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Simulation of chemical kinetics of elastomer crosslinking by organic peroxides.


The general mechanism of peroxide crosslinking is widely accepted (1-3). Upon heating, the peroxide decomposes into the primary alkoxy or other radicals, which may further react to secondary radicals. These radicals may then abstract hydrogen atoms from the elastomer chain. Crosslinks may consequentially be formed by combination of two macromolecular radicals, whereas they may be formed via combination, but also via addition of the macromolecular radical to the unsaturated site in a second elastomer chain (4), (5).

The kinetics of crosslinking has been studied in two manners. The first are phenomenological approaches, which generally tend to describe the form of the cure curve, but seldom try to encompass chemical reactions with their appropriate mechanisms involved in the crosslinking process. The macrokinetics described by the phenomenological approaches are often represented by power law and other models developed by Piloyan et al. (6), Kamal and Sourour (7), and Isayev and Deng (8). However, regardless of the number of parameters applied in the model, the parameters are to be compared only for similar systems giving us a relative criterion for overall reaction rate and order in the system in question. Furthermore, the parameters are rarely linked to the reaction mechanisms and are thus hardly useful for any insight into the course of reactions and therefore extrapolative prediction. The degree of cure, i.e. the conversion equivalent, is not referred to the conversion of specific species present in the compound, although it is a lumped representation of various crosslinks formed or dissociated during the crosslinking of elastomers. The power law model was initially applied for crosslinking of polyethylene (PE) (9) and then for several other polymers (10), nitrile butadiene rubber (NBR) (11), fluorinated elastomers (12), ethylene-propylene-diene monomer rubber (EPDM) (13-17), EPDM/PE blends (18), poly(ethylene-co-methyl acrylate)/poly (dimethyl siloxane) (PDMS) blends (19), PE/PDMS blends (20), poly (ethylene-co-vinyl acetate) (EVA)/PDMS blends (21),and epoxidized polybutadiene (BR) (22). The model developed by Piloyan, on the other hand, was applied for processing of PE (23) and crosslinking of PE foams (24). Besides the common crosslinking, the power law model was applied for the crosslinking with accompanying grafting of monomers, such as grafting of various monomers onto PE (25) or maleic anhydride onto BR (26), as well.

In contrast to the phenomenological approaches, mechanistic approaches have their foundations in the reaction chemistry and mechanisms. The first mechanistic approaches originated from the analogy between peroxide-initiated polymerization and crosslinking. Hergenrother (27) investigated the crosslinking of BR, whereas Gancarz and [sws1] Laskawski (28) examined BR modified with malcic anhydride. Sen et al. (29) applied the analogy-based model for silane grafting and moisture cross-linking of PE and poly(ethylcne-co-propylene). On the other hand, the analogy-based models were also applied for the poiymerization/crosslinking of monomers within elastomer mauix. Mateo et al. (30) adopted a modified alternative for the photoinitialed polymerization of inethacrylic monomers in BR matrix, describing kinetic, mechanistic, and structural aspects of the apparent polymerization, whereas Ghosh et al. (31) studied the modification of low density PE by grafting of acrylic monomers. The general shortcoming of this type of kinetic models is the inability lo distinguish among several possible cross-linking reaction sites such as hydrogen aroms bonded to aliphatic carbon atom and double bonds, which are subjected lo hydrogen abstraction and addition reactions, respectively. A lumped representation of the possible reaction sites is adopted instead. Moreover, polymer backbone scission reactions are not considered, whereas the [beta]-cleavage of the oxy radical is incorporated in an overall peroxide efficiency. Hamielec et al. (32) and Gloor et al. (33) remedied these limitations by considering previously unaccounted or at least not separately treated reactions for polyolefin macromolecules such as PE. Efforts to gain a deeper insight into the mechanism of peroxide-induced crosslinking of PE ensued (34). Pedernera et al. (35) and Asteasuain et al. (36) improved the kinetic model and predicted the molecular weights distribution by probability generating functions for the peroxide initialed modification of PE, respectively, whereas Cheung and Balke (37) and Tai (38) extended the model for the reactive extrusion of PE/polypropylene (PP) blends and crosslinking of EVA/polyolefin blend, correspondingly. Recently, Zhu et al, (39), (40) applied a Monte Carlo study of the grafting of malcic anhydride onto PP and PE, whereas Liu et al. (41) examined the rheokineiics of the PE cross-linking. The polyolefin backbone, although, mostly consists of repeating aliphatic carbons, which react similarly, regardless of the position in the backbone and may consequentially be considered indifferent as far as reaction site dependent kinetics is concerned. Even though the entire kinetic modeling framework for polyolefins is based on the assumption of indifferent backbone reactivity (32), (33), (35-41), Lazar et al. (42) showed that even minority structures in PE, such as double bonds, affect the crosslinking reaction kinetics, which questions the validity of the indifferent backbone reactivity supposition in particular cases. A study of the crosslinking reaction kinetics of more complex elastomers is thus more rigorous, especially for copolymerized elastomers, as individual reaction rates vary with respect to the reactive sites in the elastomer backbone. Because of the latter observation, a wholesome kinetic model framework for peroxide crosslinking of various elastomers has not been developed to the best of our knowledge. Sato et al. (43) proposed a kinetic model for crosslinking of hydrogenated NBR. However, this model also adopts a lumped representation of the crosslinkable sites rendering the determined apparent kinetic constants system specific. Masaki et al. (44) used an alternative approach for the kinetic study of the BR crosslinking in the presence and absence of vinyl acetate applying variable partial orders with respect to different components in the overall crosslinking reaction rate yet the argument of system specificity also applies in this case.

A fundamental description of crosslinking kinetics must obviously acknowledge the details of the underlying chemical mechanisms and thus the composition of the formulation. Therefore, the applicability of the fundamental kinetic approach for several systems was examined theoretically, that is the crosslinking of various elastomers with different organic peroxides, which are important members of the crosslinking agents, was studied. As model peroxides, dicumyl peroxide (DCP), t-butyl peroxide (TBP), and di(2-t-butylperoxyisopropyl) benzene (DTBPIB) were considered.


