Cross-Linking of Polymers: Kinetics and Transport Phenomena

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Ind. Eng. Chem. Res. 2011, 50, 1558–1570

Cross-Linking of Polymers: Kinetics and Transport Phenomena Blazˇ Likozar* and Matjazˇ Krajnc Chair of Polymer Engineering, Organic Chemical Technology and Materials, Faculty of Chemistry and Chemical Technology, UniVersity of Ljubljana, AsˇkercˇeVa cesta 5, 1000 Ljubljana, SloVenia

The cross-linking process of a complex system was studied at elevated temperature. Partially hydrogenated poly(acrylonitrile-co-1,3-butadiene)/dicumyl peroxide compound was selected as a representative complex polymer/cross-linking agent system. On the basis of the mechanisms of cross-linking reactions, an expression relating reaction rates and population balance equations with rheologically determined concentration of crosslinks was applied. Consequently the Eyring equation reflecting the effect of the self-diffusion on reaction rate constants was utilized and was further integrated into the classical Arrhenius equation for reaction rate constants. Diffusion coefficients acknowledging the mutual diffusion were estimated according to the free-volume theory, whereas thermodynamic properties governing the heat transfer, i.e. density, heat capacity, and thermal conductivity, were determined from independent measurements. Plateau storage modulus variation was monitored within the linear viscoelastic regime by dynamic mechanical analysis. Compounds with different initial peroxide concentrations were chosen for the study of the correspondence between model predictions and experimental observations, which proved to be very good. 1. Introduction There are several competing chemical reactions involved in the cross-linking of polymers. Each of these reactions can have a profound effect on the cure characteristics as well as on the properties of the final product. The balance between these reactions is determined by several factors, including the type of polymer, type and concentration of cross-linking agent, the cure temperature, and the presence of other compound additives.1 Acknowledging this, kinetics, transport phenomena, and viscoelasticity must unquestionably be accounted for both simultaneously and equally comprehensively, which is seldom the case due to the complexity of the process. Peroxide crosslinking was selected as a representative process, whereas it may be readily interchanged with another cross-linking type utilizing suitably modified reaction kinetics. Considering first the kinetic aspect of the cross-linking process, in addition to oversimplified phenomenological approaches, which describe only the apparent macrokinetics, much more suitable mechanistic approaches have their foundations in reaction chemistry and mechanisms. The first mechanistic approaches have originated from the analogy between peroxide initiated polymerization and cross-linking.2-6 A general shortcoming of this type of kinetic model is the inability to distinguish among several possible cross-linking reaction sites such as hydrogen atoms bonded to aliphatic carbon atom and double bonds, which are subjected to 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 while β-cleavage of oxy radical is incorporated in an overall peroxide efficiency. Many studies remedied these limitations by considering previously unaccounted for or at least not separately treated reactions for polyolefin macromolecules.7-16 The polyolefin backbone, though, 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 * To whom correspondence should be addressed. Tel.: +386 1 24 19 500. Fax: +386 1 24 19 530. E-mail: [email protected].

modeling framework for polyolefin macromolecules is based on the assumption of indifferent backbone reactivity,7,8,10-16 Laza´r et al.17 showed that even minority structures in polyethylene such as double bonds affect the cross-linking reactions’ kinetics, which questions the validity of the indifferent backbone reactivity supposition in particular cases. A study of the crosslinking reactions’ kinetics of more complex polymers thus has to be more rigorous, especially for copolymerized polymers, since individual reaction rates vary with respect to the reactive sites in the polymer backbone. Due to the latter observation, a wholesome kinetic model framework for the peroxide crosslinking of various polymers has not been developed to the best of our knowledge. Sato et al.18 proposed the kinetic model for the cross-linking of partially hydrogenated poly(acrylonitrileco-1,3-butadiene) (HPAB). However, this model also adopts the lumped representation of cross-linkable sites, rendering the determined apparent kinetic constants system-specific. Masaki et al.19 used an alternative approach for the kinetic study of the polybutadiene cross-linking in the presence and absence of vinyl acetate, applying the variable partial orders with respect to different components in an overall cross-linking reaction rate, yet the argument of system-specificity also applies in this case. Considering second the transport aspect of the cross-linking process, mass and heat transfers are predominantly determined by diffusion coefficients in the first case and thermodynamic properties, i.e., density, heat capacity, and thermal conductivity, in the second. Mutual diffusion coefficients can be considered in a general form of the free-volume theory.20 Vrentas and Vrentas21 proposed modifying this theory to address component diffusion in cross-linked polymers. Consequently, the influence of chemical cross-linking on the polymer free volume can be characterized by a single parameter, which is determined directly from volumetric data on both the cross-linked and un-crosslinked polymer. To successfully study the heat transfer in polymers, the thermodynamic properties should be predicted or determined. And whereas density is ordinarily relatively straightforward to describe, there are several models for the prediction of heat capacity, such as an assortment of equations of state with corresponding parameters and empirical correlations.22,23 Zhong et al.24,25 proposed a group contribution model

10.1021/ie1015415  2011 American Chemical Society Published on Web 01/05/2011

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

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Table 1. Formulation of Compounds weight (g)

Figure 1. Pilot-scale mold for the study of cross-linking process.

for the prediction of the thermal conductivity of polymer melts and a correlation for the prediction of the thermal conductivity of amorphous polymers, while the thermal conductivity of polymers is often considered linearly dependent on the heat capacity. Considering third the viscoelastic aspect of the cross-linking process, distinctively in terms of correlation between viscoelastic properties and the concentration of cross-links, the latter may be treated as linearly dependent on the plateau storage modulus and reciprocally on temperature.26,27 In the recent few existing simultaneous studies of the kinetics, transport phenomena, and viscoelasticity of polymers during cross-linking,28-30 either some or all the thermodynamic properties are considered constant or their dependence on temperature and the concentration of cross-links is oversimplified. Kinetics is described utilizing poorly predictive phenomenological models in terms of extrapolation, whereas diffusion is neglected on a regular basis. In this paper, a theoretical and an experimental study of kinetics and transport phenomena affecting the concentration of cross-links and other components of the polymer/cross-linking agent (hydrogenated poly(acrylonitrile-co-1,3-butadiene)/dicumyl peroxide) system compound slab is discussed, to one side of which isothermal boundary value temperature was applied. The concentration of cross-links was correlated with system viscoelasticity. Furthermore, a nonlinear system of balance equations was derived. The influence of different initial crosslinking agent concentrations on the cross-linking process was also quantitatively determined. 2. Experimental Section 2.1. Experimental Setup. For the purpose of a cross-linking study, a pilot-scale mold was constructed and is schematically presented in Figure 1. The sides and the lid of the mold were thermally insulated in order to minimize heat loss. As insulating material ceramic fibers (ZRCI Type RSMAT-3000, ZIRCAR Refractory Composites, Inc.) were utilized, with nominal thermal conductivity of 0.07 W m-1 K-1 at 315 °C and even lower at lower temperatures. Temperature at various positions was measured using K type thermocouples (TI). Pressure in the mold was measured by a bourdon tube gauge (PI), positioned on the removable lid. Compound sheets in the mold were indirectly heated by the electric heater (4 kW), which was incorporated in the bottom side. Heater power was controlled by autotuning

