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Degradation of styrene butadiene rubber (SBR) in anaerobic conditions
Polymer Degradation and Stability 111 (2015) 159e168
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Polymer Degradation and Stability journal homepage: www.elsevier.com/locate/polydegstab
Degradation of styrene butadiene rubber (SBR) in anaerobic conditions ez a, b, c, *, Sylvie Dagreou c, Bruno Grassl c, Alejandro J. Müller b, d, e Oscar Verna a
PDVSA Intevep, Department of Well Productivity, Urb. Santa Rosa, Sector el Tambor, Los Teques, 1201 Edo. Miranda, Venezuela n Bolívar, Apartado 89000, Caracas 1080, Venezuela Grupo de Polímeros USB, Departamento de Ciencia de los Materiales, Universidad Simo c Universite de Pau et des Pays de l’Adour/CNRS IPREM UMR5254, Equipe de Physique et Chimie des Polymeres, 2, Avenue du President Angot, 64053 Pau, France d Institute for Polymer Materials (POLYMAT) and Polymer Science and Technology Department, Faculty of Chemistry, University of the Basque Country n, Spain (UPV/EHU), Paseo Manuel de Lardizabal 3, 20018 Donostia-San Sebastia e IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 September 2014 Received in revised form 24 October 2014 Accepted 11 November 2014 Available online 20 November 2014
In this work, the degradation kinetics of styrene butadiene rubber (SBR) in solution was studied in anaerobic conditions. Degradation reactions in the presence of cumene hydroperoxide at different concentrations (0.20, 0.28, 0.32, 0.50, 0.60% in weight), temperatures (60, 75, 85, 100 and 120 C) and aromatic solvent (10 and 20%) were performed. The fragmentation rates of polymer chains, which define the degradation kinetics, were calculated from the change in molecular weight distribution with time. The degradation was performed in a reactor with anaerobic conditions and the characterization was performed by multiangle light scattering coupled to size exclusion chromatography (SEC-MALS). Using population balance equations, it was possible to calculate the kinetic constants for thermal and thermooxidative degradation. Analysis of the results led to the conclusion that random scission of polymer chains produced by macroradicals formed by hydrogen abstraction constituted the predominant SBR degradation mechanism. Adding alkylbenzene as a transfer agent significantly reduced the degradation. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Styrene butadiene rubber Population balance equations Polymer in solution MALS-SEC Anaerobic degradation ́
1. Introduction Physical properties of polymers depend to a large degree on molar mass, its distribution and molecular architecture, therefore, chain scission and crosslinking play an important role in polymer performance. For this reason, it is important not only to evaluate the chemical changes in a polymer reaction, but also the changes in molar mass distribution [1]. Polydienes degradation occurs at specific sites because double bonds in the main chain act as reactive points. A series of hydrogen transfer reactions may occur in polydienes when they are heated in the absence of oxygen. Transfer of allylic hydrogen can lead to chain scission. Pendant vinylene groups promote the lability of allylic hydrogen, producing stable radicals, leading to b scission processes or chain recombination [1,2]. The scheme of the b scission for polybutadiene (PB) is shown in Fig. 1a and b.
* Corresponding author. PDVSA Intevep, Department of Well Productivity, Urb. Santa Rosa, Sector el Tambor, Los Teques, 1201 Edo. Miranda, Venezuela. Tel.: þ58 212 330 7256. E-mail address: [email protected] (O. Vern aez). http://dx.doi.org/10.1016/j.polymdegradstab.2014.11.006 0141-3910/© 2014 Elsevier Ltd. All rights reserved.
̀
́
In the presence of peroxides, degradation of these polymers occurs throughout a mechanism of abstraction of the allylic hydrogen in the main chain, which may result in either a b scission or a crosslinking reaction from the recombination or termination reaction at high concentration. Thermal degradation of PB has been described with the reaction scheme shown in Fig. 1b [2]. Likewise, for polystyrene (PS) degradation, a free radical mechanism has been proposed where the radical generated from the hydrogen abstraction of the tertiary carbon can be stabilized by induction effects of aromatic rings and can then undergo chain scission as shown in Fig. 1c [3]. These scission reactions do not imply the termination of the macro-radicals, therefore, they can be considered as chain reactions that increase the number of chains in the system and lead to low molecular weight products. Peroxides and hydroperoxide homolitic decomposition with temperature lead to reactive primary alcoxy radicals, which are stabilized by abstracting labile hydrogen from the polymer chains to form more stable tertiary or allylic radicals [4,5]. Polymer degradation processes may be described by population balance equations (PBE), which can model the changes in molar
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
160
Fig. 1. a) b Scission of PB from allylic radical [2] b) Reaction scheme proposed for thermal decomposition of PB [2] c) Mechanism for free radical degradation of Polystyrene [3].
mass distribution as a consequence of polymer chain fragmentation. In this work, the kinetics aspects of the anaerobic degradation of a styrene butadiene rubber (SBR) dissolved in a paraffinic mineral oil as a function of time is evaluated. The parameters studied are: temperature, concentration of an organic hydroperoxide, and concentration of a radical transfer agent. The molar mass distribution is obtained by gel permeation chromatography (GPC) coupled with multiangle light scattering (SEC-MALS). The fragmentation kinetic is analyzed by population balance equation (PBE) theory.
change in number of molecules in time tet0 is given by the shaded region, defined as [6]:
d dt
Zb f1 ðx; tÞdx ¼ a
Z∞ xn f1 ðx; tÞdx
(3)
0
2. Model description One of the advantages of using PBE is the possibility of including phenomenological parameters that affect the kinetics of degradation reactions. It is also a way to handle continuous mixtures of particles of the same chemical features, and analyzing continuous reactions in one single integro-differential equation. If x is the molar mass of a polymer chain, the weight molar mass distribution is given by xf1(x, t), while the number molar mass distribution is f1(x, t). The molar concentration of the polymer in a molar mass range of (x,x þ dx) is f1(x, t)dx. Thus, the total number of particles in a specific time t, can be calculated as:
Z f1 ðx; tÞdx
(2)
The n-th moments of the distribution can be also defined as:
mn ¼
NðtÞ ¼
dN dt
(1)
Fig. 2 shows two density functions, one at time t ¼ t0 and the other at time t. Considering a subset of the x domain, [a,b], the
Then, the zero moment is given by the area below the curve defined by the number molar mass distribution, and represents the total molar concentration, or number of moles of polymer chains. The first moment is the total mass of the polymer or mass concentration, and may be represented by Equation (4):
Z∞ m1 ¼
xf1 ðx; tÞdx
(4)
0
Assuming mass conservation during the reaction, the first moment should be constant in time. From distributions moments it is possible to determine other important characterization parameters, such as the number average molecular weight Mn ¼ m1/m0, the weight average molecular weight Mw ¼ m2/m1; and the polydispersity index Mw/ Mn ¼ m2m0/m21.
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
161
Fig. 2. Molar mass distribution at two different times.