The complex set of reactions that characterize peroxide crosslinking involve the hemolytic cleavage of the peroxide, beta cleavage of the oxy radical, hydrogen abstraction, addition reactions, radical coupling, polymer scission, radical transfer and in particular cases dehydrogenalion, oxygenation, and acid-catalyzed decomposition of the peroxide. Crosslinks are defined as links (which are in particular cases chemical bonds) connecting two macro-molecules. Besides radical coupling, addition reactions that were mentioned as the addition of radicals on double bonds in elastomer can also combine different chains. These reactions have to be accounted for as crosslink forming ones, as well. A fundamental kinetic model for peroxide crosslinking should explicitly acknowledge these different reactions. This is a substantial task; moreover, additional modeling complexity originates from the nature and consequentially different reactivity of various sites in elastomer backbone itself, which must be incorporated in order to completely close the mass balances for each reaction. A lumped modeling approach (32), (33), (35-41), (43), (44) ignores detailed mass balances for diverse elastomers, and, therefore, cannot be a fundamental description for describing peroxide crosslinking kinetics.

Population balance equations (PBEs) were developed to describe the kinetics of peroxide crosslinking. Explicit kinetic balances were written for every member of the population. The population is defined by the collection of elastomer backbone sites or even the ones in other initially present species and in various reaction intermediates and products, available for hydrogen abstraction, addition reactions, radical coupling (crosslink formation), polymer scission, or radical transfer. The individual balances for each species were derived to include all the combinatorial possibilities for the potential chemical reactions.

The concentration of potential chemical reaction sites in elastomer backbone is in considerable excess when compared with the other reactive species. Thus, even though these reactions with the reaction sites in elastomer backbone are in effect of first order, they were modeled as inherently second order reactions. Therefore, the assumption of first order, which is valid for elastomers with high degree of unsaturation like BR and natural rubber (NR), is not true for the elastomers with low unsaturation like poly(iso-butylene) (IIR), as far as addition reactions are concerned. For an elastomer with low degree of unsaturation, an explicit accounting of the concentration of the unsaturated sites is required.

The rate of formation or breakage of all the bonds in a particular elastomer macromolecule or other species was assumed to be dependent on the position of the bond with regards to the substituents of the potential chemical reaction sites in elastomer macromolecule or in other species. For example, the different C-H bonds in ethylene-propylene elastomer (EPR) will break at different rates. The assumption of equal reactivity (32), (33), (35-41), (43), (44) is an obvious first step that can be easily relaxed to acknowledge different reactivities.

The reactivity of all the species of a particular type for a particular reaction was assumed to be dependent on the position of the potential chemical reaction site in elastomer macromolecule with regards to its substituents, as well. For example, the reactivity of secondary and tertiary carbon radical in EPR backbone with a potential radical coupling site in the elastomer backbone will differ. The assumption of similar reactivity for PE (32), (33), (35), (36), (40), (41) is consistent with the observation that the exact nature of the organic end groups has minimal effect on the reactivity of species that are well removed from the organic end groups of the macromolecule or pendant groups in otherwise linear macromolecular structure. Except for the latter case, the PBE should be generalized to acknowledge different reactivities for different species; however, the current assumption of equal reactivities (32), (33), (35-41), (43), (44) is a natural starting point.

No restriction was adopted considering minimal or maximal number of carbon atoms in any elastomer backbone. The backbone may include the active as well as the nonactive carbon atoms in regard to their potency to take part in a particular reaction. Here, active carbon atom in regard to for example hydrogen abstraction refers to the carbon atoms in the macromolecular structure excluding the quaternary carbon atom. For example there are several active and inactive carbon atoms in crosslinked elastomer matrix.

The nomenclature for various species considered in the PBE is given in Table 1. The different species considered in terms of reaction sites are the peroxide molecule ([R.sub.i][R.sub.j][R.sub.k])C-O-O-C([R.sub.l][R.sub.m][R.sub.n]), the alkoxy radical ([R.sub.i][R.sub.j][R.sub.k])C-O *, ketone derived from peroxide ([R.sub.i][R.sub.j])C-O-H, reaction site with tat least one hydrogen atom ([R.sub.i][R.sub.j][R.sub.k])C-H, which is a subset of reaction site with arbitrary number of hydrogen atoms ([R.sub.i][R.sub.j][R.sub.k][R.sub.l])C, unsaturated reaction site ([R.sub.i][R.sub.j]C = C([R.sub.k][R.sub.l]), also a subset of ([R.sub.i] [R.sub.j] [R.sub.k] [R.sub.l]) C and alkyl reaction site ([R.sub.i] [R.sub.j] [R.sub.k]) C*. Since a particular C-H bond is stronger than the other, it was assumed that hydrogen abstraction may proceed only in the case when more stable radical is formed. Hence, the formation of more stable radical is assumed in the case of addition and polymer scission reactions as well. The subscripts i-n in Table 1 denote different or potentially identical substituents. For example, linear EPR backbone prior to crosslinking has seven different potential reactive sites that is two primary, three secondary, and two tertiary reaction sites with at least one hydrogen atom, whereas the linear EPDM backbone prior to crosslinking with 1,4-hexadiene used as a termonomer has 13 that is seven like the ones in EPR backbone and additional two tertiary, two allylic, and two vinylic reactions sites with at least one hydrogen atom. The two Vinylic reaction sites also account for one unsaturated reaction site should addition reaction be considered.
TABLE 1. Definition of symbols in the population balance equations.

 Symbol Name

([R.sub.i][R.sub.j][R.sub.k])C-O-O- Peroxide molecule

([R.sub.i][R.sub.j][R.sub.k])C-O * Alkoxy radical

([R.sub.i][R.sub.i])C=O Ketone derived from peroxide

([R.sub.i][R.sub.j][R.sub.k])C-O-H Alcohol derived from peroxide

([R.sub.i][R.sub.j][R.sub.k])C-H Reaction site with at least one
 hydrogen atom

([R.sub.i][R.sub.j][R.sub.k] Reaction site with arbitrary
[R.sub.l])C number of hydrogen atoms

([R.sub.i][R.sub.i])C=C Unsaturated reaction site

([R.sub.i][R.sub.j][R.sub.k])C* Alkyl reaction site

Two specific reactions in the initial stage of EPDM crosslinking with DCP are presented in Figs. 1 and 2 and represent an example of foundation for the development of PBE. First, the reaction between the cumyloxy radical (i.e. [C.sub.6][H.sub.5]C[(C[H.sub.3]).sub.2]O *) and EPDM elastomer backbone to form radical reaction site in the EPDM elastomer backbone along with the release of [alpha], [alpha]-dimethyl benzyl alcohol (DMBA) is considered (see Fig. 1).