ingredient

compound 1

compound 2

compound 3

HPAB DCP

905 14.426

905 29.829

905 57.250

proportional/integral/derivative (PID) controller (TIC). Ten layers of 0.5 × 25 × 25 cm compound sheets were put into the mold, with thermocouples positioned among sheets along the mold vertical symmetry axis and other positions to examine eventually occurring minimal lateral heat fluxes. Approximately 5 mm thick compound sheets were prepared using the two-roll mill (Berstorff GmbH; roll diameter of 150 mm and roll length of 350 mm) with the differential roll speed ratio of 1:1.4 at 30 °C (15 min for each sheet, where cross-linking agent was added after 10 min). Heating of each compound was commenced instantaneously with bottom side set temperature of 200 °C for all compounds. The sequence with bottom side set temperature of 200 °C was utilized to obtain the temperatures at various positions and at different times during molding. It goes without saying that the interface temperature between the mold and the compound could not be increased to the set temperature instantly, yet the lag times were relatively short. Nevertheless, the initial increase of interface temperature toward the set temperature was subsequently acknowledged in the calculations. The mold lid was loose during experiments to grant constant pressure conditions. Every 15 min during an experiment a cylindrical cross-section of sheets was sampled close to the mold vertical symmetry axis, intended for the evaluation of the crosslinking degree of a compound. 2.2. Polymer and Cross-Linking Agent. HPAB used in this study was Zetpol 2020 L, provided by Zeon Chemicals (Tokyo, Japan), with a nominal density of 950 kg/m3, 36.2 wt % bound j n ) 7.72 × 104 g/mol acrylonitrile, and 91 mol % saturation. M j w ) 2.36 × 105 g/mol were determined by gel permeation and M j w determination was perj n and M chromatography (GPC). M formed using GPC instrument (Waters Alliance 2690 Separations Module, Waters Corp.) with a refractive index detector. Three columns (Styragel, Waters Corp.; 300 mm × 4.6 mm) were used in series. The HPAB solutions were prepared in tetrahydrofuran (THF), which was also used as a carrier solvent at a rate of 0.2 mL/min. The average molecular weights were calculated from the molecular weight versus retention time curve of polystyrene standards. The peroxide cross-linking agent used was dicumyl peroxide (DCP), provided by Sigma-Aldrich (St. Louis, MO, USA) with purity of 98 wt %, molecular mass of 270.4 g/mol, and density of 1560 kg/m3. Peroxide cross-linking agent was stored at low temperature. The formulation of compounds is presented in Table 1. 2.3. Determination of Thermodynamic Properties and Concentration of Cross-Links. To determine the dependence of thermodynamic properties on the concentration of cross-links, samples with known uniform distribution of concentration of cross-links within their bulk phase were prepared on a laboratory level to minimize temperature gradients within the bulk phase during the cross-linking process.31 Subsequent determination of thermodynamic properties ensued. 2.3.1. Procedure and Calculation To Determine Density of a Compound. Compound densities were determined between 30 and 180 °C with 10 °C increments by using pycnometers. Pycnometers with nominal volumes of approximately 25 mL were used. Pycnometers’ exact volumes were determined at 25 °C using distilled water as a reference liquid. For the purpose of measurements, 1.5-1.7 g of compounds and suitable amounts of silicone oil were prepared in the pycnometer and then settled

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Figure 3. Dependence of the density of HPAB polymer with different degrees of cure on temperature and the results obtained from application of eq 1 (s). The legend shows the concentration of cross-links. Figure 2. Temperature program for the determination of compounds’ heat capacities dependent both on temperature and the concentration of crosslinks.

in an oil bath at each temperature for 30 min before weighing. The pycnometer liquid was chosen on the basis of its properties, predominantly polarity, so that it simultaneously granted relatively good wetting of a compound and prevented too intensive swelling of the latter. Compound density was calculated from reference liquid density and the weight of a compound, pycnometer, pycnometer/reference liquid, and pycnometer/reference liquid/compound.32 2.3.2. Procedure and Calculation To Determine Heat Capacity of a Compound. Compound heat capacities were determined by differential scanning calorimetry (DSC) instrument (DSC 821e, Mettler-Toledo Inc.) in nitrogen atmosphere (50 mL/min). The samples were prepared by weighing 22-24 mg of a compound in the weighted 40 µL (nominal volume) aluminum crucibles without pin. Samples, sapphire standard, and empty crucible were subjected to heating/cooling programs and isothermal segments, presented in Figure 2. Indium and zinc standards were applied for the temperature calibration and for the determination of the instrument time constant. Individual segments, presented in Figure 2, served for the determination of the heat capacities of the samples. Whereas the initial dynamic segments (heating rate ranging from 10 to 50 K/min) were used for the desmearing process, the five isothermal segments during cooling (cooling rate of 10 K/min) at 0, 50, 100, 150, and 200 °C were used for the polynomial baseline correction. There were isothermal gaps between the initial dynamic segments in order to stabilize the instrument’s response between the individual segments. Compound heat capacity was calculated according to a previously developed procedure.33 2.3.3. Procedure and Calculation To Determine Thermal Conductivity of a Compound. For the prediction of thermal conductivity, dependent both on temperature and the concentration of cross-links, the glass transition temperature is the only property which has to be known.33 The dynamic segment between -150 and +200 °C (heating rate of 10 K/min) (Figure 2) was employed for the determination of the compounds’ phase transitions. 2.3.4. Procedure and Calculation To Determine Concentration of Cross-Links. The storage modulus was measured using a forced oscillation method in shear mode on dynamic mechanical analysis instrument (DMA/SDTA861e, MettlerToledo Inc.). The samples were prepared in the disk shape with thickness of 1.2-2.2 mm and diameter of 13.9-14.7 mm. A set of test measurements was performed for various samples

