POPULATION BALANCE MODEL FOR VULCANIZATION OF NATURAL RUBBER WITH DELAYED-ACTION ACCELERATOR AND PREVULCANIZATION INHIBITOR
Abstract
Our objective is to extend the population balance equation model for vulcanization of natural rubber in the presence of N-t-butylbenzothiazole-2-sulfenamide (TBBS) as an accelerator and N-(cyclohexylthio)phthalimide (CTP) as a retarder. The experiments performed by using the oscillating disk rheometer are used to track the evolution of crosslink density in natural rubber. The model quantitatively predicts the trends observed in the vulcanization mixture at various temperatures, including capturing the effects of changing the initial amounts of TBBS and CTP. This model is able to capture all the key trends involved in the retarder chemistry. Specifically, both experiments and model show that addition of CTP delays the scorching time by a few minutes, without affecting the final crosslink concentration. The model qualitatively predicts trends in the major species reported in the literature. The model can reliably predict and explain the trends of monosulfidic, disulfidic, and polysulfidic crosslinks in a conventional and efficient TBBS-accelerated system. Finally, reaction path analysis is performed for the vulcanization process, which is able to clearly identify the key reaction mechanisms in the induction, crosslinking, and postcrosslink zones during the curing process.
INTRODUCTION
Vulcanization of rubber, discovered independently by Goodyear and by Hancock in the early 19th century, is the process of curing natural rubber typically with sulfur at elevated temperatures. Sulfur crosslinks are formed between hydrocarbon chains during vulcanization. The degree of crosslinking governs the thermal and mechanical properties of rubber.
Rubber vulcanization is carried out in the presence of an accelerator and activator, which significantly reduce the time required for vulcanization. In this work, N-t-butylbenzothiazole-2-sulfenamide (TBBS) is used as an accelerator, whereas zinc oxide and stearic acid are used as activators. TBBS is a commercially important member of benzothiazole sulfenamide class of inhibitors, which were introduced in 1930s.1,2 The first step in the accelerated vulcanization chemistry is the formation of an activator–accelerator complex, which is responsible for delivering sulfur atoms to form polysulfidic links in vulcanized rubber.3–7 TBBS is a delayed-action accelerator, since crosslink formation (also referred to as scorch) starts only after an “induction time.” The amount of TBBS affects both the induction time and the rate of crosslink formation.
Although there is good amount of consensus on the general reaction pathways for accelerated vulcanization of natural rubber,3,4 some debate exists. The monomeric polysulfides of the accelerator (and their Zn complexes, when ZnO is present in the mixture) are known to be active sulfurating species. The key sulfurating agents when sulfenamides are used as accelerators are 2,2′-dithiobisbenzothiazole (MBTS) and its monomeric polysulfides (MBTPs).5,6,8 Thermal decomposition of the accelerator forms 2-mercaptobenzothiazole (MBT), which then reacts with the accelerator to give MBTS. However, recent results of Gradwell and coworkers9–11 seem to contradict this mechanism. They did not find any MBT during the induction region; MBT was only formed after scorching. They concluded that the polysulfides of TBBS, and not MBTPs, are the active sulfurating agents.
A retarder or prevulcanization inhibitor is used to give an additional degree of control over the onset of scorching. N-(cyclohexylthio)phthalimide (CTP), first invented by Coran and Kerwood,12 is a very important retarder, since it increases the induction time without significantly affecting the rate of crosslinking. Even when present in small quantities, CTP scavenges the precursors to the activator–accelerator complex during the induction region.13 It is therefore used to provide an additional control for scorching. In this work, a population balance equation (PBE) based model is developed for investigating the retarder action of CTP on accelerated vulcanization of natural rubber with TBBS used as an accelerator.
The vulcanization system is complex, and it is difficult to measure the concentration of various species undergoing chemical reactions during the process. Therefore, shear modulus of the rubber sample, which is a good indicator of the degree of vulcanization, is experimentally measured using an oscillating disk rheometer (ODR). An ODR measures the torque or the shear modulus as a function of time during vulcanization of the rubber. The vulcanization process consists of the following regions: (i) the initial induction region, (ii) the curing region, and (iii) the postcure region.
The modeling framework adopted in this work is represented in Figure 1. Most of the accelerator and retarder chemistry takes place in the induction region. This primarily leads to the formation of active sulfurating agents, which include a family of polysulfidic activator–accelerator complexes. Once formed, the sulfurating agent participates in the crosslink formation, the shear modulus starts increasing, and the mixture enters the curing region. In natural rubber, the postcuring region is associated with crosslink shortening (desulfurization) and crosslink degradation, resulting in “reversion” or a decrease in the shear modulus. Vulcanization is a complex process; the degree of crosslinking, and hence the physical properties of the product, depends on the curing mix design and temperature history. In order to reliably predict and optimize the curing process, a kinetic model that captures the dynamics of the crosslinking chemistry is required.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
Coran and coworkers provided the first set of kinetic models that described the scorch delay and captured the essential mechanism of accelerated sulfur vulcanization.8,14–16 These were lumped kinetic models, since the active sulfurating agents were all lumped together as a single species. Coran8 proposed that the crosslink formation proceeds via formation of alkenyl persulfenyl radical (R-
). The scorch delay was attributed to scavenging of these radicals (which were again lumped as a single species) by the accelerator sulfurating agent. Ding and Leonov17 modified the Coran model to include a side reaction that leads to formation of “dead-ends,” to predict the effect of cure temperature on the equilibrium modulus. This modified lumped kinetic model was able to capture the observed scorch delay and variation in shear modulus due to crosslink chemistry. Their model did not predict the postcrosslink chemistry in natural rubber. Ding et al.16 extended their model to predict the postcuring reversion observed in the cure curve for natural rubber based compounds. Though the lumped models quantitatively predict curing, they are unable to provide details about accelerator and retarder chemistry and do not explicitly account for sulfur balance in the model. Such models can emulate the degree of vulcanization as a function of time for arbitrary thermal history but fare poorly when the composition of the vulcanization mixture is changed.