In EPDM elastomer backbone, there are initially 13 different types of carbon-hydrogen bonds that can break with associated rate constants [k.sub.iha,i,j,k,l,m,n], where indexes i, j, and k represent the alkoxy radical substituents, while indexes l, m, and n represent the substituents of the reaction site with at least one hydrogen atom. Hydrogen abstraction or radical transfer can also occur between two different elastomer chains, intramolecularly or even between two other moieties, nonetheless, only if the C-H bond, and the radical in thus produced moieties are stronger and of higher stability, respectively, when compared with the reacting species. The crosslink precursors ([R.sub.i][R.sub.j][R.sub.k])C* formed according to scheme in Fig. 1 Will subsequently yield crosslinks (i.e. by coupling of two ([R.sub.i] [R.sub.j] [R.sub.k])C* radicals). However, the precursors may be formed by addition reactions as well (see Fig. 2).

There are two potential addition sites, nevertheless, the cumyloxy radical attacks the one, which is more accessible considering steric and other effects and so as to yield more stable backbone radical. The rate constant [k.sub.iar,i,j,k,l,m,n,o] is attributed to the addition reaction, where indexes i, j, and k represent the alkoxy radical substituents, whereas indexes l, m, n, and o represent the substituents of the unsaturated reaction site. Depending on the site at which the cumyloxy or other radical attacks the unsaturated reaction site, ([R.sub.i][R.sub.j][R.sub.k][R.sub.l])C and ([R.sub.i][R.sub.j][R.sub.k])C* are formed and in the case of backbone radical attack, the crosslink between elastomer chains is formed as well. Thus, the contributions to the PBEs from the reactions presented in Figs. 1 and 2 for ([R.sub.i][R.sub.j][R.sub.k])C-O*, ([R.sub.i][R.sub.j][R.sub.k])C-O-H, ([R.sub.i][R.sub.j][R.sub.k][R.sub.l])C, and ([R.sub.i][R.sub.j][R.sub.k])C* are in general terms



[d[([R.sub.i][R.sub.j][R.sub.k])C-O-H]/dt] = [([R.sub.i][R.sub.j][R.sub.k])C-O*] x [L.summation over (l = 1)][M.summation over (m = 1)][N.summation over (n = 1)][k.sub.iha,i,j,k,l,m,n][([R.sub.l][R.sub.m][R.sub.n])C-H] (2)



In Eqs. 3 and 4, the substituents of the components on the right-hand side, [R.sub.p] and [R.sub.r] (Eq 3) or [R.sub.o], and [R.sub.p] (Eq. 4) are not arbitrarily chosen, so as to yield the specific substituents of the reaction site with arbitrary number of hydrogen atoms, [R.sub.k] and [R.sub.l] (Eq. 3) or the single-specific substituent of the alkyl reaction site, [R.sub.k] (Eq. 4). Thus, only a single case of [R.sub.p] and [R.sub.r] (Eq. 3) or [R.sub.o] and [R.sub.p] (Eq. 4) has to be considered and does not require the summation of subscripts p and r (Eq. 3) or o and p (Eq. 4). The rate constants may be related to the transition state theory via the well-known Eyring equation (45); however, the individual rate constants for specific reaction site might also be determined using any other available experimental technique or theoretical calculations. Their estimation is not an essential feature of the development of the PBE yet is rather important for the predictability of the overall kinetic model. Rate constants depend on reaction site and thus as few simplifying assumptions as possible were applied. Generally, it may be predicted that the formation of more stable products is much more likely than the formation of unstable products such as certain radicals. Moreover, it was assumed that reactions most likely proceed via saddle point transition state structures than via other thermodynamically less favorable reaction paths; consequently, for example, the hydrogen abstraction reaction rate magnitude is anticipated to proceed in the following order: allylic > tertiary > secondary > primary > vinylic (3). It was also assumed that the same type of reactive sites in chemical terms, like each one of the potentially abstractable hydrogen atoms in elastomer structure, have equal reactivities. Thus, each rate constant in Eqs, 1-4 is defined solely by reactive site and its substituents, more specifically tacticity, although, was not considered in the kinetic model. Of course, omitting of stereochemistry can be relaxed if needed in order to describe the crosslinking of elastomers with various tacticities.

TABLE 2. Reaction scheme and the associated rate constants considered
in kinetic model.

Reaction Reaction rate
 number Reaction constant

 Peroxide chemistry

 1 ([R.sub.i][R.sub.j][R.sub.k])C-O-O- [k.sub.d,i,j
 C([R.sub.l][R.sub.m][R.sub.n]) k,l,m,n]
 [right arrow]
 ([R.sub.i][R.sub.j][R.sub,k])C-O * +
 ([R.sub.l][R.sub.m][R.sub.n])C-O *

 2 ([R.sub.i][R.sub.j][R.sub.k])C-O * [k.sub.i,B
 [right arrow] i,j,k]
 ([R.sub.j][R.sub.k])C=O +

 3 ([R.sub.i][R.sub.j][R.sub.k])C-O* + [k.sub.iha,i,j
 ([R.sub.l][R.sub.m][R.sub.n])C-H k,l,m,n]
 [right arrow]
 + ([R.sub.l][R.sub.m][R.sub.n])C*

 4 ([R.sub.i][R.sub.j][R.sub.k])C-O* + [k.sub.iar,i,j
 ([R.sub.l][R.sub.m])C=C([R.sub.n][R.sub.o]) k,l,m,n,o]
 [right arrow]
 ([R.sub.l][R.sub.m][R.sup.p][R.sub.r])C +

 Crosslinking chemistry

 5 ([R.sub.i][R.sub.j][R.sub.k])C* + [k.sub.ha,i
 ([R.sub.l][R.sub.m][R.sub.n])C-H j,k,l,m,n]
 [right arrow]
 ([R.sub.l][R.sub.j][R.sub.k])C-H +