with thickness of 1-3 mm and diameter of 10-15 mm to confirm that the sample’s geometry had no effect on measured properties. A linearity range check was executed so that the measurements were performed within the linear viscoelastic regime; that is within 10 N force amplitude and 10 mm displacement amplitude (values of maximal force and displacement amplitudes being lower than linear viscoelastic regime boundaries). All samples were exposed to 100 °C for 15 min, then instantaneously quenched to -50 °C, and finally maintained at ambient temperature for a while prior to measurements. Isothermal experiments were performed at 25 °C with applied frequencies ranging between 10-3 and 103 Hz, utilizing five frequencies every decade on a logarithmic scale. All experiments were performed in a nitrogen atmosphere. The concentration of cross-links was calculated according to a previously developed procedure31,34 by extrapolating storage modulus, dependent both on temperature and angular frequency, to plateau storage modulus at plateau angular frequency, dependent only on temperature, and consequently calculating the concentration of cross-links from the corresponding plateau storage modulus. In continuation the results of dynamic mechanical analysis will only be presented in terms of the concentration of cross-links, while the determination of the latter is presented in our previous studies.31,34 3. Results and Discussion 3.1. Density. The density of HPAB polymer with different degrees of cure, prepared as described in section 2.3, determined by using pycnometers, described in section 2.3.1, is presented in Figure 3. It may be observed that differently cross-linked HPAB densities are relatively similar. Furthermore, the similarity also applies to their temperature behavior as their density gradually decreases with temperature. Whereas densities of neat elastic polymers such as HPAB decrease more rapidly after reaching a certain temperature (the fastest for poly(2-chloro1,3-butadiene)),33 which was estimated at approximately 150 °C, one may observe that, in the case of cross-linked HPAB, enhanced polymer swelling at increased temperatures despite the inert liquid used for the experiments, almost does not occur. On the other hand, due to the absence of rapid density decrease with temperature, the evaporation of volatile organic compounds, present in the neat polymer or originating from the polymer main chain degradation reactions, which occur more intensively at elevated temperatures, apparently occurs only in limited extent as well. The limited extent of volatile organic compound evaporation was confirmed by the lack of bubble evolution on the compounds’ surface during the experiments, contrary to what


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Table 2. Density at the Room Temperature and Pressure (T ) 25 °C, P ) 1.013 bar) and the Coefficient of Thermal Expansion for HPAB Polymer with Different Degrees of Cure concn of cross-links (mol/m3)

density (103 kg/m3)

coefficient of thermal expansion (10-4 K-1)

0 6.16 12.3 24.7

0.91 ( 0.01 0.91 ( 0.01 0.91 ( 0.01 0.92 ( 0.01

9.8 ( 0.2 9.7 ( 0.2 9.4 ( 0.2 9.4 ( 0.2

was observed for similar neat polymers.33 These bubbles would inflate a polymer matrix, which would result in lower apparent polymer density; nonetheless, the stable HPAB polymer matrix prevents this from happening. The latter, nevertheless, occurs with similar neat polymers, affecting the apparent polymer density, but in limited extent in a cross-linked state, as weighing of the samples prior to and after the DCS experiments indicates that the greatest weight loss occurs in the case of the lowest degree of cure, that is in the case of the cross-linked compound with the concentration of cross-links of 6.16 mol/m3 (0.3 wt %), whereas for the other compounds the weight loss ranges from 0.26 wt % (the cross-linked compound with the concentration of cross-links of 12.3 mol/m3) to 0.10 wt % (the crosslinked compound with the concentration of cross-links of 197 mol/m3). While the similar neat polymers are predominantly void free up to approximately 150 °C,33 and the difficulties with density determination due to the void formation are encountered at higher temperatures, this apparently does not occur within the examined range, that is up to 180 °C, in the case of differently cross-linked HPAB. Because the phenomena of swelling and pore formation were not observed, the temperature dependence of differently cross-linked HPAB density was described using an exponential relationship, presented by eq 1. HPAB, nevertheless, did therefore not need to undergo the pretreatment33 prior to the pilot-scale cross-linking study. R(ν) ) kRν + nR ) -

∂F(T, ν) 1 F(T, ν) ∂T

(

)

P

concn of cross-links (mol/m3)

density (103 kg/m3)

coefficient of thermal expansion (10-4 K-1)

49.3 98.7 197

0.93 ( 0.01 0.95 ( 0.01 0.99 ( 0.01

9.1 ( 0.2 8.6 ( 0.2 6.7 ( 0.2

and +200 °C with the applied heating rate of 10 K/min (according to the program presented in Figure 2) for the examined differently cross-linked HPAB are presented in Figure 4. Differently cross-linked HPAB exhibits two glass transitions, the first one between -88 and -98 °C, characteristic for the (un-)cross-linked isolated partially hydrogenated poly(1,3butadiene), and the second one between -21 and -26 °C, characteristic for the (un-)cross-linked bond partially hydrogenated butadiene/acrylonitrile fraction. Analogously to similar neat polymers,33 despite the fact that the temperatures of these transitions often occur in the relationships concerning the description of the temperature dependence of polymer thermal conductivity,24,25 they are not directly related to the polymer heat capacity, which was calculated in the temperature range between 20 and 200 °C for all differently cross-linked HPAB samples. The procedure described in section 2.3.2 was used. Heat capacity of differently cross-linked polymers, calculated utilizing DSC measurements, is presented in Figure 5. To use the temperature and the degree of cure-dependent heat capacity of differently cross-linked HPAB in the differential

(1)

In eq 1 R represents the coefficient of thermal expansion, F is density, T temperature, ν the concentration of cross-links, and P is constant pressure. The coefficients of thermal expansion for the examined differently cross-linked HPAB were calculated according to eq 1. The results presented in Table 2 are rather similar to the ones obtained from the literature. Nonetheless, for the purpose of the pilot-scale cross-linking study the results obtained in this work were used because differently cross-linked polymers cannot be considered as pure substances and therefore the material properties may vary substantially from the ones found in literature. The coefficients of thermal expansion (Table 2) are relatively low, so the dependency in Figure 3 may seem linear over the examined temperature range (30-200 °C): nonetheless, it is clearly not linear over a wider temperature range. The values of parameters describing the dependence of the parameter of thermal expansion on the concentration of cross-links, kR and nR in eq 1, are (-1.5 ( 0.1) × 10-6 K-1 m3 mol-1 and (9.75 ( 0.08) × 10-4 K-1. Linearity was analogously supposed for the dependence of density at room temperature and pressure (T ) 25 °C, P ) 1.013 bar) on the concentration of cross-links (analogy with the linear combination of densities at the initial and the terminal degree of cure, utilized by Tong and Yan),35 where the values of the concentration of cross-link dependency parameters, kF and nF, were (4.4 ( 0.1) × 10-1 kg/mol and (0.9058 ( 0.0009) × 103 kg/m3. 3.2. Heat Capacity. Heat capacity of differently cross-linked HPAB (section 2.3) was determined from the DSC measurements. DSC thermograms showing the segment between -150

Figure 4. DSC thermograms showing the segment between -150 and 200 °C (heating rate, 10 K/min).