In a departure from traditional modeling efforts, Ghosh et al.4 attempted to capture the complexities actually present in the vulcanizing system. They developed a PBE model taking into account the balance of sulfur. The model comprehensively captured the accelerator chemistry, the crosslinking chemistry, and crosslink degradation. Although more complex than the lumped models, the PBEs can capture the effects of varying the initial cure compositions as well. Likozar and Krajnc18 extended their model for a system with two accelerators. They used the same mechanism as ref 4. Ghosh et al.4 assumed reaction rates to be independent of the length of the sulfur chain in the various polysulfidic links; this assumption was relaxed by Likozar and Krajnc.18
All of these models follow the same general framework depicted in Figure 1. The lumped models do not distinguish between accelerator, sulfur, and the active sulfurating agent, and each mechanism in the crosslink chemistry is represented by a single reaction. The PBE model, on the other hand, accounts for the sulfur balance and the growing polysulfidic complexes in each of the steps depicted in the schematic. While their model captures accelerator chemistry, they use a simple treatment for the action of the retarder.4,18 Specifically, the retarder reacts with MBT, which is produced via thermal degradation of TBBS and is a precursor to the formation of active sulfurating agents. Later, MBT is regenerated in the curing region ensuring that the retarder does not affect the final crosslink density. While these two reactions were included, Ghosh et al.4 did not rigorously model the formation of retarder complex.
The first objective of this work is to extend their model to capture the various chemistries observed by Gradwell and Stephenson,9 who tracked the progress of polyisoprene vulcanization by TBBS in the presence of CTP using differential scanning calorimetry and high performance liquid chromatography. They found polysulfides of 2-cyclohexyldithiobenzothiazole (CDB, a product of reaction between CTP and MBT) in the reaction mix. They also reported formation of MBT only after scorching. Extension of the model of Ghosh et al.4 is needed to capture both these observations. To this end, we first present ODR experiments with varying amounts of TBBS and CTP in the vulcanization mix, followed by development of PBE-based model to fit these data. We then evaluate the model performance to qualitatively predict the observations in ref 9. The second objective of this work is to present detailed analysis of the model predictions through sensitivity and reaction path analysis (RPA).
EXPERIMENTAL
The experiments performed using the ODR are used to track the evolution of the density of crosslinks in rubber networks. The basic construction of RPA-2000, an ODR, is as follows: The bottom die is fixed and the top die oscillates. One could vary the amplitude and frequency of oscillation of the upper disc. The temperature of the sample can be set at a predetermined degree. The blended rubber mixture is set in the die cavity and is heated to a required curing temperature. The top die oscillates at a predefined amplitude and frequency, and the torque is measured. The natural rubber mixture behaves like a viscoelastic liquid. Theory of linear viscoelasticity is used to obtain the storage and loss shear modulus. As the rubber sample cures, the torque required to bring about the same elastic deformation increases, which results in the increase of storage shear modulus, that is, the greater the density of crosslink, the greater the storage modulus (or shear modulus).
The samples were prepared by mixing natural rubber with ZnO, sulfur, TBBS, and CTP, along with carbon black as a filler, stearic acid, and antioxidants in the Banbury mixture (see Table I). ODR experiments were conducted at homothermal conditions for each of the nine samples at four different temperatures, 128, 138, 148, and 158 °C. In total nine different mixtures were prepared, with varying amounts of accelerator (0.5 to 1.7 phr) and retarder (0.07 to 0.2 phr), as shown in Table II. The first three samples contain high, medium, and low amounts of accelerator; the next three samples have varying amounts of TBBS for a constant amount of the retarder; the last three samples have varying amounts of CTP for a constant amount of the accelerator. Additionally, one more sample was prepared without adding any retarder. Although the focus of this work is on the effect of CTP as a prevulcanization inhibitor, the C-0 case serves as an additional check to verify the predictive capability of the kinetics developed here.
The addition of carbon black to a rubber mixture increases the storage modulus as a result of the filler–rubber (see ref 19) and filler–filler interactions, which are physical in nature. To extract the information on chemical crosslink density from storage modulus, one needs to correct for these physical interactions. The Guth–Gold equation is not applicable because the carbon particles agglomerate and form closed shell-like structures with rubber being present in the void (trapped rubber). During the extraction of chemical crosslink information, one needs to correct for the presence of such structures. Medalia modified the Guth–Gold equation as follows20,21:
Here Veff is the effective volume fraction of filler, Gm represents the shear modulus of the rubber mixture, and Gg represents the equivalent shear modulus of gum rubber (without filler). The effective volume fraction of filler can be determined by performing an experiment without carbon black (gum rubber) and comparing it with the same formulation with 48 phr carbon black. Specifically, we prepared a gum rubber mixture without adding filler, the sample was cured at 138 °C, and shear modulus was measured as the sample cures. With the plateau shear modulus data for filled and unfilled rubber mixture, the effective volume fraction is determined to be 0.28. The actual volume fraction of added filler is only about 0.18. Further, it is assumed that the effective volume fraction remains unchanged at all temperatures. The measured shear modulus data for each of the filled rubber samples are corrected to obtain the equivalent gum shear modulus using the effective volume fraction as 0.28. Once the equivalent shear modulus of gum rubber is estimated, the Gaussian statistical theory of rubber network (Treloar)22 is invoked to estimate the density of crosslinks, that is,
Gg represents the shear modulus of gum rubber, 
is the shear modules of initial rubber mixture or the minimum modulus in the ODR data, [Vu] represents total moles or crosslinks per unit volume, R is the gas constant, and T is the absolute temperature of the sample.