 6 ([R.sub.i][R.sub.j][R.sub.k])C* + [k.sub.p,i,j,k
 ([R.sub.l][R.sub.m])C=C([R.sub.n][R.sub.o]) l,m,n,o]
 [right arrow]
 ([R.sub.l][R.sub.m][R.sub.p][R.sub.r])C +

 7 ([R.sub.i][R.sub.j][R.sub.k])C* + [k.sub.l,i,j
 ([R.sub.l][R.sub.m][R.sub.n])C* k,l,m,n]
 [right arrow]
 + ([R.sub.l][R.sub.m][R.sub.n][R.sub.p])C

 Crosslink degradation chemistry

 8 ([R.sub.i][R.sub.j][R.sub.k])C* [k.sub.B,
 [right arrow] i,j,k]
 + ([R.sub.n][R.sub.o][R.sub.p])C*

There will of course be other reaction in addition to the ones presented in Figs. 1 and 2 that contribute to the generation or depletion of the various species and all these contributions need to be added to Eqs. 1-4. Using analogous methodology as that, outlined in Eqs. 1-4, the governing differential balances for all the species involved in the chemistry of peroxide crosslinking were developed. To facilitate the ease of development of these balances, the reaction network of peroxide crosslinking (3), (9), (18-20), (24), (27), (29), (31-37), (39-44) is summarized in tabular form in Table 2. In Reactions 1-8 in Table 2, [[R.sub.i]-[R.sub.r]] are defined as different or potentially identical sub-stituents, which is illustrated in Figs. 1 and 2, as well. The reaction scheme in Table 2 is similar to the ones adopted in literature (3), (9), (18-20), (24), (27), (29), (31-37), (39-44), except for the thorough consideration of the mechanisms by which the reactions actually proceed. In the reaction scheme of Table 2, which is the foundation of the kinetic model, it is assumed that different species may take part in analogous types of reaction, nevertheless, the reaction rates depend on the nature of the reaction site itself (Reaction 1-8 in Table 2); in contrast, for similar reaction schemes in literature (3), (9), (18-20), (24), (27), (29), (31-37), (39-44), which were applied for kinetic model framework buildup, the analogous types of reactions are assumed to proceed at the same rates, regardless of, for example the nature of radical that is produced during hydrogen abstraction, i.e. primary, secondary, allylic, etc.


In this section, the capabilities of PBE to predict the kinetics of chemical reactions during crosslinking of elastomers will be critically examined for the peroxide vulcanization of various elastomers. Principally, the time-dependent formation of the total crosslink density that is the sum of concentrations of all newly developed or cleaved intermolecular bonds for various formulations has to be described and; moreover, the temporal evolution of other occurring or depleting species predicted by the PEB has to be examined. As described in the Kinetic Model section the Predictions have to be examined considering the reaction of homolytic cleavage of the peroxide, beta cleavage of the oxy radical, hydrogen abstraction, addition reactions, radical coupling (crosslink formation), polymer scission, and radical transfer.

The model PBE for the vulcanization reactions, the development of which is facilitated by the appropriate reaction scheme (Table 2) and is indicated by the derivation of Eps. 1-4, are defined by a set of equations. The model is thus constituted of a large number of nonlinearly coupled ordinary differential equations for a system initially containing polymer chains and peroxide molecules. When other species are present, for example coagents, other equations must also be included. Although there is a large number of differential equations (or even larger, when other species are present) to be solved simultaneously in order to model the evolution of the different species that participate in the vulcanization reactions, the total number of reactions is even of a greater order of magnitude. Because of the infinite reactions of the various conformations of otherwise chemically identical species, the reactivity of diverse reaction sites has been restricted to the one of thermodynamically most stable conformations; however, the framework can be readily extended to include other frequently encountered conformations with regard to the specific formulation.

To solve the PBE, the rate constants for various reactions had to be determined utilizing molecular modeling. Making the only simplifying assumption that the rate constants are independent of species' conformation, there are eight different types of reactions and, thus, eight different types of rate constants had to be determined. The determination of all rate constants by the optimization of the geometry of reacting and evolving species, as well as the configuration of transition states is time consuming; nevertheless, once a set of rate constants is obtained the latter may be readily applied to the kinetic model permitting interchanging of specific determined constants by the ones determined by a more accurate calculating or experimental method. The rate constants [K.sub.d,i,j,k,l,m,n,] may be considered as a single constant for each type of the peroxide molecule present in formulation, as all these rate constants govern the reaction of peroxide bond scission which is relatively indifferent to the substituents [R.sub.i]-[R.sub.n]. When these substituents react with another species (i.e. peroxide or elastomer originating), the reactivity of ([R.sub.i][[R.sub.j][R.sub.k])C-O-O-C([R.sub.l][R.sub.m][R.sub.n] should be identical, as the substituent groups should have a minimal effect on the bond's reactivity, although for different peroxides this assumption may not be appropriate and was not applied, accordingly. Information about how the peroxide type affects the reactive bond is available from either experimental data or quantum chemistry calculations (46-48), and this effect on the rate constant was readily incorporated for various peroxides. The rate constants [K.sub.i[beta],i,j,k] were considered likewise (49-51), as these represent the [beta] cleavage of the oxy radical. Other constants are more or less strongly dependent on the substituents, as they essentially control the rates of peroxide, crosslinking, and crosslink degradation chemistry. Using these considerations, a total of six rate constants for all potential substituents had to be determined, that is [K.sub.iha,i,j,k,l,m,n,], [K.sub.iar,i,j,k,l,m,n,o], [K.sub.ha,i,j,k,l,m,n,], [K.sub.p,i,j,k,l,m,n,o], [K.sub.t,i,j,k,l,m,n] and [K.sub.[beta]i,j,k], The rate constants and the associated activation energies and preexponential factors were determined utilizing molecular modeling (52), (53). This was achieved by the semi-empirical Parameterized Model number 3 (PM3) method calculations, using unrestricted Hartree-Fock (UHF) spin pairing and self-consistent field controls of 0.001 convergence limit and 1000 iterations limit. The species' geometry optimization was performed using Polak-Ribiere (conjugate gradient) algorithm (54) with root-mean-square (RMS) gradient of 0.00418 kJ/(nm X mol) as the terminal condition. The transition state search for specific reactions was performed using quadratic synchronous transit algorithm (55) with RMS gradient of 0.00418 kJ/(nm X mol) as the terminal condition. The rate constants were obtained as