The weight molar mass distribution xf1(x, t) can be expressed as a function of Mn and Mw through a gamma function type as the Schulz-Zimm distribution function [7]:
RO$ þ PðxÞ!ROH þ R· ðxÞ
k kk x kx xf1 ðx; tÞ ¼ exp Mn t Mn GðkÞ Mn
HO$ þ PðxÞ!/H2 O þ R· ðxÞ
k2
(10)
k3
(5)
(11)
b scission of macroradicals where it is important to highlight that the macroradical R(x0 ) is a terminal radical [8,9]:
where k is given by: k4
R$ ðxÞ!R$ ðx0 Þ þ Pðx x0 Þ
Mn k¼ Mw Mn
(6)
In fragmentation or degradation of polymer chains, the PBE for pure fractionation reactions can be defined as [6]:
(12)
Chain radical transfer: k9a
! R$ ðx0 Þ Pðx0 Þ
(13)
k9b
vf1 ðx; tÞ ¼ vt
Z∞
vðx0 Þbðx0 ÞPðxjx0 Þf1 ðx0 ; tÞdx0 bðxÞf1 ðx; tÞ
(7)
x
where b(x0 ) is the breakage rate, defined as the fraction of particles of size x0 that break in a specific time unit. v(x0 ) is the average number of particles formed from the breakup of particle x0 . Assuming one breakup at a time for a single particle, i.e., binary breakup, the value of this function would be v(x0 ) ¼ 2. P(xjx0 ,t) is defined as the probability density of particles of size x which come from the breakup of particles of size x0 at time t. It is also called stoichiometric Kernel, and represents the distribution of sizes of chains generated from the breakup of particles of size x0 . When the breakup occurs randomly in the chain, the Kernel would be P(x,x0 ) ¼ 1/x0 as described by some authors [4]. There are different methods to solve the integro-differential equation that describe the PBE. Ramkrishna [6] describes several methods that may be applied depending on the characteristics of each system. For pure fragmentation, the moments of the density function may be used as a tool for solving PBE. Some authors [6] have described the moment solution assuming no particle growth and considering binary fragmentation n(x) ¼ 2. The reaction mechanism for thermoxidative degradation of polymer with its different kinetics constant is summarized in the next scheme of equations. Thermal degradation: k0
PðxÞ!Pðx0 Þ þ Pðx x0 Þ
(8)
Initiator decomposition: k1
ROOH!RO$ þ · OH Hydrogen abstraction by the primary radicals:
(9)
where x and x0 represent two general molecular weights with (x0 < x) From these equations the PBE can be constructed through Equation (7). Naming p(x,t) the density function of polymer in a specific time t; the PBE for thermal degradation can be represented by:
vpðx; tÞ ¼ k0 ðxÞpðx; tÞ þ 2k0 ðxÞ vt
Z∞
pðx0 ; tÞPðxjx0 Þdx0
(14)
x
Now, for degradation reaction with the added degradation agent or hydroperoxide, where p(x,t) and r(x,t) express the polymer and macro-radical concentration in time t respectively, the PBE for both macro-radical and polymer chains can be represented as:
vpðx; tÞ ¼ k2 xCRO$ ðtÞpðx; tÞ k3 xCHO$ ðtÞpðx; tÞ k9b ðxÞpðx; tÞ vt Z∞ 1 þ k9a ðxÞrðx; tÞ þ k4 x0 rðx0 ; tÞ 0 dx k0 xpðx; tÞ x x
Z∞ þ 2k0
00
pðx0 ; tÞdx
x
(15) vrðx; tÞ ¼ k2 xCRO$ ðtÞpðx; tÞ þ k3 xCHO$ ðtÞpðx; tÞ þ k9a xpðx; tÞ vt Z∞ 1 k9b xrðx; tÞ þ k4 x0 rðx0 ; tÞ 0 dx0 k4 xrðx; tÞ x x
(16)
162
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3. Experimental 3.1. Materials The polymer employed in this work was a cold polymerized Styrene butadiene rubber SBR from ISP (SBR-8113). This polymer was purified by solvent sohxlet extraction with toluene and then precipitated with 1-propanol and dried in a vacuum oven at 40 C. The mineral oil, Vassa LP-90 was used as a solvent. It is a 100% paraffinic solvent composed of C6eC22 molecules. This solvent was filtrated twice with a 0.2 mm Millipore®. An aromatic solvent was also used, named COPESOL, from Pequiven S.A. This solvent is a mixture of poly-substituted monoaromatic molecules of the alkybenzene type. Cumene hydroperoxide 80% (Trigonox K-80) from Akzo Nobel was also used with an activation energy (Ea) of 132.56 kJ/mol and an Arrhenius constant (A) of 1.15 1012 reported by the supplier. Nitrogen 98% was used to purge the reaction. 3.2. Degradation reaction The degradation assembly consists on a three-neck flask with a jacket connected to a recirculation bath for maximum temperature control and magnetic stirring. A controlled nitrogen in-out system was installed with a septum plug for needle sampling. The purified SBR was dissolved in the mineral oil at 40 C and at a concentration of 2 wt% for at least 48 h prior to the degradation reaction. Then the nitrogen purge system was mounted and the temperature was raised to the desired value. Time zero was considered when the recirculation bath reached the defined temperature. The hydroperoxide was added right before rising the temperature. Samples of 1 cm3 were taken during the reaction at different times and were dissolved in the elution solvent (in a 1:1 volume ratio) prior to the injection in the SEC-MALS. The degradation was performed without hydroperoxide at temperatures of 60 , 100 and 120 C. For 100 C, three different concentrations of cumene hydroperoxide were employed: 1.3 103 M (0.50 wt%); 8.41 104 M (0.32 wt%) and 7.36 104 M (0.28 wt%). For 120 C, two hydroperoxide concentrations were studied: 1.6 103 M (0.20 wt%) and 5.26 104 M (0.60wt %). For solutions containing 1.3 103 M (0.50 wt%) cumene hydroperoxide, three temperatures were tested: 75, 85 and 100 C. Finally SBR was dissolved in three mixtures with aromatic solvent, Vassa:Copesol (9:1; 8:2 and 7:3). The hydroperoxide concentration was kept at 0.50 wt% and the temperature was fixed at 100 C.
with the mineral oil, because the chromatography columns allow the separation of the polymer chains from the mineral oil in the sample. The oil signal eluded at higher times. No difference was observed when the elution solvent was Copesol or toluene. 4. Experimental results 4.1. Reproducibility and repeatability SBR used in this study was a heterogeneous industrially processed polymer. Before any degradation was induced, a solution of SBR was prepared and purified. At time zero, where no temperature had been applied, no differences in molecular mass should be appreciated in between different samples of the same SBR batch employed here. Nevertheless, Fig. 3 shows the large differences found for 25 similar SBR samples at time zero. These tests include two or three runs of the same sample, i.e., reproducibility and repeatability are considered in the same figure. The initial average Mw for all samples at initial time is shown in the inner table of Fig. 3. The high values of the standard deviation represent the heterogeneity of the sample. These errors have to be taken into account during the degradation analysis. It is important to point out that SEC-MALS technique is an absolute method to determine Mw, and then the conformational changes due to temperature or solvent interaction with polymer chain do not affect the results. Neither the concentration changes because of non-soluble fraction nor the polymer retention in the columns would change the results, because the equipment is able to detect absolute concentration. This is why those differences are assumed to be just a matter of sample heterogeneity. 4.2. Degradation results Fig. 4 shows an example of the change in the molecular weight distribution with time during degradation reaction with hydroperoxide and temperature. It can be seen that the molecular weight decreases asymptotically until it reaches a minimum value (see the inset in Fig. 4). This decrease is important in comparison to the values of the standard deviation observed. These results suggest that degradation process might be proportional to molecular weight as predicted. From the distribution it is possible to observe the decrease of polydispersity. This behavior is opposite to what it could be
3.3. Characterization procedures A size exclusion chromatograph (SEC) coupled with a multiangle light scattering (MALS) detector and a refractometer was used to determine the molar mass distribution. Gel permeation chromatography (GPC) columns were used as SEC device, with Mixed C and Mixed D and linear columns from Variant. An isocratic Agilent 1200 pump was used. The elution solvent was either COPESOL or filtrated HPLC grade toluene. The MALS system employed was a DAWN EOS (Wyatt Technology) coupled with a dn/dc refractometer Optilab-Rex (Wyatt Technology) to determine concentration. All measurements were carried out at 25 C and data were processed by ASTRA V software from Wyatt Technology. The dn/dc value for the SBR in the aromatic solvent was 0.0478 ± 0.0009 mL/g and 0.0456 ± 0.0011 mL/g determined in batch. This value was not affected by the mixture of the polymer
Fig. 3. Reproducibility and repeatability tests for SBR samples in solution at initial time (before degradation). The middle solid line represents the average Mw and dash lines represent one standard deviation.
