforts to deal with the typical properties of monitoring the process through UV–vis. It was found that the experimental curves can be fitted quite well with a double exponential function if the reagents have minor absorption compared to the intermediate and/or product. The connection between the observed rate constants and the rate constants of the above‐mentioned mechanism is also studied.     

Effects of UV irradiation on physico‐mechanical properties of EPDM/buckminsterfullerene composite

Ethylene propylene diene rubber‐fullerene (EPDM/C60) composite, partially crosslinked by ultraviolet (UV) radiation, was prepared and characterized for crosslink density, mechanical properties and thermal behavior. FT‐IR analysis showed peak disappearance at 1688 cm^?1, corresponding to the unsaturation of EPDM, and the appearance of new peaks relating to the formation of oxidation products of C60, such as epoxide, keto, aldehyde and carboxylic groups. Solubility studies demonstrated the dissolution of pristine EPDM in toluene even after a longer period of UV exposure, whereas EPDM/C60 composite became insoluble and/or swollen after 6 hr of UV exposure, indicating the formation of partial crosslinking between EPDM and C60. Differential scanning calorimetry (DSC) measurements revealed an increase in the glass transition temperature peak of UV‐cured EPDM. Thermogravimetric analysis (TGA) showed that UV exposure reduced the thermal decomposition temperature of EPDM/C60, pristine EPDM and dicumyl peroxide (DCP)‐cured EPDM. The modulus, tensile stress and elongation at break of EPDM/C60 composites were greatly influenced by the duration of UV irradiation. Comparison of UV‐cured EPDM/C60 composite with DCP‐cured EPDM confirmed the superior strength properties of the former system. Copyright © 2004 John Wiley & Sons, Ltd.     

Crosslink network evolution of BIIR/EPDM blends during peroxide vulcanization

Crosslink network evolution of brominated butyl rubber (BIIR)/ethylene–propylene–diene-monomer rubber (EPDM) blends during peroxide vulcanization is studied at a meso-scale level. In this work, EPDM is added as a co-agent to increase the crosslink density of BIIR vulcanization. With increasing EPDM content from 0 to 20 phr, the maximum torque of BIIR/EPDM compounds during vulcanization increases by 73%, reaching to 3.40 dNm. Vulcanization kinetic study shows that addition of EPDM favors to the crosslinking of BIIR compound. Meanwhile, the addition of 20 phr EPDM contributes to an increase in the crosslink density of BIIR/EPDM(80/20) vulcanizate, avoiding downward trend at post-cure period in comparison with BIIR only. Crosslink network evolution of BIIR/EPDM blends is divided into three periods during peroxide vulcanization at 150 °C. The role of EPDM in the crosslink network evolution is studied by proton nuclear magnetic resonance, and a “network patching” mechanism is proposed in which EPDM is implied to work as patch on damaged crosslink network resulted from the degradation nature of BIIR.     

Effective closed form starting point determination for kinetic model interpreting NR vulcanized with sulphur

In this paper, a closed form analytical approach for a recently presented kinetic model proposed in Milani and Milani (Polym Test, 2013, under review) to interpret NR sulphur vulcanization in presence of either experimental or surrogate rheometer curves is proposed. The model has kinetic base and is aimed at predicting, by means of a very refined approach, the vulcanization degree of NR vulcanized with sulphur. It needs as input only rheometer curves to fit and provides as output kinetic constants of the single reactions occurring during the crosslink process. In Milani and Milani (Polym Test, 2013, under review) a cure chemical scheme constituted by five reactions occurring in series and parallel was adopted. The chemical scheme, translated mathematically into a differential equations system, was suitably re-arranged and a single analytical equation was derived, representing rubber crosslink degree evolution upon time. The main drawback of such procedure is that the five kinetic constants corresponding to each reaction were determined through a standard non-linear least squares procedure, trying to minimize the deviation of the analytical cure curve from experimental data. Such a limitation is here superseded and a major improvement is proposed utilizing (1) a closed form solution which does not require any optimization algorithm and (2) finding analytically a starting point for the unknown kinetic constants, very near to the actual solution and thus very convenient for a successive least squares minimization. In the model, it is shown how the analytical condition deduced from the scorch point (second derivative of the rheometer curve equal to zero) and two further conditions, e.g. the time at 90 % of vulcanization and the reversion percentage, allow the simple direct evaluation of kinetic constants, providing a closed form analytical formula to predict well the state of cure of the rubber under consideration. To assess the results obtained with the model proposed, several examples on two different NRs are discussed. The approach proved to be extremely robust and much faster when compared with the model proposed by Milani and Milani (Polym Test, 2013, under review).     