MODELING OF VULCANIZATION
In this section, we review the PBE-based model for accelerated rubber vulcanization. We start with the PBE model of Ghosh et al.,4 with sequential sulfur uptake (so-called “Model-2”). The model is modified for the different accelerator (TBBS) used in this work and is extended to handle the effect of the prevulcanization inhibitor (CTP). The experimental observations of Gradwell and Stephenson9 are used to guide the extension of the model in this work to better capture the retarder chemistry and certain qualitative features of TBBS-accelerated vulcanization. The reactions included in the model are presented next; this is followed by a discussion of the other possible reaction schemes proposed by Gradwell and coworkers;9,11,23 we end this section with a description of the parameter estimation procedure.
POPULATION BALANCE EQUATIONS
Figure 1 shows a schematic of the key chemistries in the vulcanization process. Formation of active sulfurating agents (Ax) is an important step in the induction region; this complex is then responsible for the crosslink formation, via active crosslink precursors 
finally, desulfurization and main chain modifications occur in the postcrosslink stage to give the final vulcanization products (Vux). The vulcanization process is therefore split into the following submechanisms.
Accelerator Chemistry
The key reactions in the induction region are thermal degradation of TBBS to MBT and its reaction with MBT to form MBTS. MBTS then complexes with the activator to form the accelerator–activator complex. Both MBTS and its zinc complex are represented as A0. A0 then sequentially uptakes sulfur atoms (via reaction R3 below), forming a sequence of active sulfurating agents, Ax (x = 1 to 14). The reaction rates of the zinc complex are typically much faster than those of MBTS or its polysulfides. Therefore, while the mechanism in the absence of ZnO and stearic acid will remain unchanged, the kinetic parameters are applicable only in the presence of ZnO with stearic acid as an activator.
The importance of reaction R1 is contested by Gradwell and coworkers because they did not detect MBT in their vulcanization mixture in the induction region.9 However, as we will discuss later, our model with the inclusion of detailed retarder chemistry qualitatively captures these trends.
Retarder Chemistry
The main action of the retarder is to scavenge MBT formed in the induction region through the following reaction:
Here C0 represents CDB. This reaction was included in ref 4 to account for inhibition action of the retarder.
CTP is thermally unstable and degrades at vulcanization temperature to cyclohexanethiol (CHT) and phthalimide. This is an important reaction because it reduces the amount of CTP available to react with MBT through the above reaction. The degradation product, CHT, itself can react with TBBS.
Since TBBS is thermally stable at lower temperatures, its reactions with MBT (R2) and CHT (R7) are important steps in the induction region.
The following reaction, which represents formation of CDB polysulfides (CDBP or Cx) is included in our model because Gradwell and Stephenson9 reported that small amounts of CDBP were observed in their vulcanization mixture:
It should be mentioned at this stage that inclusion of this reaction was necessary to get a reasonable model performance.
The MBT that is scavenged in the induction region is regenerated to give activator–accelerator complex through the following reaction:
While the reaction of CDB (C0) with MBT was included in ref 4, we have extended this reaction to CDBP as well. All these reactions are important to qualitatively capture the trends in ref 9.
Crosslink Chemistry
Active sulfurating agents (Ax) react with rubber to give crosslink precursors (Bx) with a pendant group terminating the polysulfidic chain; these degrade thermally to give activated crosslink precursors 
finally, the activated crosslink precursors react with another hydrocarbon chain to form vulcanized rubber (Vux). The following reactions are included in the model:
The species, 
generated in the thermal degradation reaction R11 is the activated-terminated polysulfides radical.
Looping
A crosslink is formed when sulfur from a polysulfidic radical attaches with allelic groups in two different rubber chains, whereas loops are formed, through reactions similar to R12, when sulfur attaches to the same allelic rubber chain:
Sulfur Addition and Other Reactions
The model of ref 4 includes sulfur addition reactions for the free radicals formed in the crosslinking process:
as well as the following free-radical reactions
The importance of the reaction R17 should be emphasized for the sulfenamide class of accelerators, since scavenging of the active persulfenyl radicals 
is considered to be responsible for the observed scorch delay. This mechanism was first introduced in the lumped kinetic model by Coran,8,14,15 based on the observation of Campbell and Wise5,6 that crosslinking proceeds only after the active sulfurating agents are consumed.
Postcrosslink Chemistry
Natural rubber undergoes reversion as a result of desulfurization and crosslink degradation reactions. In the desulfurization step, the accelerator–activator complex removes one sulfur atom at a time from the initial crosslink chain, resulting in an increase in the concentration of monosulfidic and disulfidic links.
DISCUSSION ON KINETIC MECHANISM
The kinetic mechanism in our work includes reactions R1–R20. Reactions R6–R9 have been added to the original mechanism of Ghosh et al.4 to predict the ODR data as well as qualitatively reproduce trends observed by Gradwell and coworkers.9,23 Further discussion on their observations is presented in this section.