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
expected for random scission. During degradation, all molecules tend to a specific molar mass and then the distribution broadness also decreases. This can be explained by the asymptotic behavior of the molecular weight with time. The degradation reaction tends to stop after reaching some minimum molecular weight value. Some authors have numerically simulated this behavior as single frequency distribution function dynamics, considering that the degradation stops after some minimum molecular weight and totally random kernel. The initial frequency distribution function decreases exponentially, generating a product peak which eventually tends to a Dirac delta function for long times [10]. Fig. 5 shows the results of three thermal degradation experiments without the use of hydroperoxide at different temperatures. It can be seen that for 60 and 100 C there is a slight increase in molecular weight that falls within the standard deviation shown in Fig. 3. There was no evidence of Mw increment during the chromatography results. From the practical point of view, in this work, this increase will be considered negligible. On the other hand, for 120 C, the reduction in molecular weight is significant. Thus, there is a critical temperature at which thermal degradation begins to be noticed. Some authors suggest chain scissions will take place if the heat exceeds the dissociation energies of chemical bonds [11]. Degradation reactions with cumene hydroperoxide were carried out at different temperatures in an anaerobic environment, at a fixed concentration of cumene hydroperoxide (0.5%). Fig. 6 shows the degradation of SBR polymer chains in the presence of a constant concentration of hydroperoxide. For 75 C, degradation is negligible. Changes in molecular weight start to be significant at 85 C; at this temperature, the Mw value reaches 50% of its initial magnitude in 25,000 s. At 100 C, the polymer chains reach an asymptotic behavior at approximately 30% of the initial weight average molecular weight after 10,000 s. Degradation reactions were also performed at different concentrations of cumene hydroperoxide, at two different temperatures. At 100 C three different concentrations of the hydroperoxide were evaluated, while for 120 C only two concentrations were explored. Fig. 7 shows the degradation behavior of SBR as a function of time for all hydroperoxide concentrations. Once more, an asymptotic behavior was observed for the molecular weight reduction as a function of time. The degradation extend was large, since the samples exhibited Mw values close to less than 10% of the molecular weight of the undegraded samples. It can be also seen
Fig. 4. Distribution of molecular weights for a degradation reaction of a SBR sample in solution with 0.6% of hydroperoxide at 120 C. The inset shows how the Mw changes with reaction time.
163
Fig. 5. Thermal degradation experiments of SBR in solution for three different temperatures, without the use of hydroperoxide. The change in Mw was normalized with respect to the initial Mw of the sample.
that no significant differences in degradation can be appreciated for the different hydroperoxide concentrations (within the errors involved in the measurements). Nevertheless, it is clear that at temperatures of 100 C and 120 C, the degradation kinetic is accelerated by the addition of hydroperoxide. The effect of an alkylbenzene, as a radical transfer agent, was also evaluated at different concentrations and in the presence of the hydroperoxide. If the degradation reaction depends on hydrogen abstraction from a radical, the presence of a transfer agent should retard the degradation reaction by retaining and stabilizing radicals. Fig. 8 shows how the addition of different concentrations of the transfer agent retards significantly the degradation reaction. This is an important result in order to be able to control the degradation reaction. 5. Discussion of the results By the use of spectroscopic methods, it was not possible to follow the degradation reaction through the quantification of a chemical change in the sample. Hydroperoxide consumption was
Fig. 6. Degradation of SBR in solutions with 0.50% of cumene hydroperoxide at three different temperatures.
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ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
Fig. 8. Degradation of SBR in solution with different concentrations of an aromatic solvent (Alkylbenzene or Copesol). The experiments were performed at 100 C and with a concentration of hydroperoxide of 0.5%.
Fig. 7. SBR degradation at different hydroperoxide concentration. a): 100 120 C.
C.
b):
measured by titration and no difference was observed as compared to the homolitic decomposition of the hydroperoxide itself. The appearance or disappearance of functional groups was not enough to be quantified by available methods in our laboratory. The balance between peroxide, peroxy radicals, and polymer molecular weight should rule the time evolution of the degradation process. The hydroperoxide decomposition implies the formation of two radicals of different kinds, a hydroxy radical and a cumyloxy radical. The decomposition rate is of the Arrhenius type. In terms of the molar concentration of peroxide, c, the dissociation rate is given by kic where the coefficient ki is governed by a first order kinetics, ki ¼ Aexp(Ea/RT). This constant was calculated for different temperatures as shown in Table 1 using the values of the activation energy and the value of A, given by the supplier. Regarding the kinetic study of polymer degradation, many authors [4,9,12e20] have generated mathematical functions to describe changes in molecular weight distribution through population balance equations, considering in some cases the initiation, propagation, transfer, b scission and termination reactions. To achieve this it is necessary to neglect some phenomena, such as density changes, crosslinking or aggregation, diffusion of species or Trommdorf effect. The integro-differential relationships for the PBE are solved in terms of molecular weight using two main approximations: the long chain approximation (LCA) where initiation and termination reactions are infrequent events, thus are neglected in comparison
to hydrogen abstraction reactions. The other approximation is the quasi-steady state approximation (QSSA), where the change of radical concentration with time is too small as compared to the change of other components and could also be neglected [21]. The changes in the molecular weight also show that the degradation process is not due to a depolymerization process, but a consequence of some chain fragmentation. For a depolimerization process, no change in molecular weight distribution should be detected and the decrease in the average molecular weight would be smoother. These kinds of fragmentations represented by random scissions should be described by PBE employing the stoichiometric Kernel as the inverse function, as suggested in the literature [12], where P(xjx0 ) ¼ 1/x0 . High molecular weight chains degrade faster than short chains and this is related to the number of hydrogen atoms available for abstraction, i.e., the larger the number of hydrogen atoms present in the chain, the higher the probability of a hydrogen abstraction reaction. This behavior has been predicted by some authors [12,22,23] for hydrogen abstraction degradation mechanisms. From PBE, this behavior implies that function b(x) should be proportional to the molecular weight. The break function has been reported as linearly dependent on the molecular weight b(x) ¼ kx or even exponentially b(x) ¼ kxr [12]. Studies on the molecular weight influence on degradation indicate that random-scission rate coefficient dependence on molecular weight is linear for short chains, but second order for longer chains [23]. It may also depend on other variables, such as temperature, concentration or reaction type. A simplification frequently used in PBE is binary fractionation v(x) ¼ 2, which assumes that one chain can only suffer a single fragmentation at a time, i.e., non simultaneous fragmentation for the same molecule is considered.
