Comparison of Isothermal with Nonisothermal Kinetics for Ethylene

Article pubs.acs.org/IECR

Comparison of Isothermal with Nonisothermal Kinetics for Ethylene-Vinyl Acetate Cross-Linking Reaction in the Solid State Can-pei Liu,*,†,‡ Wei-fei Song,† Qing-xin Lu,† and Ming-feng Chen† †

College of Materials Science and Engineering, Fujian Key Laboratory of Polymer Materials, Fujian Normal University, Fuzhou 350007, China ‡ Fujian Zhiming Shoes Limited Company, Quanzhou 366200, China S Supporting Information *

ABSTRACT: Cross-linking reaction kinetics of ethylene-vinyl acetate (EVA) copolymer initiated with dicumyl peroxide is studied by differential scanning calorimetry (DSC) and a cross-linking rheometer. As the degree of cross-linking (α) is less than 0.30, EVA can flow relatively freely because of insufficient cross-linking. As α ranges from 0.31 to 1, there is more of the crosslinking reaction, and the polymer becomes fully cross-linked in the later stage of the reaction so that three-dimensional crosslinked nets have been constructed. New kinetics approaches are proposed on the basis of the cross-linking parameters measured with the cross-linking rheometer incorporated into Hill and Avrami mathematical equations. As the cross-linking rheometer is popular in manufacturing the cross-linked EVA, the approaches are simpler and more convenient than the DSC method for fast studying kinetics of the cross-linking reaction of polymer in solid or melting state. The total cross-linking reaction rate changes with α, and the apparent activation energy is 91.2−175.0 kJ/mol.

1. INTRODUCTION Ethylene-vinyl acetate copolymer (EVA) with a higher content of vinyl acetate (VA) has better elasticity, flexibility, and adhesive bonding. Consequently, its cross-linked products1,2 have widely been used in the fields of photovoltaic cell sealant, adhesive bonding, cables and tubes; and its cross-linked and foamed materials also have been vastly used in the fields of vibration attenuation and shockproof, sound insulation, heat preservation, and the shoe material industry,3−5 etc. The cross-linking process of EVA initiated with dicumyl peroxide (DCP) is very popular in industrial production under isothermal conditions. The cross-linking reaction of EVA follows a well-known free radical mechanism.1,6−9 According to Bianchi6,10,11 and to our knowledge5 of manufacturing the crosslinked EVA, the properties of the cross-linked EVA materials mainly depend on the cross-linking process. Hence it is necessary to study its isothermal kinetics for guiding the cross-linking production process. In general, it is difficult to isolate the elementary reaction from the complicated cross-linking reaction of a polymer. The crosslinking reaction kinetics of a polymer in a solid or melting state originates from standard basic experimental techniques such as differential scanning calorimetry (DSC), differential thermal analysis, and thermogravimetric analysis, as well as more sophisticated methods.2,9−13 Because the solid state cross-linking reaction has a partly inhomogeneous nature and because the reaction follows complex mechanisms involving multiple series and parallel steps with different activation energies, the isothermal and nonisothermal kinetics only provide a total measure of the cross-linking reaction rate or apparent parameters or extent of a cross-linking process that usually involves several steps with different activation energies. Bianchi et al.10 provided a nonisothermal kinetic analysis of EVA initiated by DCP in the framework of a multistep solid-state process and determined the © XXXX American Chemical Society

activation energy and the kinetic mechanisms on the crosslinking conversion. Although there were many nonisothermal kinetic studies for EVA cross-linking reaction by DSC, they involved in fitting of experimental data to the assumed reaction mathematical models and those results in literatures9−13 had large errors in industrial production. For example, the result values of Arrhenius parameters such as activation energy, preexponential factor and the rate constant of the cross-linking reaction under isothermal conditions were often inconsistent with those ones under nonisothermal ones.9−13 The differences of Arrhenius parameters between isothermal and nonisothermal experiments arose from two main sources. One major source was a result of nonisothermal kinetic analysis that involved into fitting of experimental data to the assumed reaction model. Another source arose from the fact that the temperature sensitivity of the reaction rate depended on the extent of cross-linking reaction or the degree of cross-linking to polymer product.14,15 In industrial production, a cross-linking rheometer, which is an oscillatory disc rheometer and popularly used to measure torquetime vulcanization curves in the rubber industry, is prevalently used to measure the EVA cross-linking process parameters such as torque, temperature, and time of its cross-linking reaction.1,4−6 In this paper, new approaches for kinetic analysis are proposed on the basis of Hill and Avrami mathematical equations incorporated with the cross-linking parameters measured with the cross-linking rheometer. Furthermore, the isothermal and nonisothermal cross-linking process of EVA is analyzed via our approach and DSC method, respectively. In comparison of the DSC method with a measurement of the cross-linking Received: March 20, 2014 Revised: May 21, 2014 Accepted: May 26, 2014