Figure 5. Dependence of heat capacity of HPAB polymer with different degrees of cure on temperature.

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Table 3. Parameters of Equation 2 for HPAB Polymer with Different Degrees of Cure value for given concn of cross-links (mol/m3) c (10-2 J kg-1 K-3) d (101 J kg-1 K-2) e (102 J kg-1 K-1) coefficient of determination (/)

0 mol/m3

6.16 mol/m3

12.3 mol/m3

24.7 mol/m3

49.3 mol/m3

98.7 mol/m3

197 mol/m3

-0.01 ( 0.02 0.23 ( 0.01 17.5 ( 0.3 0.997

-0.04 ( 0.02 0.25 ( 0.02 17.1 ( 0.3 0.997

-0.07 ( 0.02 0.26 ( 0.02 16.7 ( 0.3 0.996

-0.09 ( 0.02 0.28 ( 0.02 16.1 ( 0.3 0.996

-0.04 ( 0.02 0.23 ( 0.02 16.3 ( 0.3 0.996

-0.04 ( 0.02 0.23 ( 0.02 15.3 ( 0.3 0.996

-0.02 ( 0.02 0.19 ( 0.01 14.2 ( 0.3 0.996

Table 4. Parameters of Equation 3 and Equation 4 i)1 -1

i)2

i)3

coefficient of determination (/)

-3

ci (J kg K ) -0.20 ( 0.02 1.2 ( 0.2 0.9 ( 0.3 1.9 ( 0.4 -12 ( 3 12 ( 5 di (10-4 J kg-1 K-2)

0.986 0.942

energy balance, the applicability of a second degree polynomial relationship that is the relationship proposed by Burger et al.36 (the latter proved to be the best choice for similar neat polymers)33 was examined and evaluated after the suitable rearrangement for the simultaneous description of the degree of cure dependence. cp(T, ν) ) c(ν)T2 + d(ν)T + e(ν)

(2)

where c, d, and e represent the polynomial coefficients, T the absolute temperature, and ν the concentration of cross-links. The parameters in eq 2 were determined for differently crosslinked HPAB. The values of the parameters are presented in Table 3. It may be observed that eq 2 renders quite good fit to the experimental data (coefficients of determination in Table 3 are all higher than 0.996). Analogously to similar neat polymers,33 the good agreement in the case of the application of eq 2 can be attributed to the fact that there are three variable parameters in this equation. Whereas the values of these three parameters do not imply any trend whatsoever for various similar neat polymers, since the comparability of the parameters c, d, and e is lessened on account of better fitting of eq 2 to the experimental data,33 there is, for example, a clear trend of the parameter e variation upon the degree of cure. The parameter decreases upon the degree of cure increase with the exception of the obvious error, due to the applied temperature dependence of heat capacity relationship, for HPAB with the concentration of cross-links of 49.3 mol/m3. The parameters c and d first increase upon the degree of cure increase but consequentially commence to decrease again. Taking into account these features, linearity was surmised for the dependence of e on the degree of cure (analogy with the linear combination of heat capacities at the initial and the terminal degree of cure, utilized by Tong and Yan),35 similarly to the dependence of the coefficient of thermal expansion on the degree of cure, where the values of the concentration of cross-links dependency parameters, ke and ne, were -1.5 ( 0.2 J m3 kg-1 K-1 mol-1 and (1.70 ( 0.02) × 103 J kg-1 K-1. Due to the special features of the dependence of the parameters c and d on the degree of cure, the quadratic functions of the natural logarithm of the degree of cure were applied for its description, while the parameters of the latter are presented in Table 4. c(ν) ) c1 × ln(ν/(mol/m3))2 + c2 ln(ν/(mol/m3)) + c3

(3) d(ν) ) d1 ln(ν/(mol/m3))2 + d2 ln(ν/(mol/m3)) + d3

(4) 3.3. Thermal Conductivity. For neat polymers similar to HPAB, the following equation25 proved to be the best choice

for the description of the temperature dependence of thermal conductivity,33 yet it was suitably rearranged for the simultaneous description of the degree of cure dependency description: λ(T, ν) ) k3(ν)Tg + k4(ν)T

(5)

where the parameters of eq 5, that is, k3, k4, and g, depend mainly on the temperature of the glass transition and the uncross-linked polymer properties at room temperature and pressure,25 while, for the purpose of the simultaneous description of the dependence of HPAB thermal conductivity on temperature and the degree of cure, the latter had to be included in the equation. The coefficients of eq 5, namely, k3 and k4, were for the individual differently cross-linked HPAB written as (following Zhong et al.)25 k3 ) K3

λ(T°) Tg(ν) (1.0221(T°/Tg(ν))g - 0.1959(T°/Tg(ν))) g

(6) k4 ) K4

λ(T°) Tg(ν)(1.0221(T°/Tg(ν))g - 0.1959(T°/Tg(ν)))

(7) where Tg and T° represent the glass transition temperature of an individual polymer (K) and the room temperature of 298.15 K, respectively, while ν is the concentration of crosslinks. For differently cross-linked HPAB the utilized value of the exponent, g, was the one presented in our previous work.33 This value is more or less similar to the one which should be applied generally, 0.1917.25 While the coefficients k3 and k4 are distinctive for each individual polymer, the proportionality coefficients K3 and K4 should be constant, regardless of the polymer, which means that the same should apply for differently cross-linked HPAB as well; nonetheless, for the purpose of the simultaneous description of the dependence of the HPAB thermal conductivity on temperature and the degree of cure, values calculated for neat HPAB were used33 instead of the general ones.25,33 Equations 5-7 should generally be applied for amorphous polymer;25 consequentially they were suitable to be used in our case for differently cross-linked HPAB as well as the latter might have been considered amorphous in the investigated range of temperatures (Figure 4). However, as in the case of certain neat polymers,33 the same applies to differently cross-linked HPAB, that a difficulty arises if the thermograms in Figure 4 are considered. Zhong et al.25 generally do not consider elastic polymers, which often exhibit two glass transitions (the same applies to differently cross-linked HPAB). Taking into account this latter observation, the overall Tg was considered as a linear combination of the individual segments’ Tg, that is, TgA and TgB, multiplied by the mole fractions of individual segments (xA and xB), similarly to certain neat polymers.33 Equation 8 may be related to Gibbs-DiMarzio theory37 and is also identical to the


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DP/HPAB(T, ν) ) (1 - φP(T, ν)) (1 2

2χ(T, ν) φP(T, ν))D0,P/HPAB ×

(

exp -

E(T, ν) exp(-(wPVˆP* + wHPABξ(ν) VˆHPAB*(ν))/ RT

)

[wPK1,P /γP(K2,P - Tg,P + T) + wHPABδ(T, ν) K1,HPAB(ν)/γHPAB(ν) × (K2,HPAB(ν) - Tg,HPAB(ν) + T)])

Figure 6. Correlation between glass transition temperatures and the concentration of cross-links. The legend shows the studied glass transition temperatures.