Table 1 Kinetic constants for Cumene hydroperoxide decomposition for different temperatures. Temperature ( C)
ki (1/s)
75 85 95 100 105 120
1.49 5.33 1.79 3.19 5.62 2.81
108 108 107 107 107 106
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
The general degradation reaction can be divided into different reactions assuming some mechanism. After decomposition of the ROOH hydroperoxide, the resulting OH hydroxy and RO cumyloxy radicals, which are not stable, abstract hydrogens from the polymer chain P(x), generating polymer radicals or macroradicals R(x), where x represents the molecular weight as a continuous variable. 5.1. PBE for thermal degradation Assuming linear dependence of the degradation reaction with the molecular weight, PBE for Equation (14) in the case of thermal degradation can be written as:
vpðx; tÞ ¼ k0 xpðx; tÞ þ 2k0 vt
Z∞
pðx0 ; tÞdx0
(17)
x
Applying the moment operation, i.e., multiplying by xn and integration in the semi-infinite interval, the following integrodifferential equation is obtained:
vpn ðx; tÞ vt
Z∞ ¼ k0 pnþ1 ðx; tÞ þ 2k0
2 pðx0 ; tÞ4
x
Zx 0
Rearranging and knowing that pn ðx; tÞ ¼
Z
3 xn dx5dx0 ∞
Fig. 9 shows the results of applying these equations to the thermal degradation at different temperatures. It can be seen that for 100 C and 60 C the slopes are negative and very close to zero. This behavior is related to that shown in Fig. 5. For this work, these constants can be considered zero and it can be assumed that the negative values of the slope are related with experimental errors. On the other hand for 120 C, Fig. 9 shows that the value of the slope is significant, then, the thermal degradation reaction should be taken into account with a value of k0 ¼ 2.79 1010 s1. 5.2. PBE for degradation with hydroperoxide For degradation reaction with the hydroperoxide, in Equation (15) and Equation (16) the hydrogen abstraction from primary radicals (CRO) and (CHO) could be considered as a single process of hydrogen abstraction from C(t) with a kinetic constant k23. Assuming the stoichiometric Kernel as the inverse function P(xjx0 ), ¼1/x0 and applying the moment operation to Equation (15) and Equation (16):
vpn ðx; tÞ ¼ k23 CðtÞ vt
(18)
Z∞
Z∞ xnþ1 pðx; tÞ k9b 0
Z∞ xnþ1 rðx; tÞ þ k4
x pðx; tÞdx:
0
0
n
vp ðx; tÞ ðn 1Þ ¼ k0 pnþ1 ðx; tÞ vt ðn þ 1Þ
nþ1
k0 p
p0 ðtÞ p0 ðt ¼ 0Þ ¼ k0 p1 ðtÞt p0 ðtÞ 1
p ðtÞ
p0 ðt ¼ 0Þ 1
p ðtÞ
¼ k0 t
1 1 ¼ k0 t Mn ðtÞ Mn0
x
0
2
pðx ; tÞ4 0
1 x0 rðx0 ; tÞ 0 dx x Zx
3 x dx5dx0 n
(21)
(22)
0
vpn ðx; tÞ ¼ k23 CðtÞpnþ1 ðx; tÞ k9b pnþ1 ðx; tÞ þ k9a r nþ1 ðx; tÞ vt k ðn 1Þ þ 4 r nþ1 ðtÞ k0 pnþ1 ðx; tÞ ðn þ 1Þ nþ1
(20)
Then Ns can be defined as:
Ns ¼
ðx; tÞ þ 2k0
(19)
For the zero moment, n ¼ 0 substituting in the Equation (19):
Z∞ xn
0
Z∞
n
vp ðx; tÞ 2k0 nþ1 0 ¼ k0 pnþ1 ðx; tÞ þ p ðx ; tÞ vt nþ1
xnþ1 pðx; tÞ 0
Z∞ þ k9a
n
165
(23) vr n ðtÞ ¼ k23 CðtÞpnþ1 ðx; tÞ þ k9b pnþ1 ðtÞ k9a r nþ1 ðtÞ vt n k r nþ1 ðtÞ nþ1 4
(24)
On the other hand, for the hydroperoxide decomposition, the concentration of hydroperoxide with time is given by:
CROOH ðtÞ ¼ CROOH ð0Þexpk1 t
(25)
The PBE expressed in terms of the primary radical concentration are given by:
vCRO$ ðtÞ ¼ k1 CROOH ðtÞ CRO$ ðtÞ vt
Z∞
vCHO$ ðtÞ ¼ k1 CROOH ðtÞ CHO$ ðtÞ vt
k2 ðxÞpðx0 ; tÞdx0
(26)
0
Z∞
k3 ðxÞpðx0 ; tÞdx0
(27)
0
Assuming that the initiation is much faster than the hydrogen abstraction, i.e., k1 >> k3, then Equation (26) and Equation (27) can be rearranged as:
Fig. 9. Calculation of the thermal degradation kinetic constants by PBE for three different temperatures.
vCRO$ ðtÞ vCHO$ ðtÞ ¼ ¼ k1 CROOH ðtÞ ¼ k1 CROOH ð0Þexpk1 t vt vt Solving Equation (28) leads to:
(28)
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166
CRO$ ðtÞ ¼ CHO$ ðtÞ ¼ CðtÞ ¼ CROOH ð0Þ 1 expk1 t ¼ C0 1 expk1 t
(29)
Considering quasi-stationary state, the macroradical concentration with time does not change significantly, then:
vr n ðtÞ vr 0 ðtÞ vrðtÞ ¼ ¼ ¼0 vt vt vt
(30)
Replacing in Equation (24) and solving for r1(t):
r 1 ðtÞ ¼
k23 CðtÞ þ k9b 1 p ðtÞ k9a
(31)
and substituting Equation (31) in Equation (23) for zero moment (n ¼ 0):
vp0 ðx; tÞ p1 ðtÞk4 ¼ ðk23 CðtÞ þ k9b þ k0 Þ vt k9a
(32)
Now, because the initial conditions are known, and assuming that the first moment is invariable, i.e., the total mass does not change with time, Equation (32) can be solved by integration in time as:
Zt
Zt vp0 ðx; tÞ ¼
0
0
p1 ðtÞk4 ðk23 CðtÞ þ k9b þ k0 Þvt k9a
" k4 p1 ðtÞ k C ðk23 C0 þ k9b þ k0 Þt 23 0 p ðtÞ p ðt ¼ 0Þ ¼ k9a k1 # k t 1 k C e þ 23 0 k1 0
(33)
0
(34)
Assuming no thermal degradation, k0 might be neglected then Equation (34) can be expressed in terms of Ns as:
1 1 Mn ðtÞ Mn0 k4 k23 C0 k4 k9b k k C k k C t þ 4 23 0 ek1 t 4 23 0 ¼ þ k9a k9a k9a k1 k9a k1
NsðtÞ ¼
(35)
Denoting Ka ¼ k4k23/k9a and Kb ¼ k4k9b/k9a, the following relation is obtained:
NsðtÞ ¼ ðKa C0 þ Kb Þt þ
Ka C0 k1 t Ka C0 e k1 k1
Another simplification that hydrogen abstraction from the faster than macroradical chain Ka >> Kb, then the Equation (35)
NsðtÞ ¼
Fig. 10. Ka calculation for SBR degradation at 100 C and different concentrations of hydroperoxide from Equation (38), the dashed line is the fitting until 15000 s. The solid line represents the fit until 23000 s.
while the solid line, fitted the curve for longer times and leads to a Ka value of 3.88 104 with a R2 of 0.83235. Performing the same approximation but using Equation (36) directly, it is possible to calculate Ka and Kb at the same time, as shown in Fig. 11. Results from this approximation can be detailed in Table 2. At first glance, it can be confirmed that the order of magnitude of Ka is higher that Kb, in accordance to the approximation given by Equation (38). On the other hand, the difference observed in those fittings may arise from the rather large experimental errors involved, as shown in Fig. 3. For higher temperatures, i.e., 120 C, the kinetic constant for thermal degradation k0 should not be neglected, thus it should be counted as included in the Kb constant. A higher deviation from the theoretical approximation was observed for 120 C, as can be seen in the R2 values of the curve fittings. Fig. 12 shows the curve fittings for the SBR degradation at 120 C. Finally, Equation (36) was used to theoretically calculate the retardation of the reaction with addition of alkylbenzene as transfer agent. The aromatic group can stabilize a secondary radical in the a carbon of the aromatic ring through electron induction effect, involving another hydrogen abstraction step and additional
(36)
can be done is assuming that primary radicals is significantly transfer, that is, k23 >> k9b or is simplified to:
k4 k23 C0 k k C k k C t þ 4 23 0 ek1 t 4 23 0 k9a k9a k1 k9a k1
(37)
Considering that for the same temperature, all kinetic constants should be the same, then, changing the concentration of hydroperoxide allows calculating the Ka constant.