A

dx.doi.org/10.1021/ie5011788 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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parameters with the cross-linking rheometer, this study represents the kinetics results for instructing its cross-linking process of EVA in industrial production.

k = A exp(Ea /(RT ))

where t represents the cross-linking time, α is the degree of crosslinking1,10,11 that presents the extent from the linear molecular chains of EVA to a cross-linked polymer net structure in the cross-linking reaction, T is the temperature, k is the Arrhenius temperature-dependent rate constant, and f(α) is a function that represents the reaction model. n, A, Ea, and R are the order of the cross-linking reaction, the pre-exponential factor of the crosslinking reaction rate constant, the apparent activation energy, and the universal gas constant, respectively. EVA cross-linking reaction initiated with DCP can be tested in a DSC measurement. The exothermic process peaks are magnified in Figure 1 from the full DSC curves of Supporting Information, Figure S1. The temperature range of 413−495 K in the exothermic process is used to calculate α,

2. EXPERIMENTAL SECTION 2.1. Materials. Three kinds of EVA (7350M, 7360M, and 7470M) were supplied by Taiwan Plastics Corporation (Taiwan). 7350M has 18% VA and 2.50 g/10 min of melt flowing index (MFI); 7360M has 21% VA and 2.0 g/10 min of MFI; and 7470M has 26% VA and 4.0 g/10 min of MFI. DCP with 99% purity was purchased from Sinopharm Chemical Reagent Co (Shanghai, China). 2.2. Preparation of the Sample. A 100 g portion of 7350M, 7360M, 7470M, the blend of 7470M/7350M (62.5/37.5, VA = 23%) (Blend A) or 7470M/7350M (87.5/12.5, VA = 25%) (Blend B) was respectively mixed with DCP by an amount (ωDCP) of 0.60, 0.80, 1.0, 1.2, 1.4, 1.6, or 1.8 g (or in phr) for 10 min in the HAAKE rheometer mixer (HAAKE Mixer 600, Thermofisher Co., USA) which was set at 353 K. 2.3. Measurement. 2.3.1. DSC Method. The thermal crosslinking reaction of the sample was measured by a DSC thermogravimetric analyzer (DSC822e, Mettler-Toledo, Switzerland) at a heating rate (β) of 5, 10, 15, or 20 K/min from room temperature (T0) to 520 K. Nitrogen was used as a protection gas with a flow rate of 20 mL/min. Approximately 5 mg of each sample was used. 2.3.2. Cross-Linking Rheometer Method. A cross-linking rheometer (UR-2030SD, U-can Dynatex Inc., Taiwan) as typically used in the rubber industry was employed for the measurement of the cross-linking torque (M) versus the crosslinking reaction time (t) under isothermal conditions. The crosslinking rheometer was characterized by high precision and good sensitivity, and the method was standardized under China national standard of GB/T16584-1996 (equivalent to ISO 65021999). The cross-linking process caused the torque to increase in a characteristic way. The lower plate of the die of the crosslinking rhemeter was stationary, while the upper oscillated at 0.5° at a frequency of 1.67 Hz. To ensure good contact between the sample and the plate, the sample was punched out as a special shape of the die; 3.0 ± 0.1 g of each sample was used. The crosslinking curve of M−t was measured and recorded continuously with the cross-linking rheometer at a temperature (T) under isothermal conditions.

