Penultimate Model Research Articles

Penultimate model in the study of the ‘bootstrap’ effect in the methyl methacrylate-acrylamide copolymerization system

Methyl methacrylate (MMA) acrylamide (AA) system was studied in radical conventionally and non-conventionally initiated polymerization reaction with respect to the ‘bootstrap’ effect using the penultimate model propagation kinetics of vinyl copolymerization. For this, two different methods, the Nelder-Mead simplex method and a scanning method have been adapted for the determination of r 1, r′ 1, r 2, r′ 2 parameters without constraints to the reactivities of the monomers used. Among the penultimate r-parameters the ranges of their statistically equivalent values have been determined. Good agreement of the calculated copolymer microstructure with results on the experimental basis for styrene-acrylonitrile monomer system proves the reliability of the calculation procedures. The triad fractions of the MMA-AA copolymer calculated on the basis of the penultimate model showed that the penultimate model does not provide any better representation of the copolymerization of MMA with AA than the previously discussed terminal model. Spectral analysis of the triads in copolymer is proposed to verify the right kinetic model as well as the applicability of the ‘bootstrap’ model for the monomer system.

Discriminating between the Terminal and Penultimate Models Using Designed Experiments: An Overview

Statistical model discrimination methods have been applied to the modeling of copolymerization kinetics in order to improve our ability to correctly choose between competing models. The objectives were to quantify the benefits of using model discrimination methods, to examine available copolymer measurements, and to rigorously compare model discrimination techniques. Simulations were developed (for three representative free-radical systems) capable of generating data based on either the terminal or penultimate model. Using the simulations to generate data, three model discrimination methods were applied to the three copolymer systems alternatively using copolymer composition, triad fraction data, and the combination of copolymer composition and rate as the measured variable(s). The results showed that model discrimination methods can correctly discriminate often using fewer experiments than have been used in the past and that they should be able to detect smaller differences between models.

Effect of Substituents on the Radical Copolymerization of Ring-Substituted Methyl 2-Cyano-3-Phenyl-2-Propenoates with Styrene

Copolymerizations of ring-substituted methyl 2-cyano-3-phenyl-2-propenoates, RC6H4CH = C(CN)CO2CH3 (R = C2H5, CH3O, Cl, Br, F) with styrene (M 1) were studied in solution in the presence of a radical initiator. Terminal and penultimate kinetic models were applied for best prediction of the copolymer composition. The Alfrey-Price Q and e parameters calculated for the monomers correlate well with the relative reactivity (1/r 1). The relative reactivity showed a tendency to increase with increasing Hammett constant σ, cyclic voltammetric reduction potential, E p, calculated atomic charge, and 13C-NMR chemical shift on the olefinic α-carbon of the ring-substituted monomers.

A Pulsed-Laser Study of Penultimate Copolymerization Propagation Kinetics for Methyl Methacrylate/n-Butyl Acrylate

The PLP/MWD technique, based upon analysis of the molecular weight distribution (MWD) of a polymer produced by pulsed-laser polymerization (PLP), has proven to be a powerful probe for the examination of free-radical homopolymerization kinetics. The same method is applicable to the study of copolymerization propagation kinetics provided that proper calibrations are established for the measurement of copolymer MWDs, as illustrated by an examination of the methyl methacrylate (MMA)/n-butyl acrylate (nBA) system. Statistical analysis indicates that although MMA/nBA copolymer composition is well described by the terminal model, propagation kinetics are not. Like many other systems recently examined, the data are well represented by the “implicit penultimate unit effect” model of Fukuda and co-workers. However, whereas previous systems studied have propagation rates that are lower than those predicted by the terminal model, it is found that the propagation rate of MMA/nBA is greater than the terminal model predictions. The implications of these results to theoretical interpretations of the penultimate model are discussed.

Non‐uniqueness in determination of terminal and penultimate model reactivity ratios in the styrene‐methyl methacrylate free‐radical copolymerization system

AbstractThe terminal and penultimate model reactivity ratios in the free‐radical copolymerization of styrene and methyl methacrylate in bulk at 40°C were calculated by means of the simplex and scanning methods. Calculations showed that for the terminal model r1 and r2 vary in comparatively narrow ranges of 0.548–0.552 and 0.480–0.483, respectively. For the penultimate model, the most accurate reactivity ratios calculated by the simplex method, which were r11 = 0.727, r22 = 0.490, r21 = 2.890, r12 = 4.583, are surrounded with sets of reactivity ratio values of equal accuracy. The ranges of variation were found to be 0.711–0.746, 0.487–0.492, 2.810–2.970 and 4.213–5.049, respectively. Numerical values of the penultimate r‐parameters calculated with the simplex method depend, due to the structure of the multidimensional space (r11, r22, r21, r12, σ), on the initial guess for the r‐parameters. Use of the covariance matrix for the estimation of the indetermination ranges is discussed.