Intumescent flame retardation and silane crosslinking of PP/EPDM elastomer

This study deals with the silane crosslinking and intumescent flame retardation of polypropylene/ethylene‐propylene‐diene copolymer (PP/EPDM) elastomers. The effect of silane crosslinking on the flame retardancy of the PP/EPDM composites containing melamine phosphate (MP) and dipentaerythritol (DPER) was studied by limiting oxygen index, UL 94 and cone calorimetry tests. The chemical composition of the silane crosslinked and flame retarded PP/EPDM composites treated at different temperatures was studied by X‐ray photoelectron spectroscopy and real time Fourier transform infrared (FTIR) spectrometry. Thermal decomposition and crystallization behavior of the PP/EPDM composites were investigated using thermogravimetric analysis and differential scanning calorimetry, respectively. Moreover, the mechanical properties of the composites were also studied. It is found that the flame retardancy, mechanical properties, and thermal decomposition behavior of the composites are influenced by silane grafting and crosslinking. Copyright © 2008 John Wiley & Sons, Ltd.     

Addition effect of erbium(III) trifluoromethanesulfonate in the homopolymerization kinetics of a DGEBA resin

Solid bisphenol-A epoxy resin of medium molecular weight was cured using a Lewis acid initiator (erbium(III) trifluoromethanesulfonate) in three different proportions (0.5, 1 and 2 phr). A kinetic study was performed in a differential scanning calorimeter. The complete kinetic triplet was determined (activation energy, pre-exponential factor, and integral function of the degree of conversion) for each system. A kinetic analysis was performed with an integral isoconversional procedure (model-free), and the kinetic model was determined both with the Coats-Redfern method (the obtained isoconversional E value being accepted as the effective activation energy) and through the compensation effect. All the systems followed the same isothermal curing model simulated from non-isothermal ones. The “nucleation and growth” Avrami kinetic model A_3/2 has been proposed as the polymerization kinetic model. The addition of initiator accelerated the reaction having higher influence when low temperatures were applied.     

Curing degree prediction for S-TBBS-DPG natural rubber by means of a simple numerical model accounting for reversion and linear interaction

The paper presents a simple numerical model able to provide directly kinetic constants and reliable numerical rheometer curves for S-TBBS-DPG natural rubber. The approach is suitable to calculate the kinetic constants and maximum torque (MH) at any S-TBBS-DPG concentration, following a 3D mathematical interpolation/extrapolation procedure, when kinetic constants on few grid points of S-TBBS-DPG concentrations are available. In particular, the possibility to estimate with sufficient accuracy the behavior of natural rubber at any intermediate concentration of S-TBBS-DPG having engineering relevance has been proved, calibrating the model by means of simple closed form standard best fitting on few experimental data. The model used is a three kinetic parameters one, derived from the well known Han's and co-workers approach, where constants have been evaluated normalizing experimental rheometers curves following the commonly accepted Sun and Isayev method. The procedure has been validated against experimentally obtained rheometer curves by means of inverse analysis, exhibiting excellent prediction capabilities. The approach may be used for practical purposes in order to avoid expensive and cumbersome experimental investigations.     

Synthesis and characterization of siloxane resins derived from silphenylene‐siloxane copolymers bearing benzocyclobutene pendant groups

Two bis(dimethylamimo)silanes with benzocyclobutene (BCB) groups, bis(dimethylamino)methyl(4′‐benzocyclobutenyl)silane ( 2 ) and bis(dimethylamino)methyl [2′‐(4′‐benzocyclobutenyl)vinyl]silane ( 4 ), were synthesized from different synthetic routes, which were then employed to prepare two novel silphenylene‐siloxane copolymers (SiBu and SiViBu) bearing latent reactive BCB groups by polycondensation procedure with 1,4‐bis(hydroxydimethylsilyl)benzene. At elevated temperatures these copolymers were readily converted to highly crosslinked films and molding disks with network structures by polymer chain crosslinking, which followed the first‐order kinetic reaction model. The final resins of SiBu and SiViBu demonstrated excellent thermal stability with high glass transition temperatures (218 and 256 °C) and high temperatures at 5% weight loss (553 and 526 °C in N₂, 530 and 508 °C in air). After aging at 300 °C in air for 100 h, the cured resins showed weight loss lower than 4%. The films of cured SiBu and SiViBu also exhibited relatively low dielectric constants of 2.66 and 2.64, low dissipation factors of 2.23 and 2.12 × 10^?3, low water absorptions (≤0.28%), and high transparence in the visible region with cutoff wavelengths of 321 and 314 nm. Moreover, the aged films exhibited good dielectric properties and low water absorptions. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 7868–7881, 2008     

Quinoline‐containing networks from enone and aldehyde triglyceride derivatives

The crosslinking reaction of a triglyceride derivative containing α,β‐unsaturated ketones with diaminodiphenylmethane via aza‐Michael addition has been extensively studied. First, a model study with monofunctional compounds showed that the conjugated addition product undergoes a series of transformations leading to formation of a substituted quinoline. The proposed reaction pathway is presented as a variation of the Skraup‐Doebner‐Von Miller quinoline synthesis. The presence of quinolines as crosslinking points in the cured materials has been proved by means of different characterization techniques, and the properties derived from this aromatization process have been described. This new crosslinking approach has been successfully applied to an aldehyde‐containing triglyceride to obtain quinoline‐containing thermosets. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 869–878, 2010     