In eq 9 φP represents the volume fraction of the component in a compound, which was calculated from the concentration of the peroxide originating component at the specific temperature and the degree of cure (CP) (dependent on the cross-linking itself), and the molar volume of this component at the corresponding temperature (V˜P0) (determined following Hilal et al.).39,40 The initial φP at room temperature and pressure was 3.56 vol % in the case of DCP in compound 3 (Table 1). φP(T, ν) ) CP(T, ν) V˜P0(T)

empirical equation of Wood, applicability for copolymers.

38

found to have a wide

Tg(ν) ) kTgν + nTg ) xATgA(ν) + xBTgB(ν)

(8)

The relationship between the determined temperatures of the glass transition, TgA and TgB, the overall Tg, and the concentration of cross-links in eq 8, dependent on the HPAB degree of cure, is shown in Figure 6, whereas the overall Tg was calculated with the aid of eq 8 and the thermograms, presented in Figure 4. Utilizing the appropriate proportionality coefficient and ordinate intersection, kTg and nTg, a good correlation between the determined values and the supposed linear dependency is achieved. There is a clearly visible trend both for the overall glass transition temperature, Tg, and for the glass transition temperatures of the (un-)cross-linked isolated partially hydrogenated poly(1,3-butadiene) and the (un-)cross-linked bond partially hydrogenated butadiene/acrylonitrile fraction, TgA and TgB (Figure 6). The replacement of two glass transition temperatures (with the characteristic values of about -90 to -100 °C and -20 to -30 °C) with copolymer glass transition temperature Tg by means of eq 8 may seem unjustified (Tg differs from TgA and TgB more pronouncedly than these quantities change with ν); nonetheless, if either TgA or TgB were used in eqs 6 and 7, this would not change the later solution of the differential energy balance, as it may be seen in Figure 6 that Tg, TgA, and TgB change similarly with the concentration of crosslinks. 3.4. Diffusion Coefficients. In this section the determination of the temperature and the degree of cure-dependent diffusion coefficients of peroxide molecule originating components in a polymer is demonstrated, specifically taking the diffusion coefficient of DCP in HPAB as an example. Following Duda and Zielinski20 and Vrentas and Vrentas,21 the temperaturedependent diffusion coefficients in cross-linked polymers, DP/POLYMER, assume the following form, as very small initial solvent (or other components, which were in our compounds represented by the peroxide originating components) weight fractions in a polymer are approached (wP f 0), since the solvent should generally be present only in traces:

(9)

(10)

In eq 9 χ represents the polymer-peroxide originating component interaction parameter, which was calculated from the already mentioned quantities and the Hildebrand solubility parameters of the peroxide originating component (δP) and HPAB (δHPAB) at the specific temperature and the degree of cure.41,42 In the case of DCP, the initial χ at room temperature and pressure equals 2.47 (solubility parameters were determined following Pesetskii et al.43 and Rudin44). χ(T, ν) ) 0.35 +

V˜P0(T) (δP(T) - δHPAB(T, ν))2 RT

(11)

In the research of Msakni et al.,45 who studied the diffusion of DCP in a polymer melt using rheological measurements, and also in the only other existing one, in which Gupta and Sefton46 studied the crystallinity and the diffusion coefficient of DCP in low-density polyethylene with different thermal history, the values of D0,DCP/polymer and E (the first group) or the absolute values of diffusion coefficients (the second group) are determined; nevertheless, these values could not have been used, as it is the most appropriate to follow a proposed parameter estimation within one theoretical framework (in this case the free-volume theory), whereas both mentioned studies unorthodoxly disregarded this theory despite its fair approval in the field. The further three needed parameters were thus determined from the viscosity-temperature and the density-temperature data for the diffusing species (peroxide originating component). The relationship between viscosity (ηP) and temperature is derived from the Dullien expression,47 modified by Vrentas and Vrentas,48 while Tonge and Gilbert49 presented its applicability in the current form: ln(ηP(T)) ) ln

(

0.124 × 10-7V˜C2/3RT MPVˆP0(T)

)

- ln(D0,P/HNBR) +

VˆP* K1,P /γP(K2,P - Tg,P + T)

(12)

where V˜C stands for the molar volume of diffusing species at its critical temperature (for DCP determined with Joback-Lydersen group contribution method50 as 804 cm3/mol), MP for its molar mass (270.37 g/mol in the case of DCP (section 2.2)), and VˆP0

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the “universal” relationship, established by Tonge and Gilbert49 (eq 13) according to the available data.48 The initial E at room temperature and pressure is in the case of DCP ) 29.9 kJ/mol (solubility parameters and molar volumes were determined analogously as before). log(E(T, ν)/4.184) ) 0.8988 ln(log((δP(T) δHPAB(T, ν))2 V˜P0(T)/4.184)) + 2.8377

(13)

The value of the “size parameter”, ξ, was determined as follows: ξ(ν) )

ξL(ν) j /B j) 1 + ξL(ν)(1 - A

) V˜P0(0)/V˜HPAB*(ν)

j /B j) 1 + V˜P0(0)/V˜*HPAB(ν)(1 - A Figure 7. Correlation between DCP viscosity and temperature for the predicted viscosity values49,53 and the ones determined from eq 12.