Improved increments for characterization of comonomer sequencing in binary copolymers

The degree of randomness of higher order than first order was proposed to characterize the alternating structure of binary copolymers. The introduced parameter describes the alternating structure more precisely than the degree of alternation already proposed in the literature. Furthermore, the new parameter can be used in connection with standard copolymerization models where the need for such parameters is especially high. In this domain the degree of alternation fails. The usefulness of the degree of randomness was tested on model chains and two standard copolymerization models: the terminal and simplified penultimate model. This examination proves that more precise information on comonomer sequencing can be obtained when sequences longer than diads are described by means of the degree of randomness.

Penultimate Unit Effect in Free-Radical Copolymerization

Based upon new insights into the dynamics of propagation reactions, a theoretical justification of the penultimate model in free-radical copolymerization kinetics is proposed. It is suggested that the penultimate unit effect may contain a large steric contribution (a different penultimate unit may affect the frequency factor for propagation by a factor typically between 1 and 10) and that the effect on activation energies is generally small. This implies that the stabilization energy model, where only differences in activation energies play a significant role, is valid only under unusual circumstances. Furthermore, a possible explanation is given for the occurrence of both the implicit and explicit penultimate unit effects. It is shown that mainly systems in which there is a large difference in the sizes of the substituents of both monomers are expected to exhibit an explicit penultimate unit effect. Finally, it is shown that neither a purely enthalpic effect, nor the steric effect proposed here, provides an adequate explanation why, experimentally, mainly radical reactivity ratios less than unity are found.

Polymerization of an acetoxyvinyl substituted chlorocyclophosphazene

The vinyl acetate derivative, gem-isopropyl-2-(α-acetoxyvinyl)tetrachlorocyclotriphosphazene (1), has been used in radical homopolymerization and copolymerization reactions with methyl methacrylate, (MMA) and styrene. The 1,1-disubstituted olefin did not undergo radical homopolymerization. Copolymers derived from MMA and 1 contained only low amounts (<2 mol%) of 1. A maximum incorporation of 20 mol% of phosphazene monomer was achieved with styrene as comonomer. The data obtained in low- and high-conversion copolymerizations with styrene were used to calculate the monomer reactivity ratios. The results of the calculations show that the terminal model is not operative. Calculation for the penultimate model with r2=0 resulted in r1 and r′1 values of 2.8±0.2 and 0.7±0.1, respectively. For the hypothetical homopolymer of 1 a Tg value of 441 K was calculated. All the copolymers with styrene exhibit flame-retardant properties.

Preferential sorption of monomers and molecular weight effects in radical copolymerization

AbstractSpecific thermodynamic properties of monomer mixtures as a solvent for copolymers lead to preferential sorption of one of the monomers by the polymer coils and preferential solvation of growing chains. The dependence of the preferential sorption coefficient upon the molecular weight of the copolymers results in the dependence of copolymer composition and monomer reactivity ratios upon the molecular weight of the copolymers and in gradient chemical heterogeneity of the copolymers. “Non‐classical” copolymerizations can be represented by both terminal and penultimate models.

Propagation and termination processes in free radical copolymerization

AbstractThe propagation and termination processes in free radical copolymerization were discussed on the basis of current experimental data. It was shown that the penultimate unit effect (PUE) in propagation is very general, affecting not only the absolute values of the rate constant but also the monomer reactivity ratios r1 and r2. The analysis suggests that the alternating index r1r2 originates in the PUE. An adequate description of the termination process also requires a penultimate model, which leads to a general, yet simple two‐parameter rate‐of‐copolymerization equation with PUEs in both propagation and termination.

An experimental verification of statistical discrimination between the terminal and penultimate copolymerization models

In this article, the modified Buzzi–Ferraris model discrimination method was used to design experiments to discriminate between the terminal and penultimate models for styrene methyl methacrylate (STY/MMA) copolymerization. The measured variables were 13C-NMR peak areas. The peak area assignments of Aerdts8 were used. After nine experiments, the terminal model was picked over the penultimate model at 99.99% confidence. More importantly, the experimental data showed that computer simulated data used in previous studies were realistic, and the conclusions drawn from the simulation studies were valid. This experimental verification continues to show that the use of statistical model discrimination techniques can improve our ability to discriminate between competing copolymerization models. © 1996 John Wiley & Sons, Inc.