NsðtÞ 1 1 ¼ Ka t þ ek1 t C0 k1 k1
(38)
Fig. 10 shows the results for this approximation. It can be seen that deviation of the theoretical behavior occurs after 15000 s. For the dashed line, the value of Ka is 6.65 104 with a R2 of 0.90616,
Fig. 11. Ka and Kb calculation for SBR degradation at 100 C and different concentrations of hydroperoxide from Equation (36).
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna Table 2 Curve fitting for kinetic constants calculation. Ka ª 100 C until 25000 s ª100 C until 15000 s b 100 C 0.28% Equation (36) b 100 C 0.32% b 100 C 0.50% b 120 C 0.20% b 120 C 0.60% b 100 C 10% Alkylbenzene b 100 C 20% Alkylbenzene a b
3.88 6.65 1.67 6.32 1.06 1.16 7.65 6.67 7.10
104 104 104 106 106 105 105 1025 106
Kb
R2
0 0 1.23 1.09 4.06 1.05 1.17 7.10 1.00
0.83235 0.9062 0.9906 0.9804 0.9334 0.7427 0.8735 0.9617 0.9273
107 109 1015 109 109 1011 1022
Calculated from Equation (38). Calculated from Equation (36).
167
magnitude of the shift. The larger shifting effect occurs for Kb, which means that the reaction of Equation (12) is the most affected by the addition of the transfer agent. From this set of experiments, the experimental error or the raw polymer heterogeneities were too high; then, the calculation of the kinetic constant has been compromised. Nevertheless, some results from this experimental work can be highlighted as follows: No thermal degradation was observed below 100 C, but for 120 C, the effect of temperature starts to be significant. Addition of an initiator like cumene hydroperoxide leads to a significant increase in polymer degradation. In the concentration range of hydroperoxide studied, no significant differences were found for the induced degradation Addition of a transfer agent, such as an akylbenzene in the solvent, retards significantly the degradation process. 6. Conclusions
Fig. 12. PBE approximation for SBR degradation at 120 C.
intermediate radical species to the general reaction. Radicals in solvent molecules are more stable than on the polymer chain, and this effect retards the reactions of Equation (10), Equation (11) and Equation (12) shifting both k23 and k4 to lower values. Similar results have been obtained for polystyrene degradation [24] where hydrogen donor solvents decreased the polymer degradation. For Fig. 13 shows the fitting from Equation (36) and the results are summarized in Table 2. Despite the fact that no clear trend can be drawn from the kinetic constant, it is important to point out the
In this work, the characterization of the degradation reaction of SBR in solution was studied by the changes in molecular weight and its distribution, despite the fact that chemical changes were not detectable by spectroscopic techniques. The population balance equation analysis is an important tool to quantify the degradation process from the changes in molecular weight distribution. Despite large experimental errors, it can be concluded that the degradation of SBR in solution in the absence of oxygen occurs mainly by random chain scission events which are dependent on the molecular weight. The primary radicals from the hydroperoxide decomposition lead to the formation of macro-radicals in the main chain, which may generate new smaller chains from their scission. The results confirm the hypothesis of the degradation mechanism expected for SBR. A simple transfer agent as an aromatic solvent can be used during specific applications to retard SBR degradation, either to retard the SBR degradation at high temperatures or to control the degradation kinetics. Acknowledgments The authors would like to thank to PDVSA Intevep, for funding through the ID-6085-0000 project in the Department of Well Productivity. Also FONACIT (Science and Technology Ministry of the Bolivarian Republic of Venezuela) and Ministry of Foreign Affairs (France) for funding received through project: PCP 2011001409 (Postgraduate Cooperation Project). References
Fig. 13. Curve fitting for the SBR degradation with 0.50% of hydroperoxide at 100 C and different concentrations of alkylbenzene in the solvent.
[1] Chiantore O, Tripodi S, Sarmoria C, Valles E. Mechanism and molecular weight model for thermal oxidation of linear ethyleneebutene copolymer. Polymer 2001;42:3981e7. [2] Jiang DD, Levchik GF, Levchik SV, Wilkie CA. Thermal decomposition of crosslinked polybutadiene and its copolymers. Polym Degrad Stab 1999;65: 387e94. [3] Gordon Cameron G, Meyer J, McWalter I. Thermal degradation of polystyrene. 3. A Reappraisal. Macromolecules 1978:696e700. [4] Kim Y-C, McCoy BJ. Degradation kinetics enhancement of polystyrene by peroxide addition. Ind Eng Chem Res 2000;39:2811e6. [5] Madras G, Chattopadhyay S. Optimum temperature for oxidative degradation of poly(vinyl acetate) in solution. Chem Eng Sci 2001;56:5085e9. [6] Ramkrishna D. Population balances: theory and applications to particulate systems in engineering. San Diego, CA. USA: Academic Press; 2000. p. 355. [7] Tobita H. Simulation model for the modification of polymers via crosslinking and degradation. Polymer 1995;36:2585e96. [8] Chodak I, Zimanyova E. The effect of temperature on peroxide initiated crosslinking of polypropylene. Eur Polym J 1984;20:81e4. [9] Karmore V, Madras G. Kinetics for thermal degradation of polystyrene in presence of p-toluene sulfonic acid. 2002. p. 117e21.
168
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
[10] McCoy Benjamin J, Wang M. Continuous-mixture fragmentation kinetics: particle size reduction and molecular cracking. Chem Eng Sci 1994;49: 3773e85. [11] Schnabel W. Polymer degradation, principles and practical applications. 1981. p. 192e3. New York. [12] Mccoy B. Continuous kinetics of cracking reactions: thermolysis and pyrolysis. Chem Eng Sci 1996;5:2903e8. [13] Madras G, Smith J, McCoy B. Thermal degradation of poly (a-methylstyrene) in solution. Polym Degrad Stab 1996;52. [14] McCoy BJ, Madras G. Degradation kinetics of polymers in solution: dynamics of molecular weight distributions. AIChE J 1997;43:802e10. [15] Kodera Y, McCoy BJ. Distribution kinetics of radical mechanisms: reversible polymer decomposition. AIChE J 1997;43:3205e14. [16] Madras G, McCoy BJ. Oxidative degradation kinetics of polystyrene in solution. Chem Eng Sci 1997;52:2707e13. [17] Madras G, Smith JM, McCoy BJ. Thermal degradation kinetics of polystyrene in solution. Polym Degrad Stab 1997;58:131e8.
[18] Madras G, McCoy BJ. Time evolution to similarity solutions for polymer degradation. AIChE J 1998;44:647e55. [19] Sterling WJ, McCoy BJ. Distribution kinetics of thermolytic macromolecular reactions. AIChE J 2001;47:2289e303. [20] Karmore V, Madras G. Thermal degradation of polystyrene by Lewis acids in solution. Ind Eng Chem Res 2002;41:657e60. [21] Smagala TG, Mccoy BJ. Mechanisms and approximations in macromolecular reactions: reversible initiation, chain scission, and hydrogen abstraction. 2003. p. 2461e9. [22] Madras G, McCoy BJ. Kinetics and dynamics of gelation reactions. Chem Eng Sci 2007;62:5257e63. [23] Madras G, Chung GY, Smith JM, Mccoy BJ. Molecular weight effect on the dynamics of polystyrene degradation. 1997. p. 2019e24. [24] Sato S, Murakata T, Baba S. Solvent effect on thermal degradation of polystyrene 1990;40:2065e71.