⎛ α=⎜ ⎝

(1)

f (α) = (1 − α)n

(2)

∫T

T

s

⎞ ⎛ ∂H dT ⎟ / ⎜ ⎠ ⎝ ∂T

∫T

TE

s

⎞ ∂H dT ⎟ ⎠ ∂T

(4)

where Ts and TE are the onset and end point temperatures of the cross-linking reaction, respectively. In other word, Ts and TE are the onset and the end point temperatures of thermal decomposition of DCP, respectively, and T0 < Ts ≤ T ≤ TE, T0 is a starting heating temperature (e.g., room temperature), as T = Ts, α = 0, and T = TE, α = 1. Therefore, α−T curves are converted from Figure 1 according to eq 4 and they are shown in Figure 2. Equation 1 and 2 are combined with 3, and then the logarithm is taken, E ⎛ dα ⎞ ln⎜ ⎟ = ln A − a + n ln(1 − α) ⎝ dt ⎠ RT

(5)

Nonisothermal DSC measurement is made frequently with a heating rate of a certain β value from T0 to TE. Because T = T0 + βt, thus dT = β dt. After the variables α, T and t, are separated, the differential of eq 5 can be expressed as eq 6, Ea ⎛ dα ⎞ ⎟ = ln A − ln⎜β + n ln(1 − α) ⎝ dT ⎠ RT

(6)

With different heating rate (β) values, as α is given a specified value, for example, α = 0.10, 0.20, 0.30, ···, 0.9, the plots of ln(β dα/dT) versus 1/T based on eq 6 in a DSC method under nonisothermal conditions are shown in Supporting Information, Figure S2. From the slopes and intercepts of the plots, the apparent activation energy and pre-exponential factor can be calculated. They are plotted as a function of the degree of cross-linking, which are shown in Figure 3a and Supporting Information, Figure S3a, respectively. 3.2. Avrami Kinetic Analysis. It is taken into account that the cross-linking reaction of EVA initiated with DCP is similar to that of crystallization process of the polymer, and the increment of the degree of cross-linking is similar to that of the crystallinity, too.10,11 Thus, kA and nA are considered as the Avrami constant and index of the cross-linking reaction,

3. RESULTS AND DISCUSSION 3.1. DSC Kinetic Analysis. The full DSC curves of the samples are shown in Supporting Information, Figure S1. The endothermic process peaks at 310−315 K and in 337−375 K are attributed to the melting point of DCP and the softening point of EVA, respectively, while the exothermic process peak at 413− 495 K is attributed to thermal decomposition of DCP to generate free radicals to initiate EVA to cross-link. Information concerning the cross-linking reaction of EVA can be gained from the exothermic process peak in the temperature range of 413−495 K. In general, the cross-linking reaction kinetic equations of a polymer are currently accepted as dα = k f (α ) dt

(3)

ln[− ln(1 − α)] = ln kA + nA ln t

(7)

According to Figure 2, and then based on eq 7, a series of plots of ln[−ln(1−α)] versus ln t can be obtained. The slopes (nA) and intercepts (ln kA) of those plots are obtained in Supporting Information, Table S1−S6. As α is in the range of 0−0.30, nA is in the range of 4−5. It suggests that the cross-linking reaction in the α range of 0−0.30 is not very sufficient when initiated with DCP so that EVA can flow relatively freely in three-dimensions.

and the Arrhenius equation, B



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Figure 1. DSC curves of EVA cross-linking reaction under nonisothermal conditions: (a) 7350M; (b) 7360M; (c) 7470M.

Figure 2. Cross-linking degree (α) as a function of temperature in DSC method: (a) 7350M; (b) 7360M; (c) 7470M.

of the cross-linking reaction. As α is 0, 0.10, 0.20, 0.50, or 0.90, its corresponding time is very important for controlling its cross-linking reaction in industrial production. They are generally considered as the induction time (ti), scorching time (t10 or t20), half cross-linking time (t50), and technological crosslinking time (t90). According to eq 7, some of the cross-linking times at a certain α can be expressed as the time (tαA) in the following:

While α ranges from 0.31 to 1, nA is in the range of 0.85−1, namely, nA ≈ 1. It indicates that EVA has difficulty flowing and the cross-linking reaction gets more and more sufficient. EVA then becomes fully cross-linked in the reaction later stage so that three-dimensional cross-linked nets have been constructed. At a certain cross-linking temperature, the increment of the degree of cross-linking is accompanied by the cross-linking time C