An experimental verification of statistical discrimination between the terminal and penultimate copolymerization models

In this article, the modified Buzzi–Ferraris model discrimination method was used to design experiments to discriminate between the terminal and penultimate models for styrene methyl methacrylate (STY/MMA) copolymerization. The measured variables were 13C-NMR peak areas. The peak area assignments of Aerdts8 were used. After nine experiments, the terminal model was picked over the penultimate model at 99.99% confidence. More importantly, the experimental data showed that computer simulated data used in previous studies were realistic, and the conclusions drawn from the simulation studies were valid. This experimental verification continues to show that the use of statistical model discrimination techniques can improve our ability to discriminate between competing copolymerization models. © 1996 John Wiley & Sons, Inc.

The copolymerisation of tetrahydrofurfuryl methacrylate and methyl methacrylate

Abstract The bulk copolymerisations of tetrahydrofurfuryl methacrylate, THFMA, and methyl methacrylate, MMA, were carried to low conversion at 60 °C, using benzoyl peroxide, BPO, as the initiator. The compositions of the copolymers were determined using 13C NMR spectroscopy. The glass transition temperatures of the copolymers were determined using differential scanning calorimetry. From the compositional data, the statistical F-test revealed that both the terminal and penultimate models adequately describe the copolymerisation at the 95% confidence level. However, further statistical analysis showed that no significant improvement in fit was evident when considering the penultimate model over the simpler terminal model. Therefore, on the basis of the terminal model, the reactivity ratios were evaluated using a non-linear least squares analysis and were found to be rT = 1.06 and rM=0.81 for THFMA and MMA, respectively. Tacticity splittings of the methyl signal in the 13C NMR spectra were consistent with the Bernoullian distribution of stereoregularity. The iso-tacticity parameters were found to be δ TT = 0.24, δ MM = 0.22 and δ ∗ =0.20 . The glass transition temperatures of the prepared polymers were fitted to the Gordon-Taylor equation and the value of the Gordon-Taylor constant, K, was found to be 1.36 suggesting larger contributions from MMA, compared to THFMA, to the observed glass transition temperatures.

NMR Measurement of Identical Polymer Samples by Round Robin Method IV. Analysis of Composition and Monomer Sequence Distribution in Poly(methyl methacrylate-co-acrylonitrile) Leading to Determinations of Monomer Reactivity Ratios

In order to assess the reliability of NMR measurement of polymers, 1H and 13C NMR data for three copolymers of methyl methacrylate (MMA) and acrylonitrile (AN) prepared with AIBN were collected from 46 spectrometers whose resonance frequencies for 1H NMR measurements ranging from 90 to 500 MHz. 1H and 13C NMR spectra were measured in nitrobenzene-d5 at 110°C and acetonitrile-d3 at 70°C, respectively. Standard deviations (σ’s) for chemical shift measurements of the 1H and 13C NMR signals were 0.003—0.008 ppm and 0.03—0.05 ppm, respectively. Compositions of the copolymers were determined from the relative intensities of the signals due to the OCH3 (MMA) and CH (AN) protons, and the σ values for the determinations were 3.7—9.5%. The compositions determined from 13C NMR (C=O for MMA unit, CN for AN unit) agreed well with those obtained from 1H NMR. Monomer reactivity ratios rij (i,j=1 or 2) for a penultimate model were determined from monomer feed ratios and triad fractions obtained from the C=O (MMA) and CH (AN) carbon signals. Most of the σ values for rij determinations were 5—14%. While r22 and r12 are nearly equivalent, r11 and r21 are significantly different from each other, indicating a possible existence of the penultimate-unit effect in the copolymerization of MMA and AN. Terminal model reactivity ratios, r1 and r2, determined formally from the compositions of three samples by Fineman-Ross method showed large σ values (22—24%).

Penultimate Unit Effects in Free-Radical Copolymerization. Number-Average Degree of Polymerization and Copolymerization Rate in Styrene/Methyl Methacrylate/Toluene System

Recently, Fukuda et al. showed from the measurements of absolute rate constants that in the free-radical copolymerization the propagation step obeys the penultimate model and the termination step is diffusion-controlled. Also Fukuda et al. proposed the “radical stabilization energy model” in the free radical copolymerization. To examine this copolymerization model, relationships between feed monomer composition f1 and number-average degree of polymerization Xn and between f1 and copolymerization rate Rp were investigated for the copolymerization of styrene and methyl methacrylate in toluene at 40°C. The composition of the copolymer F1 was determined by 1H NMR. The number-average molecular weight of the copolymer Mn was measured to calculate Xn by osmometry and size exclusion chromatography (SEC). By the so-called terminal model (the Mayo–Lewis model), the copolymer composition data was well described while the data of Xn and Rp Were not described. Alternatively, the model proposed by Fukuda et al. described well the composition data, the Xn data and the Rp data.