One key observation in ref 9 is that with TBBS used as an accelerator, MBT was not observed in the induction region. Formation of MBT coincided with crosslink formation. Hence, Gradwell et al. postulated that MBT is formed via reaction R10 (included in our model). Since they also observed polysulfides of TBBS (represented as TBBP) in the vulcanization mixture, they postulated that TBBP initially act as sulfurating agents. They postulated that a small amount of TBBS undergoes the following reactions in the induction region to form crosslink precursor:
Another possibility is that polysulfides of CDB (Cx) themselves take part in crosslinking via the reaction
This reaction is plausible because CDBP were detected in the induction region9 and CDB itself was able to vulcanize rubber in the absence of TBBS, albeit at a much slower rate.
It is generally accepted that the polysulfides of activator–accelerator complex (Ax) are primarily responsible for the formation of crosslink precursors.24,25 A kinetic mechanism can neither conclusively confirm nor rule out the possibilities that TBBP and/or CDBP are indeed involved in crosslink formation. However, we show in the next section that the key observation regarding MBT formation is explained by our model without including the above reactions (R21–R24). Since absence of MBT in the induction region was the primary reason for postulating the above reactions, we have excluded these reactions from our model.
Second, the presence of these reactions results in some crosslink formation in the induction region, which was not observed experimentally. With the reaction R23 included, the model will not predict the observed scorch delay. For example, when we included reaction R24 (along with reactions R1–R20) and performed parameter optimization, some crosslink formation occurred in the induction region (i.e., the scorch delay was reduced). Further analysis showed that the delayed formation of MBT predicted by the model was not a consequence of these additional reactions. Therefore, the reactions discussed in this subsection are excluded from our final model.
Finally, we discuss one of the important reactions, originally proposed in ref 8 to account for scorch delay: the reaction between active persulfenyl radical precursors, 
and active sulfurating agents, Ax,
This reaction scheme is supported by the observation that crosslink formation starts after the active sulfurating agents have been depleted.5,6,26 This reaction is similar to reaction R17 of our mechanism, except that reaction R25 is for all Ay, not just A0. We have repeated our simulations with R25 included in our mechanism with the same rate constant kA-BST from R17 used for R25. With this reaction, the induction time was not affected, but only the final crosslink density changed. Moreover, if A0 gets significantly depleted before Ax (x ≥ 1), the latter will replenish A0 through the reaction R4 (note that this is a reversible reaction). To maintain consistency with the PBE-based models of Ghosh et al.,4 and Likozar and Krajnc,18 reaction R25 is not included in our model.
Thus, our model consists of reactions R1–R20, shown in the previous subsection. A detailed discussion of our model performance and predictions is presented in the next section.
MODEL EQUATIONS AND PARAMETER ESTIMATION
Based on the results of Morgan and McGill,27 the number of sulfur atoms in active accelerator complex were limited to 14, whereas that in rubber crosslinks were limited to 16 in refs 4, 18. We used the same numbers in our model. The kinetic model for vulcanization under homothermal conditions is summarized in Appendix B. The resulting model has 124 nonlinear ordinary differential equations. If we assume that the rate constants are independent of the number of sulfur atoms in the polysulfide links, the model has 20 rate constants, corresponding to each of the reactions R1–R20.
The number of parameters fitted to the data is reduced using the following assumptions made by Ghosh et al.4 Since the reactions R12 and R13 both involve reactions of activated polysulfidic radicals with rubber, the rate constants kE-R and kVu are assumed to be equal. Similar assumptions are made for sulfur addition in polysulfidic radicals (kBST-S = kE-S) and sulfur chain crossover (kA-A = kE-E). Ghosh et al.4 used the theory of Debye and Beuche28 to calculate the looping probabilities. Their study suggested that the average looping probability is approximately 10% for polysulfidic radicals with a range of 1 to 17 sulfur atoms. Therefore, the rate of loop formation is assumed to be 10 times slower than crosslink formation (i.e., kloop = 0.1kVu). All the assumptions made here are similar to the ones made in ref 4. Unlike ref 4, we allowed kdesul to be optimized along with the other rate constants.
The experimental data from the ODR are obtained for various initial amounts of accelerator (TBBS) and retarder (CTP) listed in Table II. The procedure discussed in the “Experimental” section is then used to obtain the crosslink concentrations as a function of vulcanization time. Five initial conditions are chosen as “training data,” as indicated in Table II. The MATLAB implementation of the Levenberger–Marquardt method in the function “lsqnonlin” is used to find the rate constants that minimized the following objective function:
where 
and 
represent the total crosslink concentration obtained experimentally (from ODR data) and those predicted by model, respectively, at various times for the five training set experiments.
Since the rate constants have the Arrhenius form, ki = 
, there are indeed 32 parameters (four rate constants are fixed and 2 × 16 parameters are fitted to data). The parameter estimation process is conducted for 16 parameters at a time. First, the data at temperature 128 °C are used to obtain all 16 rate constants at that temperature 
the procedure is repeated for 158 °C 
With these values, an initial estimate of the pre-exponential factors and the activation energies is obtained. The values 
and 
give unique values of frequency factor Ai and activation energy Ei for each reaction. These initial values are then used for parameter estimation, where all the training set data (at all four temperatures) are used for optimization as per Eq. 3. In this second step, the frequency factors are kept constant and only the 16 activation energies are fitted to the data. The performance of the resulting model is tested with the test data, which was not used during the parameter estimation procedure.
RESULTS AND DISCUSSION
We now discuss the model results. First, a comparison with experimental data will be presented, followed by a discussion on the transient response of various species in the model. Finally, we will present sensitivity and RPA to identify the key reaction pathways in the rubber vulcanization process.