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3.3. Rheometer Kinetic Analysis. 3.3.1. Cross-Linking Behavior of EVA. Figure 4 shows that the cross-linking torque values of EVA at 443−458 K change with the cross-linking reaction time. ωDCP is 1.0 phr, as the cross-linking temperature is over 453 K, the M−t curves (Figure 4a, Figure 4b and Figure 4c) of 7350M, 7360M, and 7470M decline in the later cross-linking stage. It implies that there are overcure situations in the later cross-linking stages. While the cross-linking temperature is 443 K and the cross-linking reaction has been performed for 600 s, M values still increase. It means that there are undercure situations. At 448 K, the cross-linking behavior is very good. Figure 4d and Figure 4e reveal that M values increase with the amount of DCP. This is the reason that activated radical centers increase with the amount of DCP so that the cross-linking reaction goes smoothly. Figure 4f shows that M values increase with VA. M values reveal the viscoelasticity and flowing abilities of EVA. The lower the M value is, the better the flowing ability of EVA is. In comparison, an M value of 7350M is the lowest. In Figure 4f, however, M values are synergistically increased because of a higher elastic effect of EVA with a higher VA. With the change of VA, ωDCP or T, some of the cross-linking times such as t10, t50, and t90 are actually measured by using the cross-linking rheometer and t10A, t50A, t90A are calculated by using eq 8. All of them are shown in Supporting Information, Figure S4. It can be seen that the measured cross-linking times are consistent with those calculated results based on eq 8. When the cross-linking temperature is elevated, t90 is all shortened. With increasing VA, t90 is shortened first and then it increases again. As ωDCP ranges from 0.90 to 1.3 phr, t90 is at a horizontal level. So, it is considered that the conditions of both 1.0 phr of ωDCP and 448 K are suitable for cross-linking EVA. Hence, when conditions are below these values, we should take a key consideration of kinetics of the cross-linking reaction as ωDCP is 1.0 phr. 3.3.2. Cross-Linking Reaction Kinetics of EVA. During measuring the cross-linking torque with the cross-linking rheometer, the maximum torque (MH) and the minimum torque (ML) values can be read out from the cross-linking curve of M−t, shown in Figure 4. At a constant temperature, α is in direct proportion to the increment of torque value in the cross-linking reaction. So, α can also be calculated using the following equation,

α=

(9)

where ML, M, and MH are the minimum torque, the torque at t, and the maximum torque, respectively. According to eq 9, the M−t curve can be changed into an α−t curve in Supporting Information, Figure S5. And then based on eq 5, ln(dα/dt) and ln(1 − α) can be calculated. The cross-linking torque values depend on the cross-linking reaction time or temperature during cross-linking, which are shown in Figure 4. The torque value change can be expressed5 as

Figure 3. Apparent activation energy as a function of the cross-linking degree: (a) DSC; (b and c) rheometer. 1/ nA ⎛1 1 ⎞⎟ tα A = ⎜ ln 1 − α⎠ ⎝ kA

M − ML MH − ML

(8)

where 0 < α < 1, and tαA are shown in Supporting Information, Figure S4. With the change of VA, ωDCP, or T, t10A, t50A, and t90A are calculated by using eq 8, and the cross-linking times such as t10, t50, and t90 are actually measured by using the cross-linking rheometer. All of them are shown in Supporting Information, Figure S4. It can be seen that the measured cross-linking times are consistent with those calculated results based on eq 8. This means that the Avrami equation can be used to calculate EVA cross-linking kinetics.

M = MH[1 − e−k(t − ti)]

(10)

where the induction time (ti) is defined as the interval from the beginning of measuring torque to that time when the torque is at the ML point. During measuring the cross-linking curves, ti is far smaller than t. It can clearly be seen that the cross-linking curves (Figure 4) at different cross-linking temperatures have certain spaces. This suggests that the cross-linking reaction rate constant D



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Figure 4. Cross-linking curves of M−t for EVA different formulations.

depends on temperature. The substitution of k in eq 10 with eq 3 gives ⎡ 1 ⎛ M − M ⎞⎤ E ln k = ln⎢ ln⎜ H ⎟⎥ = ln Ak − ak ⎢⎣ ti − t ⎝ MH ⎠⎥⎦ RT

On the torque-up phase, in the range of [ML, MH], for example, from one cross-linking degree value α1 to α2, the corresponding torque range is [Mα1, Mα2], 0 ≤ α1< α2 ≤ 1 and ML ≤ Mα1 < Mα2 ≤ MH. As α1 is quantitatively close to α2, the M−t curve is a little straight line. So,

(11) E



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Figure 5. Cross-linking reaction rate constant as a function of 1/T.