Contents lists available at ScienceDirect
Polymer Degradation and Stability journal homepage: www.elsevier.com/locate/polydegstab
Degradation of styrene butadiene rubber (SBR) in anaerobic conditions ez a, b, c, *, Sylvie Dagreou c, Bruno Grassl c, Alejandro J. Müller b, d, e Oscar Verna a
PDVSA Intevep, Department of Well Productivity, Urb. Santa Rosa, Sector el Tambor, Los Teques, 1201 Edo. Miranda, Venezuela n Bolívar, Apartado 89000, Caracas 1080, Venezuela Grupo de Polímeros USB, Departamento de Ciencia de los Materiales, Universidad Simo c Universite de Pau et des Pays de l’Adour/CNRS IPREM UMR5254, Equipe de Physique et Chimie des Polymeres, 2, Avenue du President Angot, 64053 Pau, France d Institute for Polymer Materials (POLYMAT) and Polymer Science and Technology Department, Faculty of Chemistry, University of the Basque Country n, Spain (UPV/EHU), Paseo Manuel de Lardizabal 3, 20018 Donostia-San Sebastia e IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 September 2014 Received in revised form 24 October 2014 Accepted 11 November 2014 Available online 20 November 2014
In this work, the degradation kinetics of styrene butadiene rubber (SBR) in solution was studied in anaerobic conditions. Degradation reactions in the presence of cumene hydroperoxide at different concentrations (0.20, 0.28, 0.32, 0.50, 0.60% in weight), temperatures (60, 75, 85, 100 and 120 C) and aromatic solvent (10 and 20%) were performed. The fragmentation rates of polymer chains, which define the degradation kinetics, were calculated from the change in molecular weight distribution with time. The degradation was performed in a reactor with anaerobic conditions and the characterization was performed by multiangle light scattering coupled to size exclusion chromatography (SEC-MALS). Using population balance equations, it was possible to calculate the kinetic constants for thermal and thermooxidative degradation. Analysis of the results led to the conclusion that random scission of polymer chains produced by macroradicals formed by hydrogen abstraction constituted the predominant SBR degradation mechanism. Adding alkylbenzene as a transfer agent significantly reduced the degradation. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Styrene butadiene rubber Population balance equations Polymer in solution MALS-SEC Anaerobic degradation ́
1. Introduction Physical properties of polymers depend to a large degree on molar mass, its distribution and molecular architecture, therefore, chain scission and crosslinking play an important role in polymer performance. For this reason, it is important not only to evaluate the chemical changes in a polymer reaction, but also the changes in molar mass distribution [1]. Polydienes degradation occurs at specific sites because double bonds in the main chain act as reactive points. A series of hydrogen transfer reactions may occur in polydienes when they are heated in the absence of oxygen. Transfer of allylic hydrogen can lead to chain scission. Pendant vinylene groups promote the lability of allylic hydrogen, producing stable radicals, leading to b scission processes or chain recombination [1,2]. The scheme of the b scission for polybutadiene (PB) is shown in Fig. 1a and b.
* Corresponding author. PDVSA Intevep, Department of Well Productivity, Urb. Santa Rosa, Sector el Tambor, Los Teques, 1201 Edo. Miranda, Venezuela. Tel.: þ58 212 330 7256. E-mail address: [email protected] (O. Vern aez). http://dx.doi.org/10.1016/j.polymdegradstab.2014.11.006 0141-3910/© 2014 Elsevier Ltd. All rights reserved.
̀
́
In the presence of peroxides, degradation of these polymers occurs throughout a mechanism of abstraction of the allylic hydrogen in the main chain, which may result in either a b scission or a crosslinking reaction from the recombination or termination reaction at high concentration. Thermal degradation of PB has been described with the reaction scheme shown in Fig. 1b [2]. Likewise, for polystyrene (PS) degradation, a free radical mechanism has been proposed where the radical generated from the hydrogen abstraction of the tertiary carbon can be stabilized by induction effects of aromatic rings and can then undergo chain scission as shown in Fig. 1c [3]. These scission reactions do not imply the termination of the macro-radicals, therefore, they can be considered as chain reactions that increase the number of chains in the system and lead to low molecular weight products. Peroxides and hydroperoxide homolitic decomposition with temperature lead to reactive primary alcoxy radicals, which are stabilized by abstracting labile hydrogen from the polymer chains to form more stable tertiary or allylic radicals [4,5]. Polymer degradation processes may be described by population balance equations (PBE), which can model the changes in molar
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
160
Fig. 1. a) b Scission of PB from allylic radical [2] b) Reaction scheme proposed for thermal decomposition of PB [2] c) Mechanism for free radical degradation of Polystyrene [3].
mass distribution as a consequence of polymer chain fragmentation. In this work, the kinetics aspects of the anaerobic degradation of a styrene butadiene rubber (SBR) dissolved in a paraffinic mineral oil as a function of time is evaluated. The parameters studied are: temperature, concentration of an organic hydroperoxide, and concentration of a radical transfer agent. The molar mass distribution is obtained by gel permeation chromatography (GPC) coupled with multiangle light scattering (SEC-MALS). The fragmentation kinetic is analyzed by population balance equation (PBE) theory.
change in number of molecules in time tet0 is given by the shaded region, defined as [6]:
d dt
Zb f1 ðx; tÞdx ¼ a
Z∞ xn f1 ðx; tÞdx
(3)
0
2. Model description One of the advantages of using PBE is the possibility of including phenomenological parameters that affect the kinetics of degradation reactions. It is also a way to handle continuous mixtures of particles of the same chemical features, and analyzing continuous reactions in one single integro-differential equation. If x is the molar mass of a polymer chain, the weight molar mass distribution is given by xf1(x, t), while the number molar mass distribution is f1(x, t). The molar concentration of the polymer in a molar mass range of (x,x þ dx) is f1(x, t)dx. Thus, the total number of particles in a specific time t, can be calculated as:
Z f1 ðx; tÞdx
(2)
The n-th moments of the distribution can be also defined as:
mn ¼
NðtÞ ¼
dN dt
(1)
Fig. 2 shows two density functions, one at time t ¼ t0 and the other at time t. Considering a subset of the x domain, [a,b], the
Then, the zero moment is given by the area below the curve defined by the number molar mass distribution, and represents the total molar concentration, or number of moles of polymer chains. The first moment is the total mass of the polymer or mass concentration, and may be represented by Equation (4):
Z∞ m1 ¼
xf1 ðx; tÞdx
(4)
0
Assuming mass conservation during the reaction, the first moment should be constant in time. From distributions moments it is possible to determine other important characterization parameters, such as the number average molecular weight Mn ¼ m1/m0, the weight average molecular weight Mw ¼ m2/m1; and the polydispersity index Mw/ Mn ¼ m2m0/m21.
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
161
Fig. 2. Molar mass distribution at two different times.
