DiBenedetto Equation Research Articles

Diffusion of reacting epoxy and amine monomers in polysulfone: a diffusivity model

In this work, a diffusivity model based on free volume theory is presented for the simultaneous diffusion of diglycidyl ether of bisphenol A (DGEBA) epoxy and bis(p-aminocyclohexyl) methane (PACM 20) amine monomers into amorphous polysulfone (PSU). This model is expected to predict and explain the diffusion behavior of the epoxy and amine monomers into PSU during the initial time periods. The overall free volume of the polymer system is estimated using a Kelley–Bueche approximation for free volume in a binary mixture consisting of a non-reacting thermoplastic and the reacting thermoset. The fractional free volume of the thermoset is estimated by the DiBenedetto equation. The model is valid only for low epoxy–amine concentrations and degrees of cure. The diffusivity model developed here suggests that reaction reduces the species diffusivity with increasing cure from a loss of the overall fractional free volume for diffusion. Further, a model for increased epoxy diffusivity from amine-induced PSU swelling is presented and validated using data from previous studies on the single-component diffusion of epoxy into amine-swollen PSU. By combining the reaction and swelling terms with the Arrhenius epoxy diffusivity, the epoxy diffusivity expression during the simultaneous diffusion and reaction of epoxy and amine into PSU for small times, is determined. Parametric studies on the nature of diffusivity are performed to determine the influence of the various free volume parameters on thermoset diffusion, and these studies show that the thermoset diffusivity, in general, decreases with time from reaction.

Glass-transition temperature: Conversion relationship in the polycyclotrimerization of 4,4?-thiodiphenylcyanate

Polycyclotrimerization of 4,4′-thiodiphenylcyanate was adopted as a model system for general thermosetting polymers for studying the relationship between the glass-transition temperature (Tg) and conversion (α) during network formation. Existing expressions for Tg-α relationship were used and compared. The experimental Tg-α data were well fitted to several one-parameter equations although the physical significance of parametric values thus obtained could not be unambiguously identified. Among the two-parameter models, both the Hale–Macosko–Bair equation and the so-called “original” DiBenedetto equation were well fitted by experimental data (when the mean-field crosslink density was used), yielding parametric values consistent with the original designated physical meanings within the corresponding theoretical frames. Relationships between the parameters in different theories were also discussed. Incidentally, a discontinuity of ΔCpTg at the gel point was observed (i.e., ΔCpTg is of different values in the pregel and postgel regimes, respectively). © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 726–738, 2000

Glass-transition temperature-conversion relationship for an epoxy-hexahydro-4-methylphthalic anhydride system

The DGEBA–MHHPA epoxy system has found increasing applications in microelectronics packaging, making crucial the ability to understand and model the cure kinetics mechanism accurately. The present article reports on work done to elucidate an appropriate model, modified from the empirical DiBenedetto's equation, to relate the glass-transition temperature (Tg) to the degree of conversion for a DGEBA–MHHPA epoxy system. This model employs the ratio of segmental mobility for crosslinked and uncrosslinked polymers, λ, to fit the model curve to the data obtained. A higher ratio value was shown to indicate a more consistent rate of increase of Tg in relation to the degree of conversion, while a lower value indicated that the rate of Tg increase was disproportionately higher at higher degrees of conversion. The best fit value of λ determined by regression analysis for the DGEBA–MHHPA epoxy system was 0.64, which appeared to be higher than for those previously obtained for other epoxy systems, which ranged from 0.43–0.58. The highest Tg value obtained experimentally, Tg max, was 146°C, which is significantly below the derived theoretical maximum Tg∞ value of 170. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 78: 511–516, 2000

Cure monitoring of catalysed cyanate ester resins

The cure behaviour of two bisphenol A-based cyanate ester resins, AroCy B10 and B30, catalysed by copper acetylacetonate and nonylphenol was studied. For this purpose, differential scanning calorimetric (DSC) and rheological measurements were carried out at temperatures between 130 and 170 °C. The cyanate conversion profiles are fitted with a second-order rate law in the kinetically controlled regime where a good time–temperature superposition is attained. However, it is necessary to add an empirical kinetic term to give a good description of the entire range of curing. Simultaneously, times to gelation and vitrification have been determined by dynamic rheological measurements over the same temperature range. The corresponding conversions have been calculated by correlation of rheological and DSC data, the vitrification conversion being slightly higher for the prepolymer. Independent of the cure temperature, a good correlation between cyanate conversion and glass transition temperature was obtained. The isothermal time–temperature–transformation diagrams for these systems are constructed from the kinetic model and the DiBenedetto equation, and show good agreement with the experimental data. © 2000 Society of Chemical Industry