PERFORMANCE OF THE MODEL
The final set of parameter values obtained using the parameter estimation procedure is shown in Table III. The final column of the table shows the rate constants calculated at 138 °C for comparison. In agreement with experimental observations, the rate constant for thermal degradation of TBBS is much lower than its reaction with MBT and CHT at 138 °C; higher activation energy ensures that the thermal degradation process increases rapidly at higher temperatures.
Figure 2 shows the model performance for sample H (see Table II), which contains a higher amount of TBBS among the experimental data considered in this paper. The model predicts the observed evolution in crosslink concentration accurately for most of the vulcanization time. However, the scorch delay is not captured accurately for higher TBBS concentrations (at 128 °C), while the curing and postcuring stages are captured accurately. This comparison represents “training data,” since these data were used for the parameter estimation process. Figure 3 presents a comparison for two of the test data sets (samples M and L, containing moderate and low TBBS amounts). Although these data were not used for parameter estimation, the model does a good job of predicting the measured cure curve. As can be seen from these figures, the scorch delay is captured accurately under these conditions. The model captures the experimental trends reasonably at all temperatures, though the model performance is better at 158 °C than the other temperatures. The scorch delay is captured very well at all the temperatures.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
In order to further test the model, Figure 4 compares the experimental results with model predictions for a mix without any CTP and with 1 phr TBBS. Since this paper focuses on the role of CTP as a retarder in accelerated vulcanization, our model is applicable for systems with CTP added and is not specifically designed to capture these data. Even without CTP present in the system, the model does a reasonable job in capturing the data. However, the modeling errors are slightly higher than those with CTP present in the system.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
Figure 5 shows the effect of varying TBBS concentration on the cure curve at a constant temperature of 148 °C. The symbols represent experimental data for samples T-1, T-2, and T-3, all of which were included in the training data set. As the amount of TBBS increases, the scorch delay decreases, the rate of crosslink formation in the curing region increases, and the final crosslink density increases. The model is able to predict the scorch delay accurately; the overall trends in the curing and postcuring regions are also captured reasonably. The model slightly underpredicts the final crosslink density. The model performance is best for the highest temperature of 158 °C; the mismatch between the model prediction and experiments increases at lower temperature and lower amounts of TBBS. Still, even for sample T-1 at 128 °C, the model underpredicts the final crosslink density by about 6 mol/m3 (less than 10%).
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
Changing the amount of CTP primarily changes the scorch delay slightly, as observed from Figure 6. Of the three experiments, C-3 is included in the training data set, while C-1 and C-2 are test data. The model quantitatively captures the variation in crosslink density. Model–experiment fit is accurate for other temperatures as well. The inset figure shows the effect of increasing CTP on the scorch delay. The scorch delay increases from about 10 to 13.5 min when the amount of CTP is doubled from 0.09 to 0.18 phr. The model predicts the scorch delay exactly for C-1, while it overpredicts the scorch delay for C-3 by about 1 min. The experiments show little change in the final crosslink density (at the end of 140 min) between the three samples, while the model predicts the final crosslink density to increase slightly from 75 to 78 mol/m3.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
In summary, the model quantitatively captures the effect of varying TBBS and CTP concentrations for a range of temperatures of interest in vulcanization of natural rubber. As we show in the next section, the model is also capable of providing insights into the qualitative observations reported in the literature on the same system.
TRANSIENT RESPONSE
Crosslink Concentration and Sulfurating Agents
A comparison between model prediction and experimental data for total crosslink concentration in sample H was discussed earlier (Figure 2). The contributions of various Vux species to the total crosslink concentration for the same conditions is shown in Figure 7. The crosslinks are not formed in the induction region. After the scorch delay, all the vulcanized rubber species with different lengths of sulfur crosslinks (Vux) appear approximately at the same time. Shortly thereafter the monosulfidic links predominate. This is because (i) the desulfurization reaction results in shortening of the chain length, eventually leading to monosulfidic links, and (ii) the larger number of S–S bonds makes longer crosslinks more susceptible to degradation reaction R20.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
Figure 8 shows the transient evolution of Ax and Cx. Since the maximum concentration of A0 is an order of magnitude higher than Ax, it is shown on the left axis, while the other species is on the right axis. The concentration of A0 is similar to Ax and Cx in the induction region. However after scorching, A0 concentration increases to reach an order of magnitude higher than the other species. Note that A0 represents either MBTS or its complex with ZnO (the model does not distinguish between the two). The qualitative trends of A0 match the observations of Gradwell and Stephenson.9 The initial peak in A0 corresponds to crosslink formation, where MBT is formed through reaction R10, and a second (smaller) peak corresponds to crosslink reversion (due to desulfurization reaction R19). An initial peak is observed in Ax concentrations at the time of scorching as well. This peak may be attributed to reaction R9, since it occurs at the time when concentrations of CDB polysulfides reduce significantly. After the peak, as the crosslink formation reactions proceed, the concentrations of Ax fall. There is a second peak in Ax concentration, which is attributed to the recombination of benzothiazole-terminated polysulfidic radical 
through reaction R18.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
Prediction of Major Reacting Species
Figure 9 shows the variation in concentrations of all the initial compounds and MBT vs. the vulcanization time. The crosslink density is also plotted for reference. The sulfur concentration falls quickly because the sulfur uptake reactions are relatively fast when compared with the reactions forming crosslinks. CTP decomposes faster than TBBS at the vulcanization temperature. CHT formed on thermal decomposition of CTP reacts with TBBS. Since the formation of CHT is the rate-limiting step, the model predicts negligible concentration of CHT during the vulcanization process.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
MBT, which is formed primarily as a result of the thermal decomposition of TBBS in the induction region, reacts with CTP and TBBS to form CDB and A0, respectively. MBT also reacts with CDB (and polysulfides of CDB) to form A0. Since the reactions of MBT leading to the formation of A0 are faster than the thermal decomposition of TBBS, most of the MBT formed in the induction region is consumed. As a result, MBT is not observed during the induction region. In the crosslinking region, the rate of formation of MBT increases because of reaction R10, leading to an increase in concentration of MBT. These results are consistent with the observations of Gradwell and Stephenson9,10 and Morgan and McGill.27
Effect of Initial Mix
An important criterion in tire design is the relative amounts of monosulfide, disulfide, and polysulfide crosslinks in the final vulcanizate. Loo29,30 studied vulcanization of natural rubber with sulfur using N-cyclohexyl benzothiozyl-2-sulphenamide (CBS) as an accelerator at a temperature range of 140 to 200 °C for both “conventional” and “efficient” cure. An efficient curing system is one with significantly higher amounts of the accelerator compared with the initial amount of sulfur. Monosulfidic crosslinks were more stable compared with the higher order sulfur crosslinks. Loo found that the contribution of monosulfidic crosslinks in the total crosslink concentration is higher in efficient cure.