The weight molar mass distribution xf1(x, t) can be expressed as a function of Mn and Mw through a gamma function type as the Schulz-Zimm distribution function [7]:
RO$ þ PðxÞ!ROH þ R· ðxÞ
k kk x kx xf1 ðx; tÞ ¼ exp Mn t Mn GðkÞ Mn
HO$ þ PðxÞ!/H2 O þ R· ðxÞ
k2
(10)
k3
(5)
(11)
b scission of macroradicals where it is important to highlight that the macroradical R(x0 ) is a terminal radical [8,9]:
where k is given by: k4
R$ ðxÞ!R$ ðx0 Þ þ Pðx x0 Þ
Mn k¼ Mw Mn
(6)
In fragmentation or degradation of polymer chains, the PBE for pure fractionation reactions can be defined as [6]:
(12)
Chain radical transfer: k9a
! R$ ðx0 Þ Pðx0 Þ
(13)
k9b
vf1 ðx; tÞ ¼ vt
Z∞
vðx0 Þbðx0 ÞPðxjx0 Þf1 ðx0 ; tÞdx0 bðxÞf1 ðx; tÞ
(7)
x
where b(x0 ) is the breakage rate, defined as the fraction of particles of size x0 that break in a specific time unit. v(x0 ) is the average number of particles formed from the breakup of particle x0 . Assuming one breakup at a time for a single particle, i.e., binary breakup, the value of this function would be v(x0 ) ¼ 2. P(xjx0 ,t) is defined as the probability density of particles of size x which come from the breakup of particles of size x0 at time t. It is also called stoichiometric Kernel, and represents the distribution of sizes of chains generated from the breakup of particles of size x0 . When the breakup occurs randomly in the chain, the Kernel would be P(x,x0 ) ¼ 1/x0 as described by some authors [4]. There are different methods to solve the integro-differential equation that describe the PBE. Ramkrishna [6] describes several methods that may be applied depending on the characteristics of each system. For pure fragmentation, the moments of the density function may be used as a tool for solving PBE. Some authors [6] have described the moment solution assuming no particle growth and considering binary fragmentation n(x) ¼ 2. The reaction mechanism for thermoxidative degradation of polymer with its different kinetics constant is summarized in the next scheme of equations. Thermal degradation: k0
PðxÞ!Pðx0 Þ þ Pðx x0 Þ
(8)
Initiator decomposition: k1
ROOH!RO$ þ · OH Hydrogen abstraction by the primary radicals:
(9)
where x and x0 represent two general molecular weights with (x0 < x) From these equations the PBE can be constructed through Equation (7). Naming p(x,t) the density function of polymer in a specific time t; the PBE for thermal degradation can be represented by:
vpðx; tÞ ¼ k0 ðxÞpðx; tÞ þ 2k0 ðxÞ vt
Z∞
pðx0 ; tÞPðxjx0 Þdx0
(14)
x
Now, for degradation reaction with the added degradation agent or hydroperoxide, where p(x,t) and r(x,t) express the polymer and macro-radical concentration in time t respectively, the PBE for both macro-radical and polymer chains can be represented as:
vpðx; tÞ ¼ k2 xCRO$ ðtÞpðx; tÞ k3 xCHO$ ðtÞpðx; tÞ k9b ðxÞpðx; tÞ vt Z∞ 1 þ k9a ðxÞrðx; tÞ þ k4 x0 rðx0 ; tÞ 0 dx k0 xpðx; tÞ x x
Z∞ þ 2k0
00
pðx0 ; tÞdx
x
(15) vrðx; tÞ ¼ k2 xCRO$ ðtÞpðx; tÞ þ k3 xCHO$ ðtÞpðx; tÞ þ k9a xpðx; tÞ vt Z∞ 1 k9b xrðx; tÞ þ k4 x0 rðx0 ; tÞ 0 dx0 k4 xrðx; tÞ x x
(16)
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ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
3. Experimental 3.1. Materials The polymer employed in this work was a cold polymerized Styrene butadiene rubber SBR from ISP (SBR-8113). This polymer was purified by solvent sohxlet extraction with toluene and then precipitated with 1-propanol and dried in a vacuum oven at 40 C. The mineral oil, Vassa LP-90 was used as a solvent. It is a 100% paraffinic solvent composed of C6eC22 molecules. This solvent was filtrated twice with a 0.2 mm Millipore®. An aromatic solvent was also used, named COPESOL, from Pequiven S.A. This solvent is a mixture of poly-substituted monoaromatic molecules of the alkybenzene type. Cumene hydroperoxide 80% (Trigonox K-80) from Akzo Nobel was also used with an activation energy (Ea) of 132.56 kJ/mol and an Arrhenius constant (A) of 1.15 1012 reported by the supplier. Nitrogen 98% was used to purge the reaction. 3.2. Degradation reaction The degradation assembly consists on a three-neck flask with a jacket connected to a recirculation bath for maximum temperature control and magnetic stirring. A controlled nitrogen in-out system was installed with a septum plug for needle sampling. The purified SBR was dissolved in the mineral oil at 40 C and at a concentration of 2 wt% for at least 48 h prior to the degradation reaction. Then the nitrogen purge system was mounted and the temperature was raised to the desired value. Time zero was considered when the recirculation bath reached the defined temperature. The hydroperoxide was added right before rising the temperature. Samples of 1 cm3 were taken during the reaction at different times and were dissolved in the elution solvent (in a 1:1 volume ratio) prior to the injection in the SEC-MALS. The degradation was performed without hydroperoxide at temperatures of 60 , 100 and 120 C. For 100 C, three different concentrations of cumene hydroperoxide were employed: 1.3 103 M (0.50 wt%); 8.41 104 M (0.32 wt%) and 7.36 104 M (0.28 wt%). For 120 C, two hydroperoxide concentrations were studied: 1.6 103 M (0.20 wt%) and 5.26 104 M (0.60wt %). For solutions containing 1.3 103 M (0.50 wt%) cumene hydroperoxide, three temperatures were tested: 75, 85 and 100 C. Finally SBR was dissolved in three mixtures with aromatic solvent, Vassa:Copesol (9:1; 8:2 and 7:3). The hydroperoxide concentration was kept at 0.50 wt% and the temperature was fixed at 100 C.
with the mineral oil, because the chromatography columns allow the separation of the polymer chains from the mineral oil in the sample. The oil signal eluded at higher times. No difference was observed when the elution solvent was Copesol or toluene. 4. Experimental results 4.1. Reproducibility and repeatability SBR used in this study was a heterogeneous industrially processed polymer. Before any degradation was induced, a solution of SBR was prepared and purified. At time zero, where no temperature had been applied, no differences in molecular mass should be appreciated in between different samples of the same SBR batch employed here. Nevertheless, Fig. 3 shows the large differences found for 25 similar SBR samples at time zero. These tests include two or three runs of the same sample, i.e., reproducibility and repeatability are considered in the same figure. The initial average Mw for all samples at initial time is shown in the inner table of Fig. 3. The high values of the standard deviation represent the heterogeneity of the sample. These errors have to be taken into account during the degradation analysis. It is important to point out that SEC-MALS technique is an absolute method to determine Mw, and then the conformational changes due to temperature or solvent interaction with polymer chain do not affect the results. Neither the concentration changes because of non-soluble fraction nor the polymer retention in the columns would change the results, because the equipment is able to detect absolute concentration. This is why those differences are assumed to be just a matter of sample heterogeneity. 4.2. Degradation results Fig. 4 shows an example of the change in the molecular weight distribution with time during degradation reaction with hydroperoxide and temperature. It can be seen that the molecular weight decreases asymptotically until it reaches a minimum value (see the inset in Fig. 4). This decrease is important in comparison to the values of the standard deviation observed. These results suggest that degradation process might be proportional to molecular weight as predicted. From the distribution it is possible to observe the decrease of polydispersity. This behavior is opposite to what it could be
3.3. Characterization procedures A size exclusion chromatograph (SEC) coupled with a multiangle light scattering (MALS) detector and a refractometer was used to determine the molar mass distribution. Gel permeation chromatography (GPC) columns were used as SEC device, with Mixed C and Mixed D and linear columns from Variant. An isocratic Agilent 1200 pump was used. The elution solvent was either COPESOL or filtrated HPLC grade toluene. The MALS system employed was a DAWN EOS (Wyatt Technology) coupled with a dn/dc refractometer Optilab-Rex (Wyatt Technology) to determine concentration. All measurements were carried out at 25 C and data were processed by ASTRA V software from Wyatt Technology. The dn/dc value for the SBR in the aromatic solvent was 0.0478 ± 0.0009 mL/g and 0.0456 ± 0.0011 mL/g determined in batch. This value was not affected by the mixture of the polymer
Fig. 3. Reproducibility and repeatability tests for SBR samples in solution at initial time (before degradation). The middle solid line represents the average Mw and dash lines represent one standard deviation.