A study of the kinetics of the microwave cure of a phenylethynyl-terminated imide model compound and imide oligomer (PETI-5)

The kinetic mechanism of the microwave cure of a simple phenylethynyl-terminated imide model compound, 3,4′-bis[(4-phenylethynyl)phthalimido]diphenyl ether (PEPA-3,4′-ODA) and a phenylethynyl-terminated imide oligomer (PETI-5, Mn 5000 g/mol) was studied. Dielectric properties of the model compound and PETI-5 were measured in the microwave range from 0.4 GHz to 3 GHz. FTIR was used to follow the cure of the model compound (PEPA-3,4′-ODA), while thermal analysis (DSC) was used to follow the cure of the PETI-5 oligomer. The changes in room temperature IR absorbance of phenylethynyl triple bonds at 2214 cm−1 of PEPA-3,4′-ODA as a function of cure time were measured after cure temperatures of 300, 310, 320, and 330 °C. The changes in the glass-transition temperature, Tg, of PETI-5 as a function of cure time were measured after cure at 350, 360, 370, and 380 °C, respectively. The Tg 's were determined to calculated the relative extent of cure, x, of the PETI-5 oligomer according to the DiBenedetto equation. For the model compound, the reaction followed first order kinetics, yielding an activation energy of 27.6 kcal/mol as determined by infrared spectroscopy. For PETI-5, the reaction followed 1.5th order, yielding an activation energy of 17.1 kcal/mol for the whole cure reaction, as determined by Tg using the DiBenedetto method. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 2526–2535, 2000

Glass transition temperatureversus conversion relationship in the polycyclotrimerization of aromatic dicyanates

Comparisons of the effectiveness of different theoretical equations to fit various sets of experimental Tg versus conversion data are made. Among all the theoretical expressions for the Tg versus conversion relationship, one-parameter equations give excellent fitting to the experimental data. The values of parameters obtained thereby can be merely considered as fitting parameters without realistic physical meanings. Mean-field crosslink density and elastically effective crosslink density are derived and applied to the corresponding theoretical equations for comparison in our aromatic dicyanate systems. The Hale–Macosko–Bair (HMB) equation and the so called original DiBenedetto equation combining the mean-field crosslink density, rather than elastically effective crosslink density, are applied with success to fit experimental data; hence the parameter values obtained by this method are more consistent with the original designated physical meanings within the corresponding theoretical frameworks. This is attributed to the fact that the inelastic intramolecular loops may still contribute to the increase in Tg, although they make no contribution to the gelation. An expression relating the effect of inelastic intramolecular loops on the gel fraction is also derived. Relationships between the parameters in different theories are discussed. © 2000 Society of Chemical Industry

Lignocellulosic substrates influence on TTT and CHT curing diagrams of polycondensation resins

Lignocellulosic substrates such as wood were found to have a marked modifying influence on a well-defined region of CHT diagrams during hardening of phenol–formaldehyde (PF) and urea–formaldehyde (UF) polycondensates. This was ascribed to more complex resin phase transitions due to resin/substrate interactions peculiar to these substrates. The chemical and physical mechanisms of the interactions of the resin and substrate causing such CHT diagram modifications are presented and discussed. The Di Benedetto equation describing the glass transition temperature Tg of the system as a function of the resin degree of conversion p has been slightly modified to take into account the modified CHT diagram. The modified CHT diagram can be used to good effect to describe the behavior of polycondensation resins when used as wood adhesives during their curing directly into the wood joint. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 71: 915–925, 1999

Curing conditions effects on the characteristics of thermosetting adhesives-bonded wood joints — Part 1: Substrate influence on TTT and CHT curing diagrams of wood adhesives

Lignocellulosic substrates such as wood were found to have a marked modifying influence on a well-definite region of CHT diagrams during hardening of phenol-formaldehyde (PF) and urea-formaldehyde (UF) polycondensates. This was ascribed to more complex resin phase transitions due to resin/substrate interactions peculiar to these substrates. The chemical and physical mechanisms of the interactions of resin and substrate causing such CHT diagram modifications are presented and discussed? The Di Benedetto equation describing the glass transition temperature Tg of the system as a function of the resins degree of conversion p has been slightly modified to take into account the modified CHT diagram. The modified CHT diagram can be used to good effect to describe the behaviour of polycondensation resins when used as wood adhesives during their curing directly in the wood joint.