k = A exp(-[E.sub.A]/(RT)) = [[[k.sub.B]T]/[hC.sub.TO]exp(-[DELTA][S.sup.[double dagger](T)/R)exp(-([DELTA][H.sup.[double dagger](T) + RT)/(RT)) (5)

where k, A, [E.sub.A], R, and T are the rate constant, pre-exponential factor, activation energy, gas constant, and temperature, respectively, [k.sub.B], h, and [C.sub.TO] are the Boltzmann constant, Planck constant, and total concentration of reactive sites (which should be left out in the case of monomolecular reactions, i.e. Reaction 8), correspondingly. The difference in entropy ([DELTA][S.sup.[double dagger]](T)) and enthalpy ([DELTA][S.sup.[double dagger]](T)) between the transition state and the energy minimum at the corresponding reaction coordinate was estimated at 20 temperatures to obtain the temperature-dependent rate constants alongside with preexponential factors and associated activation energies. The total concentration of reactive sites was calculated as

[C.sub.TO] = [[w.sub.P]/(1 - [w.sub.p])/[[bar.M].sub.P0] + [1/[[bar.M].sub.E0]/[[w.sub.P]/(1 - [w.sub.P])/[[rho].sub.P] + 1/[[rho].sub.E] (6)

where [W.sub.p], [[bar].M.sub.P0], [[bar].M.sub.E0] [[rho].sub.P], and [[rho].sub.E] stand for peroxide weight fraction in the formulation, initial average peroxide reaction site molecular weight, initial average elastomer reaction site molecular weight, and peroxide and elastomer density, respectively. Analogously, initial concentrations of individual reactive sites, employed as initial conditions for PBE, were calculated as follows:

[C.sub.Pi] = [[[x.sub.Pi][w.sub.P]/(1 - [w.sub.P])/[[bar.M].sub.P0]]/[[w.sub.P]/(1 - [w.sub.P])/[[rho].sub.P] + 1/[[rho].sub.E]]] (7)

[C.sub.Ei] = [[[x.sub.Ei][[bar.M].sub.E0]]/[[w.sub.P]/(1 - [w.sub.P])/[[rho].sub.P] + 1/[[rho].sub.E]]] (8)

where [C.sub.Pi], [C.sub.Ei], [x.sub.Pi] and [x.sub.Ei] represent initial concentration of individual peroxide or eleatomer reactive site and individual peroxide or elastomer reactive site molar fraction, accordingly. Although [W.sub.P] is determined by the choice of formulation, and densities may be obtained from the literature (46), (56), other varibles in Eqs. 6-8 are dependent solely on peroxide and elastomer type, respectively.

One of numerous sets of the preexponential factors and the corresponding activation energies at standard conditions for the eight rate constants applied in the kinetic model are tabulated in Table 3. Although all radicals are generally of high energy, there are variations in the energy levels among different types of radicals (3) which is evident from Table 3, as the rate of hydrogen abstraction by cumyloxy radicals proceeds at the greater rate the more stable are thus formed radicals. This has been confirmed by the increase in reactivity of the tertiary carbon in comparison with the secondary carbon in PP (39). If the obtained rate constants are compared with some of their values which may be found in the literature, it is observed that, for example, [FORMULA NOT REPRODUCIBLE IN ASCII] corresponds well with the order of magnitude of [[10.sup.6]-[10.sup.7]][1/(mol x s) applied for PE between 150 and 190 [degrees] C (39-41), as may be observed for [FORMULA NOT REPRODUCIBLE IN ASCII], as well, as the reported order of magnitude for PE at 190[degrees]C was [10.sup.8] [1/(mol x s)] (39), (40). Usually, the rate constants are obtained or applied for isothermal conditions (33), (39-41), hence occurs the limitation of their comparison over a range of temperatures. Nevertheless, some studies established the temperature. Nevertheless, some studies established the temperature dependence of the essential reactions' rates involved in PE crosslinking (35), (36). Upon comparison of these values with the ones presented in Table 3, a relatively good correspondence is observed, and when activation energy of the rate constant [FORMULA NOT REPRODUCIBLE IN ASCII] is higher for about 10 kJ/mol, this may be explained by the presence of minority structures in PE (42) which decrease the hydrogen abstraction reactivity and increase the activation energy of the reaction (3) (Table 3). Moreover, as well as in our case, the largest energy barrier applied to the polymer chain [beta]-cleavage reactions (35), (36). Peculiarly, the activation energy of chain transfer reactions was estimated unorthodoxly low at ~8 kJ/mol (35), (36), whereas in Table 3 the corresponding value is higher, according to the reactivity rule (3). Generally, there are several ambiguities in the estimation of rate constants' values not only within experimental error margin but also within several orders of magnitude (33), (35), (36), (39-41), although at least most frequently a consensus applies upon [FORMULA NOT REPRODUCIBLE IN ASCII] being larger than [FORMULA NOT REPRODUCIBLE IN ASCII] (33), (39), (40).
TABLE 3. An example of the activation energies and the preexponential
factors applied in the kinetic model.