Kinetic investigation of photopolymerization initiated by oligomeric photolatent base

Macrophotoinitiators often show polymer effect and photolatent bases make molecular amplification reaction. A main-chain oligomer containing α-aminoalkylphenone chromophore (OAK) had been synthesized with the characteristics of both photoinitiators and photolatent base hybrid structure. In order to determine whether such an effect or reaction exists in photopolymerization initiated by OAK, we used photo-differential scanning calorimeter to analyze the kinetic behavior. The experimental rate constants at different conversions were obtained by Arrhenius equation at different temperatures, the theoretic rate constants were simultaneously calculated with the use of phenomenological model. The results showed that the model correctly predicts the rate constant at the early stage, but underestimates it later. The comparative experiment between OAK and polymeric α-hydroxyalkylphenone photoinitiator (KIP150) shows that the molecular proliferation reaction caused by OAK significantly increases the experimental rate constant in the high-crosslinked stages of the polymerization. The photolatent base would be expected to provide one possible pathway in tackling photo-curing of opaque/thick material.     

Esterification of sorbitol and lauric acid with catalystp-TSA

Kinetics of the esterification of sorbitol with lauric acid in the presence ofp-TSA as a catalyst has been studied. A kinetic model of reversible second order reaction was proposed for the esterification. Parameters in the model (kinetic constants) were estimated by non-linear regression. The temperature dependence of the rate was calculated from the experimental constants estimated at various temperatures, using the Arrhenius equation. Experimental results are in good accordance with the proposed theoretical model.     

Dynamically Photocrosslinking of Polypropylene／EPDM Blends as a Thermoplastic Elastomer

The mechanical properties and the crystal morphological structures of the dynamically photocrosslinked polypropylene (PP)/ethylene-propylene-diene terpolymer (EPDM) blends have been studied by means of mechanical tests, wide-angle X-ray diffraction(WAXD), and differential scanning calorimetry(DSC). The dynamically photocrosslinking of the PP/EPDM blends can improve the mechanical properties considerably, especially the notched Izod impact strength at low temperatures. The data obtained from the mechanical tests show that the notched Izod impact strength of the dynamically photocrosslinked sample with 30% EPDM at -20℃ is about six times that of the uncrosslinked sample with the same EPDM component. The results from the gel content, the results of WAXD, and the DSC measurements reveal the enhanced mechanism of the impact strength for the dynamically photocrosslinked PP/EPDM blends as follows: (1) There exists the crosslinking of the EPDM phase in the photocrosslinked PP/EPDM blends ; (2) The β-type crystal structureof PP is formed and the content of α-type crystal decreases with increasing the EPDM component; (3) The graft copolymer of PP-g-EPDM is formed at the interface between the PP and EPDM components. All the above changes of the crystal morphological structures are favorable for increasing the compatibility and enhancing the toughness of the PP/EPDM blends at low temperatures.     

Modelling free-radical copolymerization kinetics, 3. Molecular weight calculations for copolymers with crosslinking

A comprehensive model for molecular weight calculations of free-radical crosslinking copolymerizations was developed using the pseudo-kinetic rate constants and the method of moments. The moments of copolymer chain distributions are defined in such a way so that the molecular weight averages of crosslinking copolymers can be calculated using the moments. The present model is based on a general crosslinking copolymerization scheme, accounting for chain transfer to small molecules and polymer, bimolecular termination, and crosslinking reactions. The influence of crosslinking reactions on molecular weight development is discussed. The effects of the reactivity of pendant double bonds on the moments development were further demonstrated using model simulations. The simulations results suggest that the higher-order molecular weight averages are very sensitive to the reactivity of pendant double bonds. It was found that chain transfer to polymer affects the gelation point significantly. The radical fractions must be calculated accounting for chain transfer reactions in addition to propagations in order to properly evaluate pseudo-kinetic rate constants. The present model was used to predict kinetic behavior and molecular weight development of styrene/m-divinylbenzene and styrene/ethylene dimethacrylate free-radical crosslinking copolymerizations in benzene solution at 60°C. It was found that the present model is in excellent agreement with the experimental data published in the literature. Model predictions and experimental data show that the reactivity of pendant double bonds is much lower than that of vinyl and divinyl monomers. The simulation results suggest that the assumption of the same reactivity of functional groups is likely not valid for many free-radical crosslinking copolymerizations. The present model based on a kinetics approach can be used to predict molecular weight development for vinyl/divinyl free-radical crosslinking copolymerizations and to estimate kinetic parameters in the pre-gelation period.