A study of the thermal cure of a phenylethynyl-terminated imide model compound and a phenylethynyl-terminated imide oligomer (PETI-5)

The kinetic mechanism of the thermal cure of a phenylethynyl-terminated imide model compound, 3,4′-bis[(4-phenylethynyl)phthalimido]diphenyl ether (PEPA-3,4′-ODA) and a phenylethynyl-terminated imide oligomer PETI-5 (MW 5000 g/mol) was studied. FTIR was used to follow the cure of the model compound, while thermal analyses (DSC) was used to follow the cure of the PETI-5 oligomer. The changes in IR absorbance of phenylethynyl triple bonds at 2214 cm−1 of PEPA-3,4′-ODA as a function of cure time were detected at 318, 336, 355, and 373°C, respectively. The changes in the glass transition temperature, Tg, of PETI-5 as a function of time were measured at 350, 360, 370, 380, and 390°C, respectively. The DiBenedetto equation was applied to define the relative extent of cure, x, of the PETI-5 oligomer by Tg. For the model compound, the reaction followed first order kinetics, yielding an activation energy of 40.7 kcal/mol as determined by infrared spectroscopy. For PETI-5, the reaction followed 1.5th order, yielding an activation energy of 33.8 kcal/mol for the whole cure reaction, as determined by Tg using the DiBenedetto method. However, the cure process of PETI-5 just below 90% by this method followed first-order kinetics yielding an activation energy of 37.2 kcal/mol. © 1998 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 36: 461–470, 1998

A study of the thermal cure of a phenylethynyl‐terminated imide model compound and a phenylethynyl‐terminated imide oligomer (PETI‐5)

The kinetic mechanism of the thermal cure of a phenylethynyl-terminated imide model compound, 3,4′-bis[(4-phenylethynyl)phthalimido]diphenyl ether (PEPA-3,4′-ODA) and a phenylethynyl-terminated imide oligomer PETI-5 (MW 5000 g/mol) was studied. FTIR was used to follow the cure of the model compound, while thermal analyses (DSC) was used to follow the cure of the PETI-5 oligomer. The changes in IR absorbance of phenylethynyl triple bonds at 2214 cm−1 of PEPA-3,4′-ODA as a function of cure time were detected at 318, 336, 355, and 373°C, respectively. The changes in the glass transition temperature, Tg, of PETI-5 as a function of time were measured at 350, 360, 370, 380, and 390°C, respectively. The DiBenedetto equation was applied to define the relative extent of cure, x, of the PETI-5 oligomer by Tg. For the model compound, the reaction followed first order kinetics, yielding an activation energy of 40.7 kcal/mol as determined by infrared spectroscopy. For PETI-5, the reaction followed 1.5th order, yielding an activation energy of 33.8 kcal/mol for the whole cure reaction, as determined by Tg using the DiBenedetto method. However, the cure process of PETI-5 just below 90% by this method followed first-order kinetics yielding an activation energy of 37.2 kcal/mol. © 1998 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 36: 461–470, 1998

Modelling the Cure of a Commercial Epoxy Resin for Applications in Resin Transfer Moulding

An analytical procedure has been developed for modelling the kinetics of the cure process of a commercial epoxy resin for resin transfer moulding (RTM) applications, using differential scanning calorimetry (DSC) in the isothermal and dynamic modes to obtain the experimental database. The overall reaction rate of the epoxide groups with amines was determined and fitted by an autocatalytic kinetic model. An improvement of the model to allow for diffusion limitation effects results in a good agreement between experimentally determined and predicted reaction rates. A non-linear least squares regression analysis method based on Marquardt's algorithm was used to fit the DSC reaction rate data with an appropriate model and to evaluate the activation energies and the reaction orders for this particular resin system. The Di Benedetto equation was utilised to establish the relationship between conversion and glass transition temperature (T g ), required to develop the diffusion-dominated part of the model.

