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 Such a limitation is here superseded and a major improvement is proposed allowing the utilization of an approximate but still effective closed form solution.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.     

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.     

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.     

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 ( $175^\circ \text{ C}$ ).     

On-line optimizing control of the intermittent simulated moving bed process

The I-SMB process is one of the modifications to the standard SMB process that has been demonstrated both theoretically and experimentally to exhibit rather competitive performance (Katsuo and Mazzotti in J Chromatogr A 1217:1354, 2010a, 3067, 2010b; Katsuo et al. in J Chromatogr A 1218:9345, 2011). This work aims at showing that also the I-SMB process can be controlled and optimized by using the optimizing on-line controller developed at ETH Zurich for the standard SMB process (Erdem et al. in Ind Eng Chem Res 43:405, 2004a, 3895, 2004b; Grossmann et al. in Adsorption 14:423, 2008, AIChE J 54:1942008). This is achieved by using a virtual I-SMB unit based on a detailed model of the process; past experience with the on-line controller shows that the controller’s behavior on a virtual platform is essentially the same as in laboratory experiments.     

On the determinant evaluation of quasi penta-diagonal matrices and quasi penta-diagonal Toeplitz matrices

Recently, three computational algorithms for evaluating the determinant of quasi penta-diagonal matrices have been proposed by El-Mikkawy and Rahmo (Comput Math Appl 59:1386–1396, 2010), by Neossi Nguetchue and Abelman (Appl Math Comput 203:629–634, 2008), and by Jia et al. (Int J Comput Math 89:851–860, 2013), respectively. In the current paper, two novel algorithms with less computational costs are proposed for the determinant evaluation of general quasi penta-diagonal matrices and quasi penta-diagonal Toeplitz matrices. Furthermore, three numerical experiments are given to show the performance of our algorithms. All of the numerical computations were performed on a computer with aid of programs written in MATLAB.     

A strategy for selecting the frequency in trigonometrically-fitted methods based on the minimization of the local truncation errors and the total energy error

Trigonometrically-fitted methods have been largely used for solving second-order differential problems, and particularly for solving the radial Schrödinger equation (see for instance Alolyan and Simos in J Math Chem 50:782–804, 2012; Simos in J Math Chem 34:39–58, 2003, 44:447–466, 2008; Vigo-Aguiar and Simos in J Math Chem 29:177–189, 2001, 32:257–270, 2002 and the references therein contained). It is well-known that for periodic or oscillatory problems, trigonometrically fitted methods are more efficient than non-fitted methods. A large number of different approaches have been considered in the scientific literature to obtain analytical approximations to the frequency of oscillation in case of periodic solutions, which are valid for a large range of amplitudes of oscillation. However, these techniques have been limited to obtaining only one or two iterates because of the great amount of algebra involved. In this paper we consider the use of a trigonometrically fitted method to obtain numerical approximations for the solutions. This yields very acceptable results provided that the approximation of the parameter of the method is done with great accuracy. Many trigonometrically fitted methods have been reported in the literature, but there is no decisive way to obtain the optimal frequency value. We present a strategy for the choice of the parameter value in the adapted method, based on the minimization of the sum of the total energy error and the local truncation errors in the solution and in the derivative. We include an example solved numerically that confirms the good performance of the strategy adopted.     

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.     

The first metals in Mendeleiev’s Table: Part II. A new argument against the placement of hydrogen atop the alkali metal column

Every so often an experiment trying to give reliable evidence for a metallic hydrogen solid is reported. Such evidence is, however, not too convincing. As Eric Scerri has recently reiterated, “the jury is still out on that issue” (Scerri 2012). This search stems from the common spectroscopy shared by the hydrogen atom and all the alkali metal atoms, and perhaps is guided by a desire to place hydrogen atop the alkali metals, in Mendeleiev’s Table, reinforced by the fact pointed out by Scerri (The Periodic Table, its story and its significance, Oxford University Press, Oxford, 2007, 2012) that there is no other obvious place for hydrogen in said Table. But H₂ is a light gas at room temperature, while Li, Na, K and the other alkali elements form solid metal crystals. At very low temperatures, of course, hydrogen solidifies, but it is formed by H₂ molecules (see for example, Van Kranendonk in Solid hydrogen, Plenum Press, New York, 1983). Our purpose here is to use a new argument to break this impasse: “should H be grouped with the alkali metals with which it shares a common spectroscopy, but which solidifies in a completely different fashion?” This argument has been proposed before in a couple of papers in this journal to establish a similar question for He and the alkaline earths (Novaro in Found Chem 10:4, 2008, Ramírez-Solís and Novaro in Found Chem, 2012), as is discussed in “Precedents” section.     