for its specific volume (the reciprocal of density) at the corresponding temperature (determined following Hilal et al.).39,40 In the preceding expression the correct choice of the applied units is essential, as the constant 0.124 × 10-7 was modified from its original value of 0.124 × 10-16, presented by Vrentas and Vrentas,48 considering this issue. The correct units, which have to be used in this expression, are cP (mPa s) for ηP, cm3/ mol for V˜C, g/mol for MP, J mol-1 K-1 for R, and last cm3/g for the values of VˆP0 and VˆP* (the specific volume of peroxide originating component at 0 K, for DCP determined following Hilal et al.39,40 as 0.877 cm3/g). The factor of 109 is introduced, because, in the original derivation of the relationship, SI units were not used, from which the expression utilizing the mentioned coefficient (0.124 × 10-16) ensued, but the same unit use was being enforced as proposed by Vrentas and Vrentas.48 The applied factor is appropriate for use with the units mentioned above. Nonlinear regression for the fitting of eq 12 to the data, obtained from the viscosity-temperature and the density-temperature relationships introduces three adjustable quantities, i.e., parameters, namely, D0,P/HPAB (preexponential factor) in cm2/s, K1,P/γP (the peroxide originating component free-volume parameter) in cm3 g-1 K-1, and K2,P - Tg,P (the peroxide originating component free-volume parameter) in K. The fitting was performed with the aid of the Levenberg-Marquardt algorithm;51,52 whereas the values of the initial parameter approximations were taken from literature for butyl methacrylate/poly(butyl methacrylate) (BMA/PBMA),49 the correlation between eq 12 and the predicted viscosity values, together with the determined parameter values, are presented in Figure 7. The viscosity-temperature data were determined using a relationship based on the diffusing species density and its structure.49,53 The Souders index, I, was determined by group contributions (for DCP determined following Souders53 as 880), while the temperature dependence of density was for DCP determined following Hilal et al.39,40 The predicted viscosities were compared to the measured ones (viscosity was measured using capillary viscometer under isothermal conditions at 40-80 °C since above 80 °C DCP would react too fast) and the maximal approximation error was about 5%, so the prediction of viscosity at higher temperatures seems to be justified. Activation energy, E, may be determined from the peroxide originating component and HPAB solubility parameters (δP and δHPAB) at the specific temperature and the degree of cure utilizing

(14)

where V˜P0(0) stands for the peroxide originating component (diffusing species) molar volume at 0 K (for DCP determined j /A j for the peroxide following Hilal et al.39,40 as 237 cm3/mol), B originating component aspect ratio (for DCP determined after geometry optimization, described in our previous work,31,32,34,54 specifically as the DCP molecule dimension along the primary and the secondary axis of inertia ratio, as 2.36), and V˜HPAB* for the critical free volume per mole of jumping units required for a diffusion jump. The last parameter is in some cases available in databases;48 nonetheless, this does not apply for the peroxide originating components. Consequentially, the mentioned parameter had to be estimated according to Zielinski and Duda55 from Tg,HPAB (Figure 6 and eq 8; the parameter value is 38.6 cm3/mol for neat HPAB). The value of ξ for neat HPAB is therefore 1.35. In eq 9 VˆHPAB* represents the specific volume of HPAB at 0 K calculated following Hilal et al.39,40 VˆHPAB* ) 0.938 cm3/g in the case of neat HPAB. Duda and Zielinski20 and Vrentas and Vrentas21 have presented both theoretical and experimental evidence, which have shown that δ in eq 9 is merely indirectly dependent on temperature yet is related to the specific volumes of pure cross-linked (with the specific degree of cure) and uncross-linked polymer, that is, VˆHPAB0(T,ν) and VˆHPAB0(T,0). δ was calculated with the aid of Figure 3 and eq 1, whereas this parameter assumes the value of 1 at any temperature in the case of neat HPAB. δ(T, ν) )

VˆHPAB0(T, ν) 0

VˆHPAB (T, 0)

)

FHPAB(T, 0) FHPAB(T, ν)

(15)

The free-volume parameters K1,HPAB/γHPAB and K2,HPAB Tg,HPAB may be calculated using the following expressions:49 K2,HPAB(ν) - Tg,HPAB(ν) ) C2,HPAB(ν) - Tg,HPAB(ν)

(16) K1,HPAB(ν)/γHPAB(ν) ) VˆHPAB*(ν)/(ln(10) C1,HPAB(ν) C2,HPAB(ν)) (17) where C1,HPAB and C2,HPAB represent the degree of curedependent Williams-Landel-Ferry (WLF) parameters, while Tg,HPAB was taken as the reference temperature. C1,HPAB and C2,HPAB were for differently cross-linked HPAB calculated according to the procedure presented in our previous work,32 where the glass transition temperatures presented in Figure 4 were applied as the reference temperatures for differently cross-


∂(F(T(x, t), ν(x, t)) cP(T(x, t), ν(x, t)) T(x, t)) ) ∂t I J K L ∂ ∂T(x, t) λ(T(x, t), ν(x, t)) ( ∂x ∂x i)1 j)1 k)1 l)1

(

M

N

1565

O

∑∑∑∑∑∑∑

)

×

m)1 n)1 o)1

11

∑r

p,i, j,k,l,m,n,o(T(x, t), CE(x, t), CP(x, t))

∆HR,p,i, j,k,l,m,n,o(T(x, t))

p)1

(18) ∂(F(T(x, t), ν(x, t)) V(T(x, t), ν(x, t))) )0 ∂t ∂(V(T(x, t), ν(x, t)) CE(x, t)) Figure 8. Diffusion coefficient of DCP in HPAB at different temperatures (denoted at the right-hand side) and degrees of cure (denoted in the legend).

∂t I

J

K

L

M

(19)

) ( V(T(x, t), ν(x, t)) ×

N

O

11

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑r


i)1 j)1 k)1 l)1 m)1 n)1 o)1 p)1

(20) linked HPAB. K1,HPAB/γHPAB and K2,HPAB - Tg,HPAB were 4.50 × 10-4 cm3 g-1 K-1 and 123 K, respectively, in the case of neat HPAB. Knowing the values of all parameters in eq 9 or the relationships describing their dependence on temperature and the degree of cure, any DCP diffusion coefficient values could have been calculated, as well as for the other peroxide originating species, for which the calculations were performed analogously, whereas the suitable expression in a form of eq 9 was at the same time utilized in the pilot-scale crosslinking process model. The predicted diffusion coefficients of DCP in HPAB at various temperatures, degrees of cure, and peroxide weight fractions are presented in Figure 8. Upon comparison of the individual peroxide originating component diffusion coefficient in HPAB with the research of Hong,42 Tonge and Gilbert,49 Zielinski and Duda,55 Wang et al.,56 Verros and Malamataris,57 and Guo et al.,58 it was observed that the diffusion coefficients analogously to the ones in the mentioned studies first increase with the concentration of components and then commence to decrease again. Anomalies occur with certain components, among others also with DCP (for which the diffusion coefficients are presented in Figure 8), for which there is no increase range, so that the diffusion coefficients decrease from the initial component weight fraction of 0. The latter is connected with the nature of the diffusing component and polymer, because in the case of DCP, for example, the term in eq 9, in which DCP volume fraction occurs, affects the dependence of the diffusion coefficient on the weight fraction the most. On the other hand, the first exponential term is expectedly almost insensitive to the weight fraction, whereas the second exponential term decreases with the weight fraction as well, that is, due to the relative difference in DCP and HPAB nature, which is illustrated through δP and δHPAB, but also through other parameters. 3.5. Potential and Limitations of the Developed Model. The differential energy balance (DEB) and the differential mass balances for total mass (DMBM) and different components (DMBC) for the pilot-scale cross-linking process study in the mold seen in Figure 1 (the Cartesian coordinate system, in which heat flux is predominantly oriented parallel to the x-axis and the origin of the system is located in the center of the interface surface between the bottom side of the mold and a compound) are