dM = kαM dt

and from eq 13, the cross-linking curves (Figure 4) and Arrhenius equation of eq 3, a group of Arrhenius parameters of ln k, Eak, and Ak or another group of ln kα, Eaα, and Aα can also be calculated, which are shown in Figure 5, Figure 3b and Supporting Information, Figure S3b, and the values of the linearly dependent coefficient square (r2) are listed in Supporting Information, Figure S7, respectively. As α value ranges from 0.10 to 0.90, a linear fitting of ln(dα/dt) versus ln(1 − α), the slope (n) is in 0.99−1.05, namely, n ≈ 1, shown in Supporting Information, Figure S6. Thus, the cross-linking reaction is considered as one order reaction,5 which is consistent with a p value in the range of 0.90−1.11

(12)

where kα is the cross-linking reaction rate constant in a certain cross-linking degree at a constant temperature. For the simplified calculation, Δαi = α10i − α10(i‑1) = 0.10, i = 1, 2, 3, ···, 10, so k α10i =

ln M α10i − ln M α10(i‐ 1) t10i − t10(i − 1)

(13) F



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(shown in eq S2 and Table S8) in Figure S7 in an analysis way of Avrami−Ozawa10,11 method. In a given α value, it seems that Ea can be calculated because Ea/R is the slope of the plot of ln(dα/ dt) versus 1/T based on eq 5. Herein, however, a linear fitting plot of ln(dα/dt) versus ln(1 − α) based on eq 5 gives an intercept that is ln k. Furthermore, the plot (Figure S8) of ln k versus 1/T gives its slope which is Ea/R and its intercept is the value of lnA. Thus, Ea and lnA can be calculated, the values of which are shown in Table 1. Table 1. Activation Energy and Pre-exponential Factor Based on ln k versus 1/T EVA

Ea (kJ·mol−1)

ln A

r2

7350M 7360M 7470M

119.5 ± 0.5 129.3 ± 2.4 135.5 ± 9.3

27.53 ± 2.82 30.37 ± 0.65 31.94 ± 2.47

0.9770 0.9989 0.9861

For comparison of the calculated results according to eq 11 and eq 13 with those calculated based on eq 5, as α is a certain value (α = 0.10, 0.20, 0.30, ···or 0.9), a plot of ln(dα/dt) versus 1/T based on eq 5 can be seen in Figure 6. Its slope is Eaα/R. Thus, the apparent activation energy of Eaα is calculated, shown in Figure 3c, and its intercept is lnA in Supporting Information, Figure S3c. On the basis of Figure 3, the average apparent activation energy (E̅ a, E̅ ak, or E̅ aα) can be calculated according to eq 14, 9or10

9or10

Ea̅ j = ( ∑ Δαi·Eaji)/ ∑ Δαi i=1

i=1

(14)

in which for E̅ a, j = none, i = 1, 2, 3, ..., 9; for E̅ ak, j = k, i = 1, 2, 3, ..., 9; for E̅aα, j = α, i = 1, 2, 3, ..., 10. As ωDCP = 1.0 phr, for 7350M, E̅ a based in Figure 3a, E̅ak and E̅aα based in Figure 3b, and E̅aα based in Figure 3c are 136.3, 91.2, 106.8, and 106.2 kJ/mol; for 7360M, E̅ a based in Figure 3a, E̅ ak and E̅aα based in Figure 3b, and E̅ aα based in Figure 3c are 175.0, 114.6, 125.8, and 131.4 kJ/mol; and for 7470M, E̅ a based in Figure 3a, E̅ ak and E̅aα based in Figure 3b, and E̅aα based in Figure 3c are 132.6, 128.9, 105.8, and 126.7 kJ/mol, respectively. This demonstrates that the apparent activation energy increases with increasing VA of EVA. This also demonstrates that the crosslinking reaction of 7350M takes place easier than that of 7470M. This is consistent with the cross-linking torque values of their cross-linking curves (Figure 4). The cross-linking reaction of EVA initiated with DCP mainly takes place in the main chain of the EVA molecule, while the cross-linking reaction rate constant of 7350M is smaller than that of 7360M or 7470M. t90 of 7350M is longer than that of 7470M. Consequently, the cross-linking reaction rate is fast and the cross-linking reaction time is shortened although the apparent activation energy is increased with VA. The total crosslinking reaction rate changes with the degree of cross-linking. 3.4. Combination Kinetic Analysis of Rheometer. For the purpose of calculation, the cross-linking kinetic parameters are assessed based on Hill mathematical equation and T. H. Khang16 regress: α=