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
expected for random scission. During degradation, all molecules tend to a specific molar mass and then the distribution broadness also decreases. This can be explained by the asymptotic behavior of the molecular weight with time. The degradation reaction tends to stop after reaching some minimum molecular weight value. Some authors have numerically simulated this behavior as single frequency distribution function dynamics, considering that the degradation stops after some minimum molecular weight and totally random kernel. The initial frequency distribution function decreases exponentially, generating a product peak which eventually tends to a Dirac delta function for long times [10]. Fig. 5 shows the results of three thermal degradation experiments without the use of hydroperoxide at different temperatures. It can be seen that for 60 and 100 C there is a slight increase in molecular weight that falls within the standard deviation shown in Fig. 3. There was no evidence of Mw increment during the chromatography results. From the practical point of view, in this work, this increase will be considered negligible. On the other hand, for 120 C, the reduction in molecular weight is significant. Thus, there is a critical temperature at which thermal degradation begins to be noticed. Some authors suggest chain scissions will take place if the heat exceeds the dissociation energies of chemical bonds [11]. Degradation reactions with cumene hydroperoxide were carried out at different temperatures in an anaerobic environment, at a fixed concentration of cumene hydroperoxide (0.5%). Fig. 6 shows the degradation of SBR polymer chains in the presence of a constant concentration of hydroperoxide. For 75 C, degradation is negligible. Changes in molecular weight start to be significant at 85 C; at this temperature, the Mw value reaches 50% of its initial magnitude in 25,000 s. At 100 C, the polymer chains reach an asymptotic behavior at approximately 30% of the initial weight average molecular weight after 10,000 s. Degradation reactions were also performed at different concentrations of cumene hydroperoxide, at two different temperatures. At 100 C three different concentrations of the hydroperoxide were evaluated, while for 120 C only two concentrations were explored. Fig. 7 shows the degradation behavior of SBR as a function of time for all hydroperoxide concentrations. Once more, an asymptotic behavior was observed for the molecular weight reduction as a function of time. The degradation extend was large, since the samples exhibited Mw values close to less than 10% of the molecular weight of the undegraded samples. It can be also seen
Fig. 4. Distribution of molecular weights for a degradation reaction of a SBR sample in solution with 0.6% of hydroperoxide at 120 C. The inset shows how the Mw changes with reaction time.
163
Fig. 5. Thermal degradation experiments of SBR in solution for three different temperatures, without the use of hydroperoxide. The change in Mw was normalized with respect to the initial Mw of the sample.
that no significant differences in degradation can be appreciated for the different hydroperoxide concentrations (within the errors involved in the measurements). Nevertheless, it is clear that at temperatures of 100 C and 120 C, the degradation kinetic is accelerated by the addition of hydroperoxide. The effect of an alkylbenzene, as a radical transfer agent, was also evaluated at different concentrations and in the presence of the hydroperoxide. If the degradation reaction depends on hydrogen abstraction from a radical, the presence of a transfer agent should retard the degradation reaction by retaining and stabilizing radicals. Fig. 8 shows how the addition of different concentrations of the transfer agent retards significantly the degradation reaction. This is an important result in order to be able to control the degradation reaction. 5. Discussion of the results By the use of spectroscopic methods, it was not possible to follow the degradation reaction through the quantification of a chemical change in the sample. Hydroperoxide consumption was
Fig. 6. Degradation of SBR in solutions with 0.50% of cumene hydroperoxide at three different temperatures.
164
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
Fig. 8. Degradation of SBR in solution with different concentrations of an aromatic solvent (Alkylbenzene or Copesol). The experiments were performed at 100 C and with a concentration of hydroperoxide of 0.5%.
Fig. 7. SBR degradation at different hydroperoxide concentration. a): 100 120 C.
C.
b):
measured by titration and no difference was observed as compared to the homolitic decomposition of the hydroperoxide itself. The appearance or disappearance of functional groups was not enough to be quantified by available methods in our laboratory. The balance between peroxide, peroxy radicals, and polymer molecular weight should rule the time evolution of the degradation process. The hydroperoxide decomposition implies the formation of two radicals of different kinds, a hydroxy radical and a cumyloxy radical. The decomposition rate is of the Arrhenius type. In terms of the molar concentration of peroxide, c, the dissociation rate is given by kic where the coefficient ki is governed by a first order kinetics, ki ¼ Aexp(Ea/RT). This constant was calculated for different temperatures as shown in Table 1 using the values of the activation energy and the value of A, given by the supplier. Regarding the kinetic study of polymer degradation, many authors [4,9,12e20] have generated mathematical functions to describe changes in molecular weight distribution through population balance equations, considering in some cases the initiation, propagation, transfer, b scission and termination reactions. To achieve this it is necessary to neglect some phenomena, such as density changes, crosslinking or aggregation, diffusion of species or Trommdorf effect. The integro-differential relationships for the PBE are solved in terms of molecular weight using two main approximations: the long chain approximation (LCA) where initiation and termination reactions are infrequent events, thus are neglected in comparison
to hydrogen abstraction reactions. The other approximation is the quasi-steady state approximation (QSSA), where the change of radical concentration with time is too small as compared to the change of other components and could also be neglected [21]. The changes in the molecular weight also show that the degradation process is not due to a depolymerization process, but a consequence of some chain fragmentation. For a depolimerization process, no change in molecular weight distribution should be detected and the decrease in the average molecular weight would be smoother. These kinds of fragmentations represented by random scissions should be described by PBE employing the stoichiometric Kernel as the inverse function, as suggested in the literature [12], where P(xjx0 ) ¼ 1/x0 . High molecular weight chains degrade faster than short chains and this is related to the number of hydrogen atoms available for abstraction, i.e., the larger the number of hydrogen atoms present in the chain, the higher the probability of a hydrogen abstraction reaction. This behavior has been predicted by some authors [12,22,23] for hydrogen abstraction degradation mechanisms. From PBE, this behavior implies that function b(x) should be proportional to the molecular weight. The break function has been reported as linearly dependent on the molecular weight b(x) ¼ kx or even exponentially b(x) ¼ kxr [12]. Studies on the molecular weight influence on degradation indicate that random-scission rate coefficient dependence on molecular weight is linear for short chains, but second order for longer chains [23]. It may also depend on other variables, such as temperature, concentration or reaction type. A simplification frequently used in PBE is binary fractionation v(x) ¼ 2, which assumes that one chain can only suffer a single fragmentation at a time, i.e., non simultaneous fragmentation for the same molecule is considered.
Table 1 Kinetic constants for Cumene hydroperoxide decomposition for different temperatures. Temperature ( C)
ki (1/s)
75 85 95 100 105 120
1.49 5.33 1.79 3.19 5.62 2.81
108 108 107 107 107 106
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
The general degradation reaction can be divided into different reactions assuming some mechanism. After decomposition of the ROOH hydroperoxide, the resulting OH hydroxy and RO cumyloxy radicals, which are not stable, abstract hydrogens from the polymer chain P(x), generating polymer radicals or macroradicals R(x), where x represents the molecular weight as