Figure 10 shows the evolution of monosulfide, disulfide, and polysulfide crosslinks predicted by our model for conventional (Figure 10a) and efficient (Figure 10b) cure. These represent curing at a temperature of 140 °C for an extended period of 500 min. The amount of CTP is kept at 0.15 phr, while the amount of TBBS to sulfur is 0.2 for conventional and 2.33 for efficient mix. As expected, the induction time is shorter for the efficient system. In either case, the disulfidic and polysulfidic crosslinks eventually undergo desulfurization or crosslink degradation reactions, leading to reduction in total crosslink density (reversion) and an increase in the ratio of monosulfidic to polysulfidic links. The degradation reaction R20 is responsible for decrease in the total crosslink concentration, whereas the desulfurization reaction results in an increase in concentration of shorter crosslinks at the expense of longer crosslinks. Compared with the conventional cure, the efficient TBBS-accelerated system (Figure 10a) gives a higher percentage of monosulfidic crosslinks in the curing region. Owing to the higher stability of monosulfidic crosslinks, the reversion in the postcuring region is less for the efficient TBBS-accelerated system.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
SENSITIVITY ANALYSIS
The sensitivity of the response in the total crosslink density is tracked for variations in the reaction rate constant as the parameter. The sensitivity for the response Ri to the parameter φj is defined as 
. In this work, we use a brute force sensitivity analysis (SA),31 where a parameter is increased or decreased by a factor of 2 and the normalized sensitivity coefficient is defined as
Here, 
is the total crosslink density at the end of ith experiment when varying the rate constant φj = k0,j. Figure 11 is a representative sensitivity plot for initial TBBS at 2.5 phr, CTP at 0.07 phr, sulfur at 2.5 phr and operating temperature 150 °C. The 40 rows represent the response to twofold increase (shaded bars) and decrease (blank bars) in each of the 20 rate constants. From rows 11 and 12, it is clear that changing the rate constant kTBBS-CHT does not affect the final crosslink concentration, whereas the crosslink concentration changes when any of the other rate constants are varied.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
The sensitivity analysis is then repeated for various initial amounts of TBBS and CTP (see Table IV) and at three different temperatures (130, 140, and 150 °C). The amount of sulfur is kept constant at 2.5 phr. The sensitivity matrix is thus built for 20 rate constants and 36 responses (for increase/decrease in a rate constant, at six inlet concentrations, and three temperatures: 2 × 6 × 3), with Si,j representing one element of the matrix. Principal component analysis is performed on the sensitivity matrix. The eigenvalues of ST S are computed: the largest eigenvalue is 3.32 and the three smallest eigenvalues are 1.99 × 10−8, 6.1 × 10−5, and 3.4 × 10−5.
Mhadeshwar and Vlachos31 suggest using a threshold of 10−5 to determine the number of important reactions in the mechanism. As one could expect from Figure 11, one reaction may be eliminated from the mechanism without affecting the final result from the model. Indeed, the eigenvectors of ST S confirm that reaction R7
does not affect the overall crosslink density. The values of rate constants given in Table III show that the decomposition of CTP to give CHT is much slower than this reaction. Thus, reaction R6 is the rate-limiting step, and one may combine reactions R6 and R7 to the following equivalent reaction
Note that the above reaction does not imply a direct reaction between TBBS and CTP. Another consequence of this is that the amount of CHT predicted by the model in the induction region is very low, not exceeding 0.006 mol/m3. This result matches the results of Gradwell and Stephensen,9 who did not observe any CHT in their experiments. While they attributed this observation to volatility of CHT, our model predicts an alternate reason for their observation: CHT is consumed at a much faster rate than it is formed.
REACTION PATH ANALYSIS
We now present RPA to identify the key reaction steps as the vulcanization process progresses. The relative contributions of various reactions to the formation or consumption of various reacting species is tracked. The net rate of formation or consumption is given by
Note that we sum over all reactions where the stoichiometric coefficient for the species is positive (formation) or negative (consumption). The contribution of a reaction in the rate of formation (or consumption) of a species is the ratio of the reaction rate in that particular reaction to the net rate of formation:
RPA is performed at four different times, t = 3, 12, 24, 48 min for sample H at 138 °C. These correspond to the induction region, early crosslinking region, late crosslinking region, and postcrosslink, respectively (see Figure 9). The amount of TBBS is the largest in this sample, compared with the others used experimentally. RPA was repeated for the other samples as well; the results are qualitatively similar to those presented here.