αmax(t − ti)n kHn + (t − ti)n

Figure 6. Plot of ln(dα/dt) versus 1/T at a certain cross-linking degree values: (a) 7350M; (b) 7360M; (c) 7470M.

regress of that, αmax ≈ 1 (in Table 2), so ln k = −(n ln kH + ln αmax) ≈ −n ln kH. Thus, eq 15a can be changed into eq 15b,

(15a)

where αmax and kH are the two constants in the Hill mathematical equation. On the basis of eq 15a, those data in Table 2 are regressed from Supporting Information, Figure S5. Owing to the

α= G

αmax(t − t i)n k(t − t i)n = kHn + (t − t i)n 1 + k(t − t i)n

(15b)



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Table 2. Kinetic Analysis and Hill Mathematical Equation Fitting Parameters Hill equation fitting

Arrhenius kinetic parameters

EVA

T (K)

ωDCP (phr)

αmax

kH

n

r2

−ln k

Ea (kJ·mol−1)

ln A

r2

7350M

443 448 453 458 443 448 453 458 443 448 453 458 448 448 448 448 448 448 448 448 448 448

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.60 1.00 1.40 1.80 0.60 1.00 1.40 1.80 1.00 1.00

1.08 1.03 1.01 1.00 1.08 1.03 1.01 1.00 1.07 1.03 1.01 1.00 1.01 1.03 1.04 1.04 1.02 1.03 1.04 1.04 1.03 1.03

73.75 58.66 40.51 29.70 71.47 46.82 34.12 22.87 65.59 44.72 34.16 24.24 45.00 58.66 53.37 59.87 36.81 44.72 49.99 52.48 41.62 43.65

1.23 1.43 1.76 2.04 1.21 1.41 1.60 1.93 1.23 1.45 1.64 1.95 1.61 1.43 1.46 1.50 1.41 1.45 1.44 1.47 1.34 1.37

0.9990 0.9987 0.9986 0.9991 0.9985 0.9981 0.9985 0.9986 0.9977 0.9973 0.9977 0.9987 0.9990 0.9987 0.9988 0.9987 0.9980 0.9973 0.9978 0.9980 0.9979 0.9977

5.290 5.823 6.515 6.918 5.166 5.423 5.648 6.041 5.146 5.511 5.791 6.217 6.129 5.823 5.807 6.138 5.084 5.511 5.633 5.822 4.996 5.156

188.2 ± 11.6

56.4 ± 3.2

0.9878

96.0 ± 9.2

31.2 ± 2.5

0.9727

117.8 ± 6.8

37.1 ± 1.8

0.9900

7360M

7470M

7350M

7470M

Blend A Blend B

Table 3. Combination Kinetic Analysis Parameters 0.005 ≤ α ≤ 0.325

0.325 < α ≤ 0.995

0.005 ≤ α ≤ 0.995

EVA

T (K)

ωDCP (phr)

−ln k*

m

r2

−ln k*

m

r2

−ln k*

m

r2

7350M

443 448 453 458 443 448 453 458 443 448 453 458 448 448 448 448 448 448 448 448 448 448

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.60 1.00 1.40 1.80 0.60 1.00 1.40 1.80 1.00 1.00

15.04 13.15 17.76 16.69 14.08 14.56 12.23 9.90 15.48 15.71 15.08 12.94 13.76 13.14 12.83 16.60 12.46 15.09 20.76 18.04 14.12 15.73

3.90 3.51 5.04 5.03 3.45 4.00 3.45 2.97 4.05 4.36 4.50 3.95 3.62 3.51 3.27 4.35 3.51 4.12 5.87 5.03 4.00 4.50

0.9873 0.9954 0.9974 0.9985 0.9822 0.9973 0.9784 0.9703 0.9911 0.9853 0.9980 0.9835 0.9840 0.9957 0.9796 0.9941 0.9978 0.9818 0.9892 0.9946 0.9931 0.9931

10.75 11.15 10.91 9.57 11.88 11.11 10.38 8.165 11.77 11.39 10.47 9.76 12.91 13.54 10.80 11.89 11.63 12.97 12.88 10.39 11.24 11.76