 Preexponential factor
 ([s.sup.-1] or Activation energy
 Rate constant [[s.sup.-1]/[C.sub.TO]]) (kJ/mol)

[k.sub.d], DCP (46) 2.69 X [10.sup.12] 120.3

[k.sub.i[beta]], 1.26 X [10.sup.12] 64.0
DCP (51)

[k.sub,iha], 5.84 X [10.sup.13] 50.3

[k.sub.iar], 1.79 X [10.sup.16] 55.1

[k.sub.ha], 1.64 X [10.sup.15] 57.6

[k.sub.p], 1.68 X [10.sup.15] 69.1

[k.sub.t], 1.84 X [10.sub.13] 27.6

[k.sub.[beta]], 4.47 X [10.sub.14] 161.3

[k.sub.iha], DCP, 7.22 X [10.sub.13] 37.5

[k.sub.iha], DCP, 6.88 X [10.sup.13] 41.9

[k.sub.0], DCP, 6.61 X [10.sup.13] 43.2

[k.sub.iha], 6.38 X [10.sup.13] 46.2

[k.sub.iha], DCP,- 6.05 X [10.sup.13] 50.2

[k.sub.iha], DCP, 5.84 X [10.sup.13] 50.3

[k.sub.iha], DCP, 5.55 X [10.sup.13] 54.7
-H, -H

[k.sub.iha], DCP, 5.25 X [10.sup.13] 65.3

[k.sub.iha], DCP,= 4.93 X [10.sup.13] 68.1

[k.sub.iha], DCP,= 4.64 X [10.sup.13] 70.8

[k.sub.iha], DCP, 4.32 X [10.sup.13] 71.0

[k.sub.iha], DCP, = 4.05 X [10.sup.13] 72.9

[k.sub.iha], DCP,-TBP 3.78 X [10.sup.13] 74.7

In Fig. 3, the temporal behavior (presented as the temperature dependence in which way the results will be presented henceforth as well) is predicted by the population balance model, solved by application of a variable order Adams-Bashforth-Moulton PECE solver (57), for the total concentration of crosslinks (v) when the system temperature increases linearly at the heating rate of 1 K/min for different formulations with varying amounts of different peroxides and with varying fractions of either propylene or butadiene in linear ethylene-propylene or ethylene-butadiene copolymer, respectively. The model is able to adequately describe the vulcanization process, including the initial induction period, cure and the postcure (i.e. equilibration). The model PBE are able to describe the important features of the net evolution of crosslinks during the vulcanization process for a wide range of peroxide and elastomer compositions; moreover, certain features of the crosslinks' development are quite shows that initially there is basically no evolution of crosslinks, subsequently followed by initially slow, but steady, and then eventually a more rapid increase in the crosslink density. The predicted temporal evolution of crosslinks is defined by the independently determined parameters, as the set of rate constants influences particular features of the cure curve and significantly interchangeably affect all aspects of the crosslink evolution. The fact that the model predicts that crosslinks are not formed at lower temperatures is not surprising, considering that crosslink can only form after the peroxide has reacted to such an extent that crosslink precursors like ([R.sub.i][R.sub.j][R.sub.k])C* are formed in sufficient concentrations. The crosslinks form in greater extent when certain amount of peroxides is consumed to accumulate sufficient number of ([R.sub.i][R.sub.j][R.sub.k])C*. For a specific kind of peroxide, higher initial concentration results in a faster increase in radical concentration and thus earlier apparent crosslinking point. However, in Fig. 3 the apparent starting points of the curves seem to be rather similar for a single type of peroxide. Nevertheless, this merely appears as such, as the concentration of crosslinks is greater for higher initial amounts of the specific peroxide throughout the process. The decisive factor is of course the activation energy of the peroxide bond dissociation reaction, which tends to enforce on temperature, far more noticeable than the actual initial amounts of the specific peroxide. The effect of peroxide amount on the apparent crosslinking process initiation point is therefore not null, the effect of the peroxide type is much more profound. The explanation for the behavior in the initial period of the v versus temperature predictions is as follows. First, the concentrations of crosslinks is inferred from the forming bonds among macromolecular chains, where the kinetic analysis is performed at linearly increasing temperature; however, the system requires a finite time and a critical temperature to achieve sufficient amount of primary radicals to inflict substantial crosslinking, and thus, regardless of the definition of initial conditions at t = 0, the crosslinking initiation is determine primarily by the peroxide type, and hence the activation energy and the preexponential factor of the latter. According to Fig. 3a, the crosslinking initiation of DTBPIB is in between the ones when TBP or DCP is present in the system, the latter providing the earliest induction. This corresponds quite well with the peroxide dissociation description and parameters available in literature [3, 46-49] and moreover, DTBPIB should most intensively dissociate in the temperature region between the most intensive dissociations of TBP and DCP, as its peroxide group is bonded to both cumyl and t-butyl substituents (Fig. 3a). At t = 0 the concentration of crosslinks is at is its minimum (see Fig. 3), whereas visible crosslinking occurs approximately at temperatures 133, 151, and 168[degrees]C for DCP, DTBPIB, and TBP, respectively. As the system is being heated even before this initial time period, the vulcanization reactions will have already started prior to these temperatures; consequently, the delay presented in Fig. 3 may be shortened upon choosing a less stable peroxide or increase in heating rate. Second, the initial gradual increase in the concentration of crosslinks followed by a sharper rise as indicated in Fig. 3 appears to proceed similarly regardless of the type of either peroxide or elastomer, indicating the peroxide dissociation kinetics being vital for the overall reaction progress (3), (18), (20), (42), (43). During the early stages when peroxide is available, peroxide radicals are also formed only to react further. As peroxide radicals react at higher reaction rates when compared with the dissociation, there is an initial slow rise in v production followed by an difference, as the reactivity of peroxides varies noticeably, yet the reactivity of evolving primary radicals is more loxy radicals in the PBE may be omitted but is not warranted. The model also predicts equilibration process (decrease in recombination rate (41)), analogously to HNBR (11), (43), EPDM (14), PE and its blends', (18), (23), (40-42), PDMS and its blends', (19), (20) and BR (28) data found in literature. The only mechanism for reversion that is included in model is the crosslink degradation chemistry as discussed in the Kinetic Model section. The rate of scission reactions is anticipate to be a function of the reactive site and temperature; however acknowledging all potential reactive sites, the rate is mainly affected by the system temperature which has to be reasonably high for scission reactions to prevail due to their high activation energy (Table 3). Moreover, the predicted equilibration or even slight increase of the crosslink concentration depends upon the details of the reactive sites' concentration distribution: consequently, the reactions that precede the formation of crosslinks can affect the postcure region. All things considered that the model is able to predict all important features of the crosslinking process for a range of various elastomers and peroxides and their concentrations, using a reaction mechanism that is chemically reasonable. Additionally, the decrease or increase inv upon increasing the fraction of propylene or butadiene in the corresponding copolymer with ethylene, respectively, may be interpreted through specific reactivity. That is PP is less effectively crosslinked with peroxides than PE (3), owing to its relatively difficultly abstracted primary carbon bonded hydrogen atoms even though the tertiary carbon bonded ones are more reactive (Table 3) (37-39), and is therefore often used in blends with PE (37), (38). BR, on the other hand, has reactive double bonds in its structure and even more importantly, hydrogen atoms bonded to allylic carbon which are one of the most reactive toward hydrogen abstraction (Table 3) (27), (28), (30), (44) and all that makes BR one of the most effectively crosslinkable elastomers (3).