Comparison of microwave and thermal cure of an epoxy/amine matrix

AbstractAn investigation was carried out into the effect ofa microwave cure on an epoxy prepolymer with a cycloaliphatic diamine mixture, as compared to a standard thermal cure. The microwave waveguide and process (propagation mode TE01) were adjusted to obtain large homogeneous samples. The extent of reaction, x, was measured during the microwave processing by size exclusion chromatography and differential scanning calorimetry. A good estimate of x was found using a modified DiBenedetto equation correlating x and the glass transition temperature Tg. The homogeneity of the samples was checked during the last steps of cure, showing the efficiency of the microwave processing and waveguide. The influence of the nature of the mold (metallic or dielectric) on the reaction kinetic was also investigated. Samples cured by both thermal and microwave processing were characterized by dynamic and static mechanical properties and then compared with those of fully crosslinked networks, i.e., postcured at a high temperature.

Isothermal curing of an uncatalyzed dicyanate ester monomer: Kinetics and modeling

AbstractThe isothermal cure of a dicyanate ester monomer has been studied under different atmospheres (air and argon) without any catalyst, by following the evolutions of both glass transition temperature and conversion. Independently of both atmosphere and temperature of cure, there is a unique one‐to‐one relationship between these two parameters, correctly described by a restated DiBenedetto's equation. The chemical kinetics are satisfactorily modeled with two competing reactions schemes, which are second‐order and second‐order autocatalyzed, the latter displaying more relative importance under air‐curing conditions. Whatever the cure temperature may be, gelation occurs around 0.60 conversion under air, whereas the theoretical xgel = 0.50 is experimentally observed under argon. The time–temperature‐transformation diagrams have been established. © 1993 John Wiley & Sons, Inc.

Curing reaction of o‐cresol novolac epoxy resin according to hardener change

AbstractThe investigations of cure kinetics and glass transition temperature (Tg) versus reaction conversion (α) of o‐cresol novolac epoxy resin with the change of hardener were performed. All kinetic parameters of the curing reaction such as the reaction rate order, activation energy, and frequency factor were calculated. The curing mechanisms were classified into two types. One was an autocatalytic mechanism and the other was a nth order kinetic mechanism. The constants related to the chain mobility of polymer segments were obtained by using the DiBenedetto equation. We have tried to correlate the relationships between curing mechanism and molecular structures of hardeners from these results. © 1993 John Wiley & Sons, Inc.

Cure kinetics of elementary reactions of a DGEBA/DDS epoxy resin: 1. Glass transition temperature versus conversion

The influence of elementary reactions (linear polymerization and crosslinking) on the development of the glass transition temperature ( T g) of diglycidyl ether of bisphenol A/diaminodiphenylsulfone (DGEBA/DDS) epoxy resin system has been analysed quantitatively according to a proposed model. The model was derived on the basis of the reaction mechanisms observed in the curing system and the assumption that the T g has individual linear relationships with the degree of conversion of the two reactions. The conversions of elementary reactions during the cure were obtained experimentally by measuring the changes in the concentrations of epoxy, primary and secondary amine groups with time by near-infra-red spectroscopy. The T g values were measured by differential scanning calorimetry (d.s.c.). In addition, and for the purpose of comparison, the overall conversion of the system was also obtained from d.s.c. in a dynamic scanning mode. The conversion-time data from the two different techniques were consistent. The proposed model successfully predicted the separate increases in T g due to the linear polymerization and due to the crosslinking reactions. The Di Benedetto equation and the viscoelastic model of Gan et al. are also discussed.