In-situ imaging sensors for bioprocess monitoring: state of the art

Over the last two decades, more and more applications of sophisticated sensor technology have been described in the literature on upstreaming and downstreaming for biotechnological processes (Middendorf et al. J Biotechnol 31:395–403, 1993; Lausch et al. J Chromatogr A 654:190–195, 1993; Scheper et al. Ann NY Acad Sci 506:431–445, 1987), in order to improve the quality and stability of these processes. Generally, biotechnological processes consist of complex three-phase systems—the cells (solid phase) are suspended in medium (liquid phase) and will be streamed by a gas phase. The chemical analysis of such processes has to observe all three phases. Furthermore, the bioanalytical processes used must monitor physical process values (e.g. temperature, shear force), chemical process values (e.g. pH), and biological process values (metabolic state of cell, morphology). In particular, for monitoring and estimation of relevant biological process variables, image-based inline sensors are used increasingly. Of special interest are sensors which can be installed in a bioreactor as sensor probes (e.g. pH probe). The cultivation medium is directly monitored in the process without any need for withdrawal of samples or bypassing. Important variables for the control of such processes are cell count, cell-size distribution (CSD), and the morphology of cells (Höpfner et al. Bioprocess Biosyst Eng 33:247–256, 2010). A major impetus for the development of these image-based techniques is the process analytical technology (PAT) initiative of the US Food and Drug Administration (FDA) (Scheper et al. Anal Chim Acta 163:111–118, 1984; Reardon and Scheper 1995; Schügerl et al. Trends Biotechnol 4:11–15, 1986). This contribution gives an overview of non-invasive, image-based, in-situ systems and their applications. The main focus is directed at the wide application area of in-situ microscopes. These inline image analysis systems enable the determination of indirect and direct cell variables in real time without sampling, but also have application potential in crystallization, material analysis, polymer research, and the petrochemical industry.

The Homfly polynomial of double crossover links

Motivated by double crossover DNA polyhedra (He et al. in Nature 452:198, 2008; Lin et al. in Biochemistry 48:1663, 2009; Zhang et al. in J Am Chem Soc 131:1413, 2009; Zhang et al. in Proc Natl Acad Sci USA 105:10665, 2008; He et al. in Angew Chem Int Ed 49:748, )2010, in this paper, we construct a new type of link, called the double crossover link, formed by utilizing the “ $n$ -point star” to cover each vertex of a connected graph $G$ . The double crossover link can be used to characterize the topological properties of double crossover DNA polyhedra. We show that the Homfly polynomial of the double crossover link can be obtained from the chain polynomial of the truncated graph of $G$ with two distinct labels. As an application, by using computer algebra (Maple) techniques, the Homfly polynomial of a double crossover tetrahedral link is obtained. Our result may be used to characterize and analyze the topological structure of DNA polyhedra.     

Comment on “A note on the principal measure and distributional (p,q)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction”

In García Guirao and Lampart (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this note, we give an example which shows that the proofs of Theorems 3.1 and 3.2 in [J Math Chem 51:1410–1417, 2013] are incorrect, and two open problems.     

The proof of a conjecture on the comparison of the energies of trees

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph. In this paper, we first present a new method to directly compare the energies of two bipartite graphs, then also present some new techniques to compare the quasi-orders of some bipartite graphs. As the applications of these methods, we prove that a conjecture proposed by Wang and Kang (J Math Chem 47(3):937–958, 2010) is true. At the same time, our results also provide the simplified proofs of the main results of Wang and Kang (J Math Chem 47(3):937–958, 2010) and Li and Li (Electron J Linear Algebra 17:414–425, 2008).     