As the model is able to predict concentration of the diverse types of crosslinks with various substituents, the model predictions shown in Fig. 3 are only a lumped concentration of crosslinks, i.e. the substituent dependence of the crosslinks is not considered. Similar lumped concentrations for other important species including

([R.sub.i][R.sub.j][R.sub.k])C-O, ([R.sub.i][R.sub.j])C=O, ([R.sub.i][R.sub.j][R.sub.k])C-O-H, etc. are shown in Fig. 4. as well the consumption of the peroxide ([R.sub.i][R.sub.j][R.sub.k])C-O-O-C(R.sub.j][R.sub.m][R.sub.n]) and the reaction site with at least one hydrogen atom ([R.sub.i][R.sub.j][R.sub.k])C-H. As expected, the concentration of peroxide decreases monotonically during the course of reactions (42). Although the radical species ([R.sub.i][R.sub.j][R.sub.k])C-O are initially zero, they are formed immediately upon initiation of reactions. The Concentration of ([R.sub.i][R.sub.j][R.sub.k])C-O* reaches the maximum (about 0.02 mol/[m.sup.3], which corresponds with experimental observations [34, 44]) approximately at the temperature when the depletion of the peroxide is the greatest. In this order of occurrence ([R.sub.i][R.sub.j][R.sub.k])C*, ([R.sub.i][R.sub.j][R.sub.k])C-O*, ([R.sub.i][R.sub.j])C = C([R.sub.k][R.sub.l] and ([R.sub.i][R.sub.j][R.sub.k])C-O-C(R.sub.j][R.sub.m][R.sub.n]) show peaks in concentration between 140 and 160[degrees]C. During the early stages of the cure, the concentration of ([R.sub.i][R.sub.j][R.sub.k])C* is greater than that or ([R.sub.i][R.sub.j][R.sub.k])C-O*, which may appear to be somewhat surprising as ([R.sub.i][R.sub.j][R.sub.k])C* is formed through abstraction and addition reactions of ([R.sub.i][R.sub.j][R.sub.k])C-O*, However, this is a consequence of the fact that ([R.sub.i][R.sub.j][R.sub.k])C-O* is in the first stage rapidly consumed by coupling reactions in addition to both beta cleavage and abstraction/addition reactions with ([R.sub.i][R.sub.j][R.sub.k][R.sub.l]) C. Thus, although ([R.sub.i][R.sub.j][R.sub.k])C* is formed from ([R.sub.i][R.sub.j][R.sub.k])C-O* at the beginning, its concentration quickly exceeds the concentration of ([R.sub.i][R.sub.j][R.sub.k])C-O*, as it grows simultaneously from hydrogen abstraction and addition reactions during the early stages of cure, when peroxide is present in significant concentrations. At longer times when most of the peroxide is depleted, the concentration of ([R.sub.i][R.sub.j][R.sub.k])C-O* reaches its maximum yet is still lower than that of ([R.sub.i][R.sub.j][R.sub.k])C*, as expected. Liu et al. (41) predicted macroradical concentrations which are several orders of magnitude greater than the one in Fig. 4b and are dependent on the applied system testing conditions; nonetheless, the direct measuring techniques (34), (44) and model predictions by Pedernera et al. (35) seem a more reliable comparison. The vinyl group will generally be consumed during the course of crosslinking if it is initially constituting the elastomer backbone (30), (44), but in Fig. 4b, it may be observed that firstly the double bonds' concentration increases due to scission reactions, as the unsaturation is initially not present in the theoretically chosen linear PE. After the maximum concentration, however, the amount of unsaturation commences to decrease as expected (30), (35), (44). The evolving concentration of vinyl groups, should they not be initially present in the macromolecular structure, is of an expected order of magnitude that is [10.sup.-2] mol/[m.sup.3] (Fig. 4b, (35)).

The dependence of ([R.sub.i][R.sub.j][R.sub.k])C-H concentration shows an initial delay until ~ 140[degrees]C is reached, and then a rapid decrease to a constant value ensues at ~ 165[degrees]C. The initial behavior of the ([R.sub.i][R.sub.j][R.sub.k])C-H concentration (i.e. induction period) is not surprising, as the reactive precursor ([R.sub.i][R.sub.j][R.sub.k])C-O* is formed very rapidly upon the initiation of reaction; however, the substantial decrease in ([R.sub.i][R.sub.j][R.sub.k])C-H concentration is consistent with noticable ([R.sub.i][R.sub.j][R.sub.k])C-O* concentration builtup. Specifically, once ([R.sub.i][R.sub.j][R.sub.k])C-O* has been formed, it reacts with alkyl reaction sites (i.e. ([R.sub.i][R.sub.j][R.sub.k])C*), and only after a maximum in concentration of the latter has been reached ([R.sub.i][R.sub.j][R.sub.k])C-O* can considerably abstract hydrogen atoms. Finally, the concentration of the lumped ([R.sub.i][R.sub.j][R.sub.k])C-H species decreases slowly to a constant value due to formation of a stable network and the presence of equilibrium reactions (Reactions 5-8). The equilibrium reactions and network formation occurs throughout the cure process; however, the rate of crosslink formation is initially much greater than the rate of crosslink degradation so that the concentration of v increases for temperatures lower than 150[degrees]C (Fig. 5 and corresponding curve in Fig. 3). The PBE model results shown in Fig. 3 qualitatively agree with the lumped parameter simulations (32), (33), (35-41), (43), (44). However, in contrast to these lumped parameter models which allow for the initial concentration of peroxide and elastomer as the input, the developed model predicts the formation of various species from the initial concentrations of all different peroxide and elastomer reaction sites. In addition, the developed populations that several minority species are formed or consumed during the course of reactions even in a simple case of, for example, PE crosslinking (42).