Measurement of the extent of reaction of an epoxy–cycloaliphatic amine system and influence of the extent of reaction on its dynamic and static mechanical properties

AbstractAn investigation was carried out into the effect of the extent of reaction on the mechanical properties, in a mixture of an epoxy prepolymer and a cycloaliphatic diamine, above the gel point. First, the most adequate technique to approach the extent of reaction was chosen between differential scanning calorimetry, Fourier transformation infrared, and size exclusion chromatography. It was found that the differential scanning calorimetry gave a good estimate of the extent of reaction, measuring the glass transition temperature and using a modified DiBenedetto equation. Dynamic mechanical properties and static mechanical properties are shown to be highly dependent upon the extent of reaction. The β relaxation was found to be responsible for the fracture properties at room temperature through the changes in extent of reaction. © 1992 John Wiley & Sons, Inc.

A viscoelastic description of the glass transition-conversion relationship for reactive polymers

The glass transition temperature,T g is a sensitive and practical parameter for following cure of reactive thermosetting systems. A new equation was developed for predicting theT g -conversion relationship based on the Dillman-Seferis viscoelastic compliance model. It assumes that the changes inT g are primarily due to changes in relaxation time as chain extension and crosslinking reduce the mobility of a polymer network. Such information is essential in combining kinetic and viscoelastic measurements, which monitor transformations of thermosets during cure. The equation derived from the viscoelastic model was shown to be applicable for a variety of experimental data. The success of the methodology was further demonstrated by comparing well-established relations, such as the Fox equation and the Di-Benedetto equation, to predictions made possible by adjusting two viscoelastic model parameters. Finally, the fitting power of the proposed equation was shown by fitting published epoxy data from the literature as well as experimental data on a relatively new resin system such as dicyanates used as a model in this study.

Relationships between glass transition temperature and conversion

The Couchman approach to the expression of the compositional variation of glass transition temperature (or the Di-Benedetto equation) can be used to evaluate the effect of molecular weight on Tg for a stepwise linear polymer. An expression for the KL constant in the Fox-Flory equation is proposed. It is also possible to estimate the TgL corresponding to the hypothetical linear structure defined in a three-dimensional network. For different epoxy-diamine systems, we have found a good concordance with the results obtained previously using an additive law.

Glass transition temperature versus conversion relationships for thermosetting polymers

AbstractOwing to enthalpy relaxation, values of the glass transition temperature (Tg) for partially reacted polymers may depend on the thermal history of samples and the heating rate used for measurements. Use of theoretical relations between Tg and the extent of reaction (x) of a thermoset must take this fact into account. The original DiBenedetto equation has been reevaluated as a convenient constitutive equation for expressing Tg versus x. An extension of Couchman's approach for the expression of the compositional variation of Tg enabled us to derive the same functionality as given by the DiBenedetto equation. Thus, the DiBenedetto equation may be regarded as based on entropic considerations applied to a model of the thermosetting polymer consisting of a random mixture of a fully reacted network with the initial monomers in an amount which depends on the particular conversion level. These two equations have been applied with success to different diepoxy‐diamine copolymers.

Effects of the structure of the aromatic curing agent on the cure kinetics of epoxy networks

The curing of an epoxy prepolymer based on the diglycidyl ether of bisphenol A (DGEBA) with different aromatic diamines, i.e. 4,4′-diaminodiphenylmethane (DDM), 4,4′- and 3,3′-diaminodiphenylsulphone (DDS), bis[4,(4-aminophenoxy)phenyl]sulphone (BAPS) and 2,2′-bis[(4-aminophenoxy)phenyl]propane (BAPP), was analysed using size exclusion chromatography, viscosity measurements and differential scanning calorimetry as the main techniques. The diamines chosen have high melting temperatures. The diamines DDM, 4,4′-DDS, 3,3′-DDS and BAPP dissolve easily in DGEBA, but an unprocessable product results with BAPS. As determined by size exclusion chromatography and viscosity measurements, the reactivities of the diamines are in the order: 4,4 ′- DDS<3,3 ′- DDS< BAPS< DDM Using the fact that the molecules of aromatic diamines disappear more rapidly than the molecules of diepoxy, we have calculated a ratio of the reactivity of secondary to primary amine hydrogens ( k 2 k 1 in the range 0.8–0.9 for DDS diamines. All of the results have been plotted on a time-temperature-transformation diagram. The adjustable parameter λ appearing in Di Benedetto's equation may be estimated from the gel T g value or from ΔC p measurements. This adjustable parameter is approximately 0.4 for DDM and 4,4′-DDS, and approximately 0.6 for 3,3′-DDS and BAPP.