(

∂(V(T(x, t), ν(x, t)) CP(x, t))

∂ D (T(x, t), ν(x, t)) × ) ∂t ∂x P/HNBR ∂(V(T(x, t), ν(x, t)) CP(x, t)) ( V(T(x, t), ν(x, t)) × ∂x

I

J

K

L

M

N

O

)

11

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑r


i)1 j)1 k)1 l)1 m)1 n)1 o)1 p)1

(21) with the accompanying initial and boundary conditions: T(x, 0) ) Ti(x) T(0, t) ) T0(t) ∂T(x, t) ∂x

|

(22)

)0 x)L

F(T(x, 0), ν(x, 0)) V(T(x, 0), ν(x, 0)) ) F(Ti(x), νi(x))0) V(Ti(x), νi(x))0) (23) CE(x, 0) ) CEi(x) ) CEi CP(x, 0) ) CPi(x) ) CPi ∂(V(T(x, t), ν(x, t)) CP(x, t)) ∂t I

J

K

L

M

N

|

(24)

) (V(T0(t), ν(0, t)) × x)0 O

11

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑r

p,i, j,k,l,m,n,o(T0(t), CE(0, t), CP(0, t))

i)1 j)1 k)1 l)1 m)1 n)1 o)1 p)1

∂CP(x, t) ∂x

|

)0 x)L

(25)

where most of the variables have already been explained in sections 3.1-3.4, whereas some are more thoroughly explained in the Supporting Information and our previous work.31-34,54 t represents time and x the dimension, i.e., the vertical distance from the heated bottom side of the mold, where the distance under the mold lid is denoted as L. rp,i, j,k,l,m,n,o is the p type reaction rate (Supporting Information, eqs SI1-SI11), dependent on temperature and the concentrations of different polymer (CE) and peroxide (CP) originating reaction sites with possible bond substituents Ri-Ro (Supporting Information, Table SI1), ∆HR,p,i, j,k,l,m,n,o is the temperature-dependent enthalpy change of the reaction rp,i, j,k,l,m,n,o (calculated similarly to the difference in enthalpy between the transition state and the energy minimum at the corresponding reaction coordinate, ∆Hq, described in our previous work),31,34,54 and V is the compound volume. Balances

1566


Figure 9. Experimentally determined temperatures (a-c) and model predictions using explicit finite difference method (d-f) for HPAB/DCP compounds (TS ) 200 °C): compounds 1 (a and d), 2 (b and e), and 3 (c and f) in Table 1.

analogous to the ones for CE and CP were established for peroxide originating species-peroxide originating species ([P-P]) and peroxide originating species-polymer ([P-E]) bonds, and ν, as well, taking into account Supporting Informationeqs SI9-SI11. As in our previous work,33 the lateral and the top mold sides were thermally insulated providing conditions for the study of the predominant heat transfer in one dimension. Ti(x), CEi(x), CPi(x), [P-P]i(x), [P-E]i(x), and νi(x) are the initial

distributions of temperature and the concentrations of various reaction sites in polymer and peroxide, P-P and P-E links, and cross-links within a compound, respectively, which were usually quite uniform, and may have been in the case of the mentioned concentrations represented by single values (0 for [P-P]i(x), [P-E]i(x), and νi(x)). Ti(x) and T0(t), the temperature at the compound/mold bottom side interface, were determined as in our previous work.33


1567

Figure 10. Experimentally determined concentrations of cross-links (a-c) and model predictions using explicit finite difference method (d-f) for HPAB/ DCP compounds (TS ) 200 °C): compounds 1 (a and d), 2 (b and e), and 3 (c and f) in Table 1.

As in our previous work,33 there is a vital distinction between the set temperature (TS) and the temperature at the compound/ mold bottom side interface. The first is merely nominal, while the second was used for all calculations. The relationships for the description of the temperature and the degree of cure-dependent thermodynamic properties and diffusion coefficients (eqs 1, 2, 5, and 9) were applied in the differential energy balance (eq 18) and the differential mass balances for total

mass (eq 19) and different components (eqs 20 and 21); the balances, however, could not have been solved quasi-exactly33 due to complexity of the system but only with the implementation of the explicit finite difference method. The explicit finite difference method was validated with implicit and semiimplicit finite difference (element) method to ensure simulation accuracy. DEB, DMBM, and DMBC represented by eqs 18-21 were rewritten by application of the mentioned method. Appropriate

1568


Figure 11. Methodology and the flow chart of the numerical simulation for integrated analysis.

coordinate and time intervals, ∆x and ∆t, were chosen so as to grant the numerical stability of the finite difference method solution. The measured temperatures, the experimentally determined concentration of cross-links (section 2.3.4), and the explicit finite difference method solution results for HPAB/DCP compounds (compounds 1-3 in Table 1) are presented in Figure 9 (temperature) and Figure 10 (concentration of cross-links). The methodology and the flow chart of the numerical simulation for integrated analysis is outlined in Figure 11 analogously to Jia et al.59 The agreement of the explicit finite difference method solution of DEB, DMBM, and DMBC with the experimental values was poorer for the concentration of cross-links in comparison to temperature (Figures 9 and 10), as in the first case a noticeable error occurs during the cross-linking process compound sampling. By comparison of the overall variance (statistical dispersion, calculated as the square of the summated differences between the model predictions and the experimental values) it was concluded, that the agreement was good regardless of the compound in question. In Figures 9 and 10 the denser mesh (constituted of more points) in the case of the results obtained by the explicit finite difference method model (Figures 9d-f and 10d-f) was only applied to illustrate the fact that the model predictions are continuously distributed. Nevertheless, the model behavior analogous to the one of the experimental values may be observed, especially close to the heated bottom side of the mold, where the model excellently corresponds the enforced temperature oscillation on the interface between the bottom side of the mold and a compound, which is a consequence of the regulation of the actual plate temperature with

