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.     

Direct and closed form analytical model for the prediction of reaction kinetic of EPDM accelerated sulphur vulcanization

A novel direct model with kinetic base for the prediction of the final vulcanization level of EPDM cured with sulphur is presented. The model bases on a preliminary characterization of rubber through standard rheometer tests and allows an accurate prediction of the crosslinking degree at both successive curing times and different controlled temperatures. Both the case of indefinite increase of the torque and reversion can be handled. The approach proposed bases on a previously presented exponential model, where a calibration of three kinetic constants at fixed temperature by means of non-linear least square fitting was required. Here the exponential model is superseded and kinetic constants are evaluated through simple closed form formulas. 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 the exponential model for real compounds at different temperatures is finally provided.     

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).     

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.     

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.     

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.     

A closed form solution for the vulcanization prediction of NR cured with sulphur and different accelerators

The paper presents a novel efficient closed form approach to determine the degree of vulcanization of natural rubber (NR) vulcanized with sulphur in presence of different accelerators. The general reaction scheme proposed by Han and co-workers for vulcanized sulphur NR is re-adapted and suitably modified taking into account the single contributions of the different accelerators, focusing in particular on some experimental data, where NR was vulcanized at different temperatures (from 150 to \(180\, ^{\circ }\hbox {C}\)) and concentrations of sulphur, using TBBS and DPG in the mixture as co-agents at variable concentrations. In the model, chain reactions initiated by the formation of macro-compounds responsible for the formation of the unmatured crosslinked polymer are accounted for. It is assumed that such reactions depend on the reciprocal concentrations of all components and their chemical nature. In presence of two accelerators, reactions are assumed to proceed in parallel, making the assumption that there is no interaction between the two accelerators. Despite there is experimental evidence that a weak process by which each accelerator affects the other, the reaction chemistry is still not well understood and therefore its effect cannot be translated into any mathematical model. In any case, even disregarding such interaction, good approximations of the rheometer curves are obtained. From the simplified kinetic scheme adopted, a closed form solution is found for the crosslink density, with the only limitation that the induction period is excluded from computations. The main capability of the model stands however in the closed form determination of kinetic constants representing the velocities of single reactions in the kinetic scheme adopted, which allows 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 accelerators, 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 analyzed.     

Vulcanization kinetics of graphene/styrene butadiene rubber nanocomposites

This paper presents the influence of graphene on the vulcanization kinetics of styrene butadiene rubber （SBR） with dicumyl peroxide. A curemeter and a differential scanning calorimeter were used to investigate the cure kinetics, from which the kinetic parameters and apparent activation energy were obtained. It turns out that with increasing graphene loading, the induction period of the vulcanization process of SBR is remarkably reduced at low graphene loading and then levels off; on the other hand, the optimum cure time shows a monotonous decrease. As a result, the vulcanization rate is suppressed at first and then accelerated, and the corresponding activation energy increases slightly at first and then decreases. Upon adding graphene, the crosslinking density of the nanocomposites increases, because graphene takes part in the vulcanization process.     

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 ( $175^\circ \text{ C}$ ).     

Kinetic estimation of the adsorbate distribution on the surface from adsorbed amounts

A phenomenological multilayer adsorption model for a well-dispersed, homogeneous, nonporous adsorbent and a molecular adsorbate is presented. The model provides explicit kinetic expressions associating the adsorbed amounts to the fraction of the surface occupied and reduces to the first- and second-order adsorption models for special cases. Parameters of the model are a pair of true rate constants related to the adsorbate-adsorbent and adsorbate-surface adsorbate affinities. A general graphical procedure and analytical equations for special cases are provided to estimate the rate constants from kinetic adsorption data. Data from the adsorption of sodium stearate onto alpha-alumina from water were used to test the model. The predicted values of the rate constants suggested that the stearate was distributed homogeneously on the alumina surface and essentially adsorbed as a monolayer before starting to form the second layer.     

Vulcanization kinetics study of natural rubber compounds having different formulation variables

Vulcanization kinetics of natural rubber (NR) compounds with efficient vulcanization system was studied through phenomenological approach using the experimentally cure data obtained from a moving die rheometer. The cure kinetic parameters were defined using the proposed models by Claxton?CLiska and Deng?CIsayev with the support of curve fitting software. The effects of the amount of accelerators, sulfur and silica in the formulations on the cure characteristics and cure kinetic parameters at high cure temperatures were investigated. Kinetic data results showed that the above two models were able to describe the curing behaviour of the studied compounds satisfactorily. It showed that the fitting of the experimental data with Claxton?CLiska and Deng?CIsayev could provide a good platform to investigate the cure kinetics of the prepared NR compounds.     

Techniques used for determining cure kinetics of rubber compounds

Controlling and assuring the quality of the manufacture of high precision engineering rubber components has led to the need to simulate fundamental industrial processes such as compression molding and injection molding using CAE tools. Both compression and injection molding techniques for the fabrication of rubber products involve crosslinking or vulcanization which is invariably assisted by temperature and pressure. Vulcanization is a chemical process and therefore its simulation necessarily involves characterization of kinetic parameters. The kinetics of curing or vulcanization is somewhat complex as it depends upon the compound formulation, temperature and in some cases pressure. The present paper reports and discusses the application and utility of different techniques for characterizing the cure behavior of rubber compounds. Kinetic data has been fitted to various mathematical models in order to see which of the models can best represent the crosslinking behavior of selected rubber compounds. Finally, the kinetic data is used to simulate the injection molding process for relatively simple geometries.     

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.     

Mechanistic modeling of reversion phenomenon in sulphur cured natural rubber vulcanization kinetics

A novel kinetic model of natural rubber sulphur vulcanization is proposed. The modeling approach takes into account current knowledge on the different polysulfidic structures present during vulcanization, and the associated individual reactions. A simplified scheme is proposed, giving a mechanistic view of the reversion phenomenon, which results in a decrease of the elastic modulus (related to the sulphur crosslink density) for long vulcanization times at high temperature. The resulting set of differential equations is solved by an appropriate numerical method to predict the evolution of the degree of vulcanization for isothermal cure conditions.     

Kinetics of the crosslinking reaction of nonionic polyol dispersion from terpene-maleic ester-type epoxy resin

The bulk crosslinking reaction kinetics of a novel two-component waterborne polyurethanes (2K-WPUs) composed of a bio-resin-based polyol dispersion and a hydrophilically modified hexamethylene diisocyanate tripolymer are investigated by freeze–drying and differential scanning calorimetry (DSC) technique at different heating rates. The data fit for the above two components is implemented with the nth-order kinetics equation and Málek’s mechanism function method, respectively. The kinetic parameters of crosslinking reaction are determined by the kinetic analysis of the data obtained from the thermal treatment, and then the kinetic model is built. The results indicate that the nth-order model deduced from Kissinger and Crane equation has great distinction with the experimental data, while the Málek analytic mechanism shows that the crosslinking process of the crosslinking reaction follows an autocatalytic reaction. The two-parameter (m and n) autocatalytic model (S–B model) can well describe the crosslinking reaction process of the studied 2K-WPU. The DSC curves derived from the experimental data show a good agreement with the theoretical calculation under 5–20 °C min^?1 heating rate. The results provide theoretical basis for the choice of the manufacturing process and the optimization of processing window.     

Using bulk convection in a microtensiometer to approach kinetic-limited surfactant dynamics at fluid-fluid interfaces

The effects of carbon chain length and temperature were investigated on adsorption kinetics and surface tension of a group of slightly volatile, short carbon chain molecules: 1-octanol, 1-hexanol, and 1-butanol. Experiments were performed in a closed chamber where simultaneous adsorption from both sides of the vapor/liquid interface was considered. The dynamic (time dependent) and steady-state surface tensions were found to decrease with temperature ranging from 10 °C to 35 °C. It was shown that, at the final steady-state, the effect of adsorption from the vapor phase was much more important than that from the liquid phase especially for short carbon chain molecules (e.g., 1-butanol). The modified Langmuir equation of state and modified kinetic transfer equation, which account for adsorption from both sides of a vapor/liquid interface, were used to model the experimental data of the steady-state and dynamic surface tension, respectively. Modeling results showed that the equilibrium constants and adsorption rate constants were increased with temperature and carbon chain length. The maximum surface concentration showed a decrease with temperature and an increase with carbon chain length. Some variations in the fitting parameters were observed in the dynamic modeling. These variations may be due to the experimental errors or the limitations of the proposed model.     

硅橡胶硫化反应场的数值模拟

以化学反应动力学原理为基础,实验确定了硅氢加成反应的动力学模型;引入 了硫化率增量的概念,获得了全量硫化率和硫化率增量的数值计算式;根据统计理 论,获得了起始分子无规分布时的交联结构参数的数值计算式;介绍了非稳态硫化 反应场的有限元模拟步骤。在此基础上,设计了硅橡胶硫化反应过程的有限元模拟 软件;应用该软件,根据输入的初边值条件,可计算任一空间点在任一时刻的交联 反应参数和交联结构参数,进而可预测制品性能、优化反应参数,或优化设计高分 子材料。最后给出了一个典型算例,并实验验证了该模拟理论及算法的合理性。     

Natural rubber/styrene-butadiene rubber blends prepared by solution mixing: Influence of vulcanization temperature using a Semi-EV sulfur curing system on the microstructural properties

Blends of natural rubber (NR) and styrene-butadiene rubber (SBR) were prepared by solution mixing and vulcanized with sulfur and accelerator in a Semi-EV system at 433 K and 443 K in order to study the vulcanization kinetic and the influence of vulcanization temperature on final structure of the blends. The vulcanization kinetic studied through the variation in rheometer curves was analyzed using the Ding and Leonov model, which takes into account the reversion effect during the cure process. The average free nanohole volume and the fractional free volume of samples with different NR/SBR ratio were estimated using positron annihilation lifetime spectroscopy (PALS). Also, the crosslink density was determined by means of swelling tests in a solvent. For all the compounds, a correlation between the free nanohole volume and the delta torque obtained from the respective rheometer curves was established.