Model discrimination via designed experiments: Discrimination between the terminal and penultimate models based on rate data

Simulations have been used to study the application of model discrimination methods to the problem of discriminating between the terminal and penultimate copolymerization models on the basis of composition and rate data. Both models are reviewed and the three model discrimination methods chosen for study are explained. Composition data have been used in conjunction with rate data due to the high correlation between estimated parameters based solely on rate data. This correlation has been discussed. The simulation programs are then explained including the generation of simulated data. Data are presented to show that the simulation programs can duplicate the experimental observations of other researchers. The results show that both the modified Buzzi-Ferraris method and the exact entropy method are able to correctly discriminate between the models in greater than 95% of the simulations. Differences between the three model discrimination methods are discussed along with differences between the three copolymer measurements that have been studied.

A high resolution NMR investigation into the microstructure of HEMA and EEMA copolymers

13C NMR spectra (125 MHz) of 2-hydroxyethyl methacrylate (HEMA) and 2-ethoxyethyl methacrylate (EEMA) copolymers, prepared by bulk free radical copolymerisation, were analysed in terms of the comonomer composition and relative stereochemical configuration. The monomer reactivity ratios were evaluated using a nonlinear least squares analysis, considering both the terminal and penultimate models. Although both models adequately describe the experimentally determined copolymer compositions at a 99.5% confidence level or better, the statistical F-test showed that there is no significant improvement in the fit of the data for the penultimate model over that for the terminal model. The best estimates for the terminal model reactivity ratios are rH = 1.202 and rE = 0.724. The copolymers were found to be mainly syndiotactic. Using the Bernoullian distribution of stereoregularity, the coisotacticity parameters ΘHH = 0.25, ΘEE = 0.23 and Θ* = ΘHE = ΘEH = 0.21 were determined. Predictions for stereochemical and comonomer distributions in a copolymer prepared to high conversions are presented.

The copolymerisation of 2,4-dicyanobut-1-ene with N-vinyl carbazole

The copolymerisation of the monomers 2,4-dicyanobut-1-ene and N-vinyl carbazole has been studied. The presence of a charge transfer complex from these two monomers with a 1:1 stoichiometry has been established, and a value of 0.22 dm 3 mol −1 at 298 K for the equilibrium constant for complex formation has been obtained from NMR measurements. The copolymer composition data have been analyzed allowing for the participation of the complex in the propagation reactions; the results have been compared with the predictions of two simpler terminal models and with the penultimate model.

Model discrimination via designed experiments: Discriminating between the terminal and penultimate models based on triad fraction data

AbstractComputer simulations have been used to study the application of statistical model discrimination techniques to the modelling of copolymerization reactions based on triad fraction data. Since there are six triad fraction measurements, the problems of parameter estimation and model discrimination become multivariate problems. The multivariate forms of three model discrimination methods (exact entropy, Hsiang and Reilly, and Buzzi‐Ferraris et al.) are presented. Programs have been developed to simulate the application of these techniques to the systems styrene‐acrylonitrile, styrene‐methyl methacrylate and styrene‐butyl acrylate. The simulation programs are explained, including the method used to simulate triad fraction data. Evidence is presented that the simulation programs are capable of duplicating the type of measurements that would be obtained in the laboratory. The simulation results show that model discrimination methods are able to accurately and reliably discriminate between the terminal and penultimate models. The use of simulated model discrimination methods leads to reliable discrimination in fewer experiments than have been used in past work. Also, model discrimination methods are able to detect smaller penultimate effects than those found by Hill et al. for styrene‐acrylonitrile. In addition, the results show that the use of four of the six triad fractions, versus one copolymer composition measurement, should lead to more precise reactivity ratio estimates and an increased ability to discriminate between the terminal and penultimate models. Our work suggests that use of model discrimination methods will indeed lead to improvements in the modelling of copolymerization reactions.

A new linear method to calculate monomer reactivity ratios by using high-conversion copolymerization data: penultimate model with r2=0

The new iterative linear least-squares method for calculation of monomer reactivity ratios for the terminal model, proposed by Mao and Huglin, is applied to the penultimate model system with r 2=0. The principle and the calculation procedure are similar for the two different models. All advantages of the new method when applied to the terminal model, i.e. fast convergence, exact calculation and the fact that any initial estimation of reactivity ratios always leads to the same correct results, remain valid when applied to the penultimate model. The superiority of the new method over the Kelen-Tüdős and extended Kelen-Tüdős procedures is even more evident for the penultimate model than for the terminal model.