2.47 2.76 2.86 2.79 2.72 2.76 2.80 2.56 2.73 2.85 2.87 2.96 3.10 3.17 2.74 2.91 2.90 3.13 3.10 2.71 2.82 2.93

0.9902 0.9911 0.9869 0.9822 0.9768 0.9864 0.9852 0.9823 0.9674 0.9800 0.9895 0.9917 0.9828 0.9823 0.9834 0.9816 0.9775 0.9768 0.9817 0.9752 0.9824 0.9805

12.50 11.76 12.38 11.88 12.71 11.76 10.64 8.850 12.12 11.33 10.54 12.48 11.60 11.66 12.50 13.80 10.69 12.37 13.78 13.17 11.51 11.96

2.85 2.90 3.14 3.27 2.88 2.89 2.85 2.69 2.99 3.04 3.10 2.87 2.88 2.87 3.03 3.22 2.77 3.04 3.28 3.16 2.88 2.97

0.9777 0.9894 0.9854 0.9789 0.9766 0.9867 0.9910 0.9869 0.9820 0.9830 0.9842 0.9735 0.9853 0.9887 0.9895 0.9887 0.9827 0.9812 0.9743 0.9840 0.9849 0.9830

7360M

7470M

7350M

7470M

Blend A Blend B

The cross-linked extent of EVA can be expressed by α and the un-cross-linked extent is expressed by (1−α). Thus, the ratio of α versus (1−α) depends on the cross-linking reaction time under isothermal conditions.

As 0 < α < 1, combining eq 7 with 16 gives ⎫ ⎧⎛ α ⎞ ⎟[ − ln(1 − α)]⎬ ln⎨⎜ ⎝ ⎠ ⎭ ⎩ 1−α

⎛ α ⎞ ⎟ = − n ln k ln⎜ H + n ln(t − t i) ⎝1 − α ⎠ = ln k + n ln(t − t i)

= ln(kAk) + (nA + n)ln(t − t i) (16)

(17)

Also, it is supposed that H


Industrial & Engineering Chemistry Research g(α) =

⎛ α ⎞ ⎜ ⎟[ − ln(1 − α)] ⎝1 − α ⎠

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New kinetic analyses are constructed based on the crosslinking parameter measurement with the cross-linking rheometer incorported into Hill and Avrami mathematical equations. For the purpose of instructing industrial production, the cross-linking rheometer is a more useful and convenient method for fast investigating kinetics of the cross-linking reaction of a polymer in solid or melting state.

(18)

Because of t ≫ ti during rheometer measuring, t in eq 7 can be approximatively replaced with (t − ti) and k* = kA·k, m = nA + n, t* = t − ti, consequently, ln[g (α)] = ln k* + m ln t *

■

(19)

The value of eq 18 or the left side of eq 19 can be calculated based on Supporting Information, Figure S5 or based on the crosslinking curves of Figure 4 and eq 9. The left side of eq 19 versus ln t* can be linearly fitted. Hence, its slope and intercept can be considered as m and lnk*, respectively, and they are listed in Table 2 and Table 3. With VA increase, the differences of the cross-linking reaction orders are reduced, while the cross-linking reaction rate is increased. Both the order and the rate of the crosslinking reaction are increased with elevating reaction temperature, but they are reduced with increasing ωDCP. 3.5. Comparison of Cross-Linking Rheometer with DSC. The cross-linking rheometer method can be used to predict the kinetics under isothermal experimental conditions. However, kinetic assessments of both the cross-linking rheometer and DSC are different. The first, the cross-linking rheometer presents a simpler, faster and more convenient mesurement than DSC because it takes a shorter time to measure the cross-linking parameters with the cross-linking rheometer. The cost of the cross-linking rheometer is less than that of DSC. On the other hand, the cross-linking rheometer is very popular in the rubber industry. The second difference is that the sample amount with the cross-linking rheometer is larger than that used in DSC. This means that the cross-linking rheometer is a more reasonable choice than DSC for instructing industrial production. The apparent activation energy is assessed in our approaches in the range of 91.2−135.5 kJ/mol by rheometer under isothermal conditions and 132.6−175.0 kJ/mol by DSC under nonisothermal conditions, which is equivalent to that reported by Bianchi10,11 by 92−178 kJ/mol and 87−105 kJ/mol, respectively. To our knowledge of manufacturing the cross-linked EVA, our assessment better approaches the production actual practice.