Figure 12 shows the key reaction pathways active at 3 min. In agreement with the literature, thermal degradation of TBBS R1 is not significant, accounting for only 12% of the total reaction flux of TBBS. However, this reaction is important in the induction region because it primarily produces MBT. Reaction R2, between TBBS and MBT to form A0, accounts for two-thirds of the total rate of consumption of TBBS for sample H, whereas the remaining 21% of the flux is due to the reaction of TBBS with CHT (R7). This trend is reversed when the relative amount of CTP is increased: for example, in sample T-1 (where the amount of TBBS is 0.5 phr), the reaction R7 accounts for 65% of the net TBBS flux and the rate of formation of A0 is 8 times lower than that in sample H.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
The role of retarder in the induction region is also tracked. Thermal degradation of CTP (R6) and its reaction with MBT (R5) are both equally important. The “pseudo-steady-state ratio,” defined as the ratio of the net rate of reaction of a species to the net rate of formation or consumption
is used to explain two of the key trends reported by Gradwell and Stephenson.9 If the ratio is close to zero, the rates of formation and consumption of the species k are nearly equal and the species is said to be in pseudo-steady-state. The pss for both MBT and CTP are found to be less than 0.001 at 3 min, implying that their concentrations remain nearly constant, at a relatively low value, in the induction region. This explains the experimental observation of ref 9 that neither MBT nor CTP could be detected in the induction region.
The role of activator–retarder complexes A0 and Ax is elucidated for the entire vulcanization process in Figure 13. The time stamps in the brackets indicate the times at which fluxes in the specific reaction pathway are significant. Initially, A0 is primarily formed as a result of the reaction between TBBS and MBT. In the induction region and early crosslinking region, when some amount of sulfur is present in the vulcanization mixture, the sulfurization reaction R3 is the primary reaction for formation and consumption of Ax. During the latter part of the process (i.e., 24 min and thereafter), the reaction of Ax with rubber (R10) becomes significant. At this stage, sulfur is depleted and the active sulfurating agents, Ax, are primarily formed via recombination (R18) of activator-terminated polysulfide radicals, 
These radicals, in turn, are primarily formed through reaction R11. Thus, in the crosslinking and postcrosslink regions, Ax reacts with rubber to give Bx and MBT (R10), 
is formed through degradation of Bx (R11), which recombines with 
to give back Ay.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
Finally, Figure 14 shows the reactions involving 
and Vux at 24 min and the net reaction pathways for vulcanized species. In the initial stages (before ∼15 min), the rates of reactions involving either of these species are low. The sulfur addition reaction R16 is the predominant reaction during this period (though the rate is still low due to lower concentrations of 
). After the scorch delay, these reactions pick up. Most of 
is formed by thermal degradation of Bx. The formation of rubber crosslinks is 10 times faster than the formation of loops because of the assumption that kloop = 0.1 kVu. The reaction R12 between 
and rubber is the predominant reaction leading to the formation of Vux, whereas desulfurization (R19) and crosslink degradation (R20) reactions are both important in the reversion process. The maximum rate of reaction R12 is more than two orders of magnitude higher than the rates of reactions R19 or R20. Eventually, as 
is consumed, the rate of reaction R12 falls below the rates of R19 and R20, indicating the start of postcrosslink phase in the vulcanization process.
, and Vux at 24 min from the start of vulcanization of sample H at 138 °C. The percentages mentioned refer to the relative contribution in rate of formation or consumption of species 
and Vux.
Citation: Rubber Chemistry and Technology 85, 2; 10.5254/rct.12.89971
In the postcrosslink region, the crosslink degradation reaction is more predominant for higher order crosslinks, whereas the desulfurization reaction becomes important for disulfidic, trisulfidic, and tetrasulfidic crosslinks. For example, the degradation reaction contributes 85% of the net reaction flux for consumption of Vu10, while it contributes 33% for Vu2. However, owing to the significantly lower concentration of Vu10, the rate of degradation of Vu10 is much lower than that of Vu2. The desulfurization reaction R19 produces A1, which can then react with rubber to give B1, 
, and eventually Vu1. Only a small amount of 
is produced at this stage, with the net rate of formation being nearly two orders of magnitude lower than the rate of formation of 
. We observe that while Vu10 is formed almost exclusively by desulfurization of Vu11, Vu1 is formed via crosslinking reaction R12. At 24 min, both the reactions have nearly equal contributions for the formation of Vu5. Finally, it should be noted that over the entire vulcanization experiment, the maximum rate of formation of crosslinks is two orders of magnitude higher than the maximum rate of consumption of crosslinks. In other words, crosslink degradation is a much slower process than crosslink formation. This is also expected from Figure 7, where the concentration of Vux falls gradually in the postcrosslink region.
In summary, RPA is able to throw light on the various active processes during different stages of the vulcanization process. It may indeed be used to gain quantitative insights into vulcanization.