regard to the set one. The overall results of simulation can be confirmed in terms of temperature and the concentration of cross-links (Figure 9 and 10); nonetheless, to confirm the results of simulation in terms of the concentrations of other species, CE and CP, these should be determined experimentally. It is shown in Figure 10 that the concentration of cross-links varied at different distances from the heated plate. The measurement of the concentration of cross-links required a relatively large sample (disk shape with thickness of 1.2-2.2 mm and diameter of 13.9-14.7 mm), while other measurements such as density or heat capacity by DSC required only rather a small amount of sample (1.5-1.7 g and 22-24 g, respectively). The sample used for DSC could therefore have a much more homogeneous concentration of cross-links, different from the average value, measured by DMA. Therefore, DMA samples were gradually decreased to the same size as was used for the density measurements (disk shape with thickness of 1 mm and diameter of 1 mm) to see if there is any correlation between the properties such as the concentration of cross-links and sample size. It was observed that regardless of sample size the determined concentration of cross-links was the same, so that the sample size could not have been the reason for the minor quantitative disagreement between simulation and experiments mentioned before. The standard dynamic mechanical analysis of the mentioned samples with the thickness of 1.2-2.2 mm and diameter of 13.9-14.7 mm was performed as described in section 2.3.4, and these results were used for the calculation of the concentration of cross-links, while smaller samples were only used to check the results because for too small samples (usually with the thickness of about 1 mm and diameter of about 1 mm) observational error was often not negligible. The fraction of heat loss in comparison to the complete amount of heat transferred to bulk compound phase (η) was calculated according to our prior work.33 The overall heat loss expectedly increased with the increase of the initial peroxide concentration in a compound, as the increased heat loss fraction occurred due to the greatest extent of overall evolved heat, because of the reactions pertinent to the cross-linking process. Nevertheless, η even for the experiments for the greatest initial peroxide concentration compound (compound 3 in Table 1) never surpassed 0.71%, the latter being its highest value. Negligible heat loss premise may thus be justified, seeing that it only scarcely affects the measured temperatures. 4. Conclusions The cross-linking process of a representative polymer/crosslinking agent compound (HPAB/DCP compound) was studied on a pilot-scale mold. Temperature and the degree of curedependent density, heat capacity, and thermal conductivity were determined separately using pycnometers and differential scanning calorimetry, correspondingly. Diffusion coefficients were calculated according to the free-volume theory utilizing the determined densities as well. Previously proposed expressions for the description of temperature-dependent behavior were extended and applied to describe the degree of cure-dependent behavior simultaneously as well. The exponential dependency of density on temperature was valid throughout the examined range because enhanced compound swelling, evaporation of volatile organic compounds, or even both phenomena did not occur. Explicit finite difference method was applied to the differential energy balance (DEB) and the differential mass balances for total mass (DMBM) and different components (DMBC), in which the relationships for description of temper-


ature and the degree of cure-dependent thermodynamic properties and diffusion coefficients were applied. The general observation was that the goodness of fit did not change much by applying various initial concentrations of the cross-linking agent in a compound. The predicted thermodynamic properties and diffusion coefficients and the values obtained from literature for similar systems were comparable, as it might have been observed for the determined dependence parameters as well. Although the explicit finite difference method rendered poorer agreement with the experimental results for the concentration of cross-links in comparison to temperature, it overall coincided quite well with the experimentally measured values. The most important feature of the developed model is the simultaneous treatment of kinetics and transport phenomena in relation to viscoelasticity at both high theoretical as well as numerical level, whereas ordinarily interpretation would have been limited to a single aspect and even then sometimes oversimplified as far as cross-linking kinetics are concerned. The developed model, however, may be readily applied for a plethora of compound formulations varying in polymer, cross-linking agent, etc., process conditions, such as temperature and pressure, and even chemically differing compounds upon modifying the kinetic terms. The obtained knowledge may contribute to the further recognition of the mechanisms and the development of cross-linking technology of HPAB and polymers in general. Supporting Information Available: Overview of the kinetic aspect of the cross-linking process presenting the definition of symbols in the kinetic terms, the reaction scheme, and the associated rate constants considered in the kinetic terms and the kinetic terms themselves. This information is available free of charge via the Internet at http://pubs.acs.org/. Literature Cited (1) Dluzneski, P. R. Peroxide Vulcanization of Elastomers. Rubber Chem. Technol. 2001, 74, 451. (2) Hergenrother, W. L. Characterization of Networks from the Peroxide Cure of Polybutadiene. II. Kinetics and Sol-Gel. J. Polym. Sci. 1973, 11, 1721. (3) Gancarz, I.; Łaskawski, W. Peroxide-Initiated Crosslinking of Maleic Anhydride-Modified Low-Molecular-Weight Polybutadiene. I. Mechanism and Kinetics of the Reaction. J. Polym. Sci. 1979, 17, 683. (4) Sen, A. K.; Mukherjee, B.; Bhattacharyya, A. S.; De, P. P.; Bhowmick, A. K. Kinetics of Silane Grafting and Moisture Crosslinking of Polyethylene and Ethylene Propylene Rubber. J. Appl. Polym. Sci. 1992, 44, 1153. (5) Mateo, J. L.; Calvo, M.; Bosch, P. Photoinitiated Polymerization of Methacrylic Monomers in a Polybutadiene Matrix (PB): Kinetic, Mechanistic, and Structural Aspects. J. Polym. Sci., Part A: Polym. Chem. 2001, 39, 2444. (6) Ghosh, P.; Chattopadhyay, B.; Sen, A. K. Modification of Low Density Polyethylene (LDPE) by Graft Copolymerization with Some Acrylic Monomers. Polymer 1998, 39, 193. (7) Hamielec, A. E.; Gloor, P. E.; Zhu, S. Kinetics of Free Radical Modification of Polyolefins in ExtruderssChain Scission, Crosslinking and Grafting. Can. J. Chem. Eng. 1991, 69, 611. (8) Gloor, P. E.; Tang, Y.; Kostanska, A. E.; Hamielec, A. E. Chemical Modification of Polyolefins by Free Radical Mechanisms: A Modeling and Experimental Study of Simultaneous Random Scission, Branching and Crosslinking. Polymer 1994, 35, 1012. (9) Zhou, W.; Zhu, S. ESR Study of Peroxide-Induced Cross-Linking of High Density Polyethylene. Macromolecules 1998, 31, 4335. (10) Pedernera, M. N.; Sarmoria, C.; Valle´s, E. M.; Brandolin, A. An Improved Kinetic Model for the Peroxide Initiated Modification of Polyethylene. Polym. Eng. Sci. 1999, 39, 2085. (11) Asteasuain, M.; Sarmoria, C.; Brandolin, A. Peroxide Modification of Polyethylene. Prediction of Molecular Weight Distributions by Probability Generating Functions. Polymer 2002, 43, 2363.

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ReceiVed for reView July 19, 2010 ReVised manuscript receiVed November 2, 2010 Accepted November 16, 2010 IE1015415

Cross-Linking of Polymers: Kinetics and Transport Phenomena

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