ASSOCIATED CONTENT

S Supporting Information *

The full DSC curves of EVA compounds containing DCP (Figure S1); curves of ln(βdα/dT) versus 1/T in DSC method (Figure S2); pre-exponential factor (ln A) as a function of the degree of cross-linking (Figure S3); Avrami analysis (Table S1−S6); cross-linking time (t10, t50, and t90; and t10A, t50A, and t90A) as a function of VA, ωDCP or T (Figure S4); rheometer analysis for the degree of cross-linking (Figure S5); regression linearly dependent coefficient square (r2) of ln k or ln kα versus 1/T (Table S7); linear fitting of ln(dα/dt) as a function of ln(1 − α) (Figure S6); Avrami−Ozawa method. Linear fitting of ln β versus [ln(T − Ts) − ln β] (Figure S7); nonisothermal cross-linking kinetic parameters based on Avrami−Ozawa analysis (Table S8); linear fitting of intercept (ln k) versus 1/T (Figure S8); combination mathematical kinetic equations. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +86 591 83464353. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Natural Science Foundation of Fujian Province of China, No. 2012J01030, and by Jinjiang Science and Technology Foundation of Fujian Province, No. 2013J0243.

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4. CONCLUSIONS As α is less than 0.31, nA is in the range of 4−5. The cross-linking reaction of EVA initiated with DCP is not sufficient and EVA can flow relatively freely in three-dimensions. While α ranges from 0.31 to 1, nA is in the range of 0.85−1. The cross-linking reaction gets more and more sufficient and then the EVA becomes fully cross-linked in the later cross-linking stage so that threedimensional cross-linked nets have been constructed. The cross-linking reaction of EVA initiated with DCP mainly occurs at the main chain of the EVA molecule. The cross-linking reaction is approximatively considered as a first order reaction, but differences of the cross-linking reaction orders are reduced and the cross-linking reaction rate is increased with VA increase. The apparent activation energy and the cross-linking reaction rate constant also increase with an increase of VA of EVA. The apparent activation energy, which is in the range of 91.2− 135.5 kJ/mol by rheometer under isothermal conditions and 132.6−175.0 kJ/mol by DSC under nonisothermal conditions, and pre-exponential factor depend on the degree of cross-linking. As a result, under isothermal conditions, the apparent activation energy, pre-exponential factor or the total cross-linking reaction rate is a function of the degree of cross-linking. I

NOTATION A = the pre-exponential factor of the cross-linking reaction DCP = dicumyl peroxide DSC = differential scanning calorimetry Ea = the apparent activation energy of the cross-linking reaction Eak = the apparent activation energy on the Arrhenius temperature-dependent rate constant of the cross-linking reaction in rheometer measurement Eaα = the apparent activation energy on a certain degree of cross-linking in rheometer measurement EVA = ethylene-vinyl acetate copolymer ∂H/∂T = the heat flow function in DSC measurement k = the Arrhenius temperature-dependent rate constant of the cross-linking reaction kA = the Avrami constant of the cross-linking reaction kα = the Arrhenius temperature-dependent rate constant on a certain degree of cross-linking of the cross-linking reaction in rheometer measurement M = the torque value corresponding to the crosskicking time t during rheometer measurement n = the order of the cross-linking reaction nA = the Avrami index of the cross-linking reaction dx.doi.org/10.1021/ie5011788 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX


Article

p = the linearly fitted parameter by Avrami−Ozawa method phr = parts per hundred of EVA r2 = the linearly dependent coefficient square R = the universal gas constant t = the cross-linking time T = the temperature TE = the onset temperature of the cross-linking reaction in DSC measurement Ts = the end point temperature of the cross-linking reaction in DSC measurement VA = content of vinyl acetate domains in the EVA copolymer α = the degree of cross-linking of EVA in the cross-linking reaction β = the heating rate in DSC measurement ωDCP = the weight percentage of the corresponding component DCP

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(15) Vyazovkin, S.; Wight, C. A. Isothermal and nonisothermal kinetics of thermally stimulated reactions of solids. Inter. Rev. Phys. Chem. 1998, 17, 407−433. (16) Khang, T. H.; Ariff, Z. M. Vulcanization kinetics study of natural rubber compounds having different formulation variables. J. Therm. Anal. Calorim. 2012, 109, 1545−1553.

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