As the PBEs explicitly account for different reaction sites, the evolution of the individual species ([R.sub.i][R.sub.j][R.sub.k][R.sub.l])C and ([R.sub.i][R.sub.j][R.sub.k]C* are also predicted. In Fig. 5 the time evolution of different crosslinks and ([R.sub.i][R.sub.j][R.sub.k][R.sub.l])C is given for an exemplary case. The crosslink distribution during cure commences to evolve noticeable at ~ 133[degrees]C and is subsequently biased toward straightforward crosslinks among elastomer chains (i.e. E-E bonds); however, with increasing temperature the peroxide-elastomer links (P-E) are formed as well, with ensuing elastomer-peroxide-elastomer crosslinks shifting the total concentration of crosslinks toward higher concentrations. Elastomer-peroxide-elastomer crosslinks are indirect crosslinks, for which peroxide originating species act as an intermediate link(s) between two macromolecular chains and should not be mistaken for the direct elastomer-elastomer crosslinks, for which peroxide originating species solely act as a precursor, however, they are not part of the link between the chains. The formation of P-P and P-E links reaches its effective termination around 155[degrees]C followed by E-E and v crosslinks at about 165[degrees]C, As subsequent recombination of crosslinks is stable and equilibrated by scission reactions, the overall concentration of different crosslinks levels off.

Concentrations of total crosslinks, elastomer-elastomer bonds, and peroxide-elastomer bonds at the terminal extent of crosslinking predicted by the kinetic model for nine different formulations cured at 1 K/min are shown in Fig. 6. Should formulations containing 100 mol/[m.sup.3] of DCP and TBP, and 50 mol/[m.sup.3] of DTBPIB be compared, one may observe that as t-butoxy radical impact is increased holding the total peroxide group concentration constant, the portion of P-E links increases. Moreover, by comparing the formulations containing increasing amounts of specific peroxide the relative ratio of P-E links per E-E links increases, as well. The increase in total number of crosslinks with increasing peroxide concentration is expected, as additional peroxide will introduce greater probability for higher terminal crosslink formation in the vulcanizate. The increase in the terminal E-E and P-E links' concentration with increasing initial peroxide concentration is also intuitive, As the peroxide concentration increases crosslinks are redistributed in favor of the elastomer-peroxide-elastomer crosslinks due to the relatively increased frequency of peroxide originating species within the polymer matrix, resulting in an increase in P-E/E-E concentration ratio even though the overall crosslink density remains approximately the same, latter being dependent primarily on the initial peroxide concentration. Specifically, increasing the peroxide concentration will result in larger extent of the peroxide chemistry and crosslinking chemistry reactions (Reactions 3-7; Table 2) among the peroxide originating species and in larger extent the peroxide originating species and in larger extent of the crosslink degradation chemistry reactions (Reaction 8; Table 2) among the elastomer originating species, with the result that greater fraction of various elastomer-peroxide-elastomer crosslinks are formed. Predicting how the distribution of crosslink kinds is affected by the amount and type of peroxide is necessary in order to predict how the properties of the crosslinked material vary with both amount and type. These results clearly indicate that the population balance method with its explicit accounting of the reactive sites distribution is necessary to describe the important features of the vulcanization process.



In Figs. 7 and 8, the predictions of the kinetic model for the evolution of the total concentration of crosslinks are presented for the formulations considered in Fig. 3a, however, when system's temperature increases at 2 K/minand 0.5 K/min, respectively. The predictions are again reasonable, describing the analogous response as a function of composition when compared with Fig. 3a. The effect of heating rate on the vulcanization kinetics for a single formulation is shown in Fig. 9. The kinetic model predicts a more gradual increase in the total concentration of crosslinks with an increase in heating rate. This occurs because the temperature of the system increases more rapidly, and therefore, more narrow time intervals are rendered for the crosslinking reactions which consequentially leads to the crosslinking reactions which consequentially leads to the lesser extent of these reactions and, hence, a lower total concentration of crosslinks.




PBE of the kinetic model are able to predict most of the features of the peroxide crosslinking, including induction, crosslinking, and postcrosslinking periods for various elastomer formulations with different initial concentrations of various peroxides; moreover, the kinetic model is able to capture the increase in crosslink density with increasing concentration of different peroxides and the variation in the terminal crosslink density with alternating elastomer type or its composition. The key feature of the kinetic model that enables a reasonably accurate description of the experimentally observed crosslinking characteristics was the explicit incorporation of the nature of various elastomer backbones and peroxide originating species. The kinetic parameters that occur in PBE were determined separately utilizing molecular modeling and were indirectly employed to predict the temporal evolution of the total crosslink density, and the temporal evolution of the distributions and cumulative values of the depleting, intermediate and evolving species. There are scarce experimental measurements of individual species like ([R.sub.i][R.sub.j][R.sub.k])C-H, ([R.sub.i][R.sub.j][R.sub.k][R.sub.l])C, and ([R.sub.i][R.sub.j][R.sub.k])C* and of the distribution of all different crosslink types for the peroxide-elastomer systems being studied, yet the principal features of the simulations coincide with several experimental observations available in the literature. Final concluding remarks would be that the developed model allows for various peroxide and elastomer formulations, isothermal, dynamic or variable temperature regimes, variable crosslinking process conditions such as pressure which may indirectly enter the model parameters through the reactive sites' concentrations, and corresponds well with the experimentally observed crosslinking features noted in literature. Nevertheless, the model should be currently applied for as many of product and process properties and characteristics in question as possible in order to remedy its potential shortcomings which are bound to come to light. However, one of the crucial advantages of the kinetic model is that individual parameters such as rate constants may easily and readily be interchanged by a new set, should such necessity arises.


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Blaz Likozar, Matjaz Krajnc

Chair of Polymer Engineering, Organic Chemical Technology and Materials, Faculty of Chemistry and Chemical Technology, University of Ljubljana, Askerceva cesta 5, 1000 Ljubljana, Slovenia

Additional Supporting Information may be found in the online version of this article.

Correspondence to: M. Krajnc; e-mail:

Contract grant sponsor: Slovenian Ministry of Higher Education, Science and Technology; contract grant numbers: L2-6686, Pl-0191.

DOI 10.1002/pen.21218

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Date:Jan 1, 2009
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