CONCLUSIONS
A population balance model for vulcanization of natural rubber with N-t-butylbenzothiazole-2-sulfenamide (TBBS) as an accelerator and N-(cyclohexylthio)phthalimide (CTP) as a retarder in the presence of sulfur and zinc oxide was developed in this paper. The model is an extension of the one proposed by Ghosh et al.,4 with appropriate modifications to capture the retarder chemistry. Experiments for several mix designs were performed for various amounts of TBBS and CTP for semiefficient vulcanization of natural rubber, and the effect of accelerator and retarder on vulcanization was tracked through ODR experiments. The model accurately predicted the temporal evolution of crosslink concentration, as well as qualitatively captured the trends reported by Gradwell and Stephenson9 for a similar system. The model also reliably predicted the formation of monosulfidic, disulfidic, and polysulfidic crosslinks.
SA and RPA were performed to better understand the key reaction pathways. The retarder action is due to its reaction with MBT and TBBS in the induction region. At the initial period, sulfur uptake reactions were dominant in the active sulfurating complex (Ax) and the activated crosslink precursor radicals 
. As a result, all the vulcanized rubber species having different numbers of sulfur atoms appear simultaneously during scorching. The mechanism then shifts toward one mediated by accelerator-terminated polysulfide radicals (Ex), resulting in higher rates of formation of lower sulfur crosslinks in vulcanized rubber. Thus, the key reaction steps are
-
in the induction region; -
in the early crosslinking region (with rates of reaction being similar for all x); -
Ex → Ax, Ex → Bx dominant in middle and late crosslinking region (with lower sulfur chains forming faster); and
-
Vux → Vux−1 and Vux→ D dominant during the postcrosslinking region.
-
The retarder action is due to
with the latter reaction occurring in the crosslinking region.
The retarder reacts with TBBS and MBT to give CDB, resulting in an increase in the induction time. Since thermal degradation of TBBS is a slow reaction at the vulcanization temperature, MBT gets consumed in the induction region as soon as it is formed. Consequently, MBT may not be detected in the system until after scorching. Thus, another key outcome of this work is to show that the observation of Gradwell and Stephenson9 (that MBT is formed only after crosslinking is initiated) is consistent with the role of Ax (polysulfides of MBTS and their Zn complexes) as active sulfurating agents.
While it is generally accepted that desulfurization is the key reaction that results in increase in relative amounts of monosulfide crosslinks compared with polysulfide, our results show that sulfur atoms recovered by active sulfurating agents are channeled back into the monosulfide and disulfide links.
In conclusion, the model is able to quantitatively capture the effects of accelerator and retarder chemistry during vulcanization, and the detailed analysis of the model exposes key features of the vulcanization chemistry.
— Model framework for accelerated sulfur vulcanization in presence of a prevulcanization inhibitor (retarder) adopted in this work. The general structure proposed by Morrison and Porter7 and implemented for population balance kinetics by Ghosh et al.4 is extended to include the effect of the retarder.
— Comparison between model predictions (lines) and experimental data (symbols) for evolution of crosslink concentration at four different temperatures for the training data set sample H (TBBS = 1.7 phr, CTP = 0.07 phr, sulfur = 2.5 phr, ZnO = 5 phr).
— Comparison between model predictions (lines) and experimental data (symbols) for evolution of crosslink concentration at four different temperatures. The comparison is presented for test data. (Left) sample M (TBBS = 1.2 phr, CTP = 0.11 phr); (right) sample L (TBBS = 0.7 phr, CTP = 0.2 phr). Sulfur was kept at 2.5 phr and zinc oxide at 5 phr.
— Model predictions (lines) and experimental data (symbols) for vulcanization in the absence of retarder at two different temperatures. The amount of TBBS is 1 phr, sulfur is 2.5 phr, and ZnO is 5 phr.
— Effect of varying TBBS concentration on evolution of crosslinks at a constant temperature of 148 °C. Symbols represent experimental data, and the lines represent model predictions.
— Effect of varying CTP concentration on evolution of crosslinks at a constant temperature (128 °C) and initial quantity of TBBS (1 phr). The induction region is zoomed in and shown in the inset. Symbols represent experimental data, and the lines represent model predictions.
— Prediction of concentrations of various crosslinks for sample H at 138 °C. This condition represents conditions denoted by circles in Figure 2.
— Transient evolution of concentrations of active sulfurating agents (Ax, left) and CDB polysulfides (Cx, right) for the conditions in Figure 7. The abscissa is truncated to highlight the key transients. A0 is an order of magnitude larger than any other species in the two plots.
— Prediction of concentrations of sulfur (left) and TBBS, CTP, and MBT (right) for conditions in Figure 7. The total crosslink concentration is also shown (right) for comparison.
— Prediction of concentrations of monosulfides, disulfides, and polysulfides crosslinks for (a) conventional (sulfur = 2.5 phr, TBBS = 0.5 phr, CTP = 0.15 phr) cure system and (b) efficient (sulfur = 1.5 phr, TBBS = 3.5 phr, CTP = 0.15 phr) cure system at 140 °C.
— Results of brute force sensitivity analysis performed at TBBS = 2.5 phr, CTP = 0.07 phr, and 150 °C. Each row represents the sensitivity coefficient for increase or decrease in individual rate constant by a factor of 2.
— RPA during the induction region for sample H at 138 °C. The thickness of the arrows corresponds to the relative importance of various reactions. All of these reactions are completed in about 12 min.
— RPA for (a) A0 and (b) Ax for sample H at 138 °C. The times from the start of vulcanization process are specified in brackets. The thickness of lines corresponds to the contribution of the reaction in formation or consumption of the species.
— RPA for the fate of species Bx, , and Vux at 24 min from the start of vulcanization of sample H at 138 °C. The percentages mentioned refer to the relative contribution in rate of formation or consumption of species and Vux.
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