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 Such a limitation is here superseded and a major improvement is proposed utilizing (1) a closed form solution which does not require any optimization algorithm and (2) finding analytically a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).     

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.     

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.     

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.     

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.     

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 ( $175^\circ \text{ C}$ ).     

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.     

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.     

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.     

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.     

A method,based on statistical moments,to evaluate the kinetic parameters involved in unstable enzyme systems

The evaluation of individual rate constants involved in any reaction mechanism of an enzymatic systems first requires experimental monitoring of the time course of the concentration or product rate creation or of any enzyme species. The experimental progress curves obtained must then be fitted to the corresponding theoretical symbolic equation. Nevertheless, in some cases, e.g. when the equation involves two or more exponential terms, this fit is not easy and sometimes impossible. Simplification of the equation is usually required by assuming, for example, that the system has reached the steady-state, assuming an initial steady-state of a segment in the scheme of the reaction mechanism or assuming rapid equilibrium in one or more of the reversible steps, if there are any. But, obviously, simplified equations produce either fewer individual rate constants or global constants consisting of algebraic associations of individual rate constants or individual rate constants or global constants that might considerably differ from the real ones due to the approaches made. In this contribution, we suggest an alternative procedure for evaluating the rate constants of enzyme reactions corresponding to enzyme systems where one or more of the species involved is unstable or where one or more of the enzyme species is irreversibly inhibited, or both. The procedure is based on the numerical determination of statistical moments from experimental time progress curves. The fitting of these experimentally obtained moments to the corresponding theoretical expressions allows us, in most cases, to evaluate of all of the rate constants involved, with only a small error. To verify the goodness of the suggested procedure, it was applied to an unstable enzyme system which had previously been analysed with other methods. Finally, it is indicated how this procedure could also be extrapolated for application to any stable or unstable enzyme system.     

Experimental probes of silver metal nanoparticle formation kinetics: Comparing indirect versus more direct methods

The kinetics of noble metal nanoparticle formation in bottom-up syntheses are important for controlling and optimizing these methods. Hence, experimental probes that are easily accessible to most laboratories are also of interest. We collected kinetic curves for the formation of silver nanoparticles in a modified Turkevich method with citrate acting as the reducing and stabilizing agent by (i) measuring the change in silver nanoparticle surface plasmon resonance by UV-visible spectroscopy, a somewhat indirect method, and then also by (ii) measuring the change in silver ion concentration by ion-selective electrode potentiometry and/or atomic absorption spectroscopy, two more direct methods. The resulting sigmoidal kinetic curves were curvefitted with the Finke-Watzky two-step kinetic model of slow, continuous nucleation and fast autocatalytic growth to extract average rate constants. We found that the kinetic curves obtained by following the change in silver ion concentration were apparent mirror images of those constructed by following the change in nanoparticle surface plasmon resonance, and that their respective curvefits displayed the same sigmoidal characteristics. The resultant values of the rate constants for nucleation and growth overlapped within experimental error between the methods and showed similar trends over the range of citrate concentrations studied. The use of multiple probes in this work to follow the kinetics of nanoparticle formation helps fill a need for the comparison and evaluation of different methods available to scientists, particularly those considered easily accessible.     

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.     

A simple kinetic method for the determination of the reaction model from non-isothermal experiments

In a recent article, we obtained an approximate solution for the evolution of a transformed fraction under isochronal conditions for a large variety of single-step transformations. We verified that this solution is accurate and can, in many instances, be used instead of the exact numerical solutions of the corresponding differential equations. In this article we want to examine the possibilities offered by an analytical solution in the analysis of thermoanalytical curves. We will show that for single-step transformations, our model predicts that under the proper time scaling the thermograms obtained at different heating rates merge into a single curve. This ‘universal curve’ is exclusively related to the kinetic model. In addition, the universal curve can be obtained from experimental thermograms by means of a simple transformation. In this way, the dependence of the experimental curves on the rate constant and the kinetic model can easily be separated, making it possible to independently determine the kinetic parameters and the kinetic model. In addition, one can easily check the validity of the kinetic analysis as well as calculate a reliable statistical measure of the goodness of the single-step assumption.     

Mixing efficiency and instrumental delay effect on recorded kinetic curves

The effect of reactant addition time and instrumental response time on recorded kinetic curves was considered. The mixing-time. effect was considered for first- and second-order reactions in the case that a simple function of the concentration is measured, and for first-order reactions in the more complex case of non-adiabatic enthal-pimetric measurements. For any ratio of addition time to half-transformation time the proposed equations allow calculation of the correct rate constant and the error in the calculated initial concentration extrapolated from the experimental curve and show which portion of the experimental curve must be disregarded owing to the misleading effect of the addition time. The distortion due to the response-time of a thermistor used as concentration transducer has been calculated from a simplified model. The experimental kinetic measurements performed by quasi- and non-adiabatic enthalpimetry agree very satisfactorily with the theoretical data.     

Determination of kinetic constants of the cationic copolymerization of isobutylene with isoprene

A theoretical kinetic model has been developed for cationic isobutylene–isoprene copolymerization in methyl chloride with an AlCl₃ catalyst. Kinetic constants of this process have been derived from experimental data available on copolymerization kinetics (isobutylene conversion curve) and on molecular weight characteristics of the isobutylene–isoprene copolymer (butyl rubber). The adequacy of the theoretical kinetic model of the isobutylene–isoprene copolymerization process has been demonstrated by comparing the calculated molecular weight characteristics and degree of unsaturation of butyl rubber to the corresponding independent experimental data.     

Constitutive characterization of rubber blends by means of capillary-viscometry

A new procedure for the material characterization of rubber blends is proposed. It is based on a generalized Newton-Raphson procedure. Such a model has to consider the effect of wall slippage on the viscous properties as well as the non-linear coupling of the viscosity and shear strain rate, which is typical for rubber blends and polymers, respectively. The main goal of this method is the reduction of the experimental effort for determination of the viscosity function. Application of the new method is shown for four rubber materials.An experimental investigation of the viscous properties of rubber blends by means of capillary-viscometry serves as the basis of the work. Because of the pseudo-plastic material behavior of the proposed material, characterization is represented by a coupled system of non-linear equations. Describing their solution requires a numerical integration algorithm.Verification of the developed method for parameter identification was done by comparison of the obtained viscosity curves and the respective velocity profiles in capillary dies with the one obtained by the commonly used material characterization based on correction methods.     

Radiation grafting of methyl methacrylate on radiation crosslinked natural rubber film

Summary Grafting ofmethyl methacrylate (MMA) on radiation crosslinked natural rubber (NR) film has been investigated by mutual radiation grafting. The effect of experimental parameters like radiation dose, dose-rate, additives like acids and inorganic salts, solvents, monomer concentration, cross-linking density of the natural rubber film on the grafting extent has been studied.From the kinetic studies, a kinetic equation showing almost parabolic and linear dependence of grafting on concentration and dose rate, respectively, was deduced.Preliminary thermal stability studies of grafted films indicated that grafting of MMA does not enhance the thermal stability of NR.     

Kinetic model for a polymerization process where the rate constants of both propagation and termination by monomer depend on chain-length

A kinetic model with non-constant propagation and termination rate constants is given for a sort of anionic polymerization. The expression of molecular weight distribution and other molecular parameters are derived by a non-steady state procedure. The effect of the reaction conditions on the molecular parameters are illustrated by numerical examples. The theoretical curves of molecular weight distribution with two or three peaks are similar to those of polymers generated in the anionic polymerization of polar monomers in nonpolar solvents.