Cure temperature influence on natural rubber—A small angle X‐ray scattering study

To analyze the natural rubber behavior during vulcanization under different cure treatments, an experimental investigation using small angle X‐ray scattering was performed. To achieve this, a set of samples were prepared using sulfur and N‐t‐butyl‐2‐benzothiazole sulfenamide as accelerator and then cured at temperatures between 403 and 463 K reaching their optimum mechanical properties considering rheometer tests. The crosslink density of the samples was evaluated by means of the swelling technique in solvent. In the usual Lorentz corrected representation of the X‐ray scattered intensity, a maximum was observed in the plots corresponding to the cured samples, revealing a highly correlated structure. This maximum shifted toward higher values of the scattering vector when the cure temperature of the samples increased. This behavior is discussed in terms of the crosslinks type present in the vulcanized rubber network at different cure temperatures. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2966–2971, 2007     

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.     

Thermal properties of nanocomposites based on polyethylene and n-heptaquinolinum modified montmorillonite

The aim of the research was obtaining and application of smectic clay modifying agent. The organophilic clay is used as nanofiller in polymer nanocomposites [1]. A microwave-assisted reaction led to obtaining N-heptaquinolinum, which is amphiphilic compound, containing both hydrophobic (alkyl and aromatic) and hydrophilic sections in its structure [2]. N-heptaquinolinum was used as a montmorillonite clay modifying agent. Modification was carried out in formulated way [3, 4]. Modification efficiency was determined by X-ray diffraction (XRD) analysis and elementary analysis. Organophilic clay (Ch7) was introduced, using the extrusion method, into polyethylene matrix in different mass relations (1.5, 3 and 5?%) [3]. The structure of obtained materials was studied by means of XRD and SEM. To evaluate potential applications thermal properties of received nanocomposites were tested with thermogravimetric analysis and differential scanning calorimetry. The thermal stability of PE/clay composites can be improved in the case of loading 1.5 and 5?mass%.     

Activation enthalpies of pericyclic reactions: the performances of some recently proposed functionals

We have assessed the performances of three recently proposed functionals, RC (Ragot and Cortona in J Chem Phys 121:7671, 2004), TCA (Tognetti et al. in J Chem Phys 128:034101, 2008), and RevTCA (Tognetti et al. in Chem Phys Lett 460:536, 2008) by calculating the activation enthalpies for ten pericyclic reactions and eighteen 1,3-dipolar cycloadditions. We have found that the local functional (RC) gives results only marginally better than the local-density approximation ones, while the two GGA functionals, TCA and RevTCA, both strongly improve the results with respect to PBE. The performances of RevTCA, in particular, are not far different from those of a hybrid functional such as B3LYP.     

The development of radioactive glass surrogates for fallout debris

The production of glass that emulates fallout is desired by the nuclear forensics community for training and measurement exercises. The composition of nuclear fallout is complex, with widely varying isotopic compositions (Fahey et al., Proc Natl Acad Sci USA 107(47):20207–20212, 2010; Bellucci et al., Anal Chem 85:7588–7593, 2013; Wallace et al., J Radioanal Nucl Chem, 2013; Belloni et al., J Environ Radioact 102:852–862, 2011; Freiling, Science 139:1058–1059, 1963; Science 133:1991–1999, 1961; Bunney and Sam Government Report: Naval Ordinance Laboratory, White Oak, 1971). As the gaseous cloud traverses from hotter to cooler regions of the atmosphere, the processes of condensation and nucleation entrain environmental materials, vaporized nuclear materials and fission products. The elemental and isotopic composition of the fission products is altered due to chemical fractionation (i.e. the fission product composition that would be expected from fission of the original nuclear material is altered by differences in condensation rates of the elements); the fallout may be enriched or depleted in volatile or refractory fission products. This paper describes preliminary work to synthesize, irradiate and fractionate the fission product content of irradiated particulate glass using a thermal distillation 2 h after irradiation. The glass was synthesized using a solution-based polymerization of tetraethyl orthosilicate. (Izrael, Radioactive fallout after nuclear explosions and accidents, 2002) Uranium was incorporated into the glass particulate at trace concentrations during polymerization. The particulate was subjected to a short thermal neutron irradiation then heated to 1,273 K approximately 2 h after the end of irradiation. Fission products of ^{133, 134, 135}I, ^{132, 134}Te, ¹³⁵Xe, ¹³⁸Cs and ^{91, 92}Sr were observed to be distilled from the particulate. The results of these preliminary studies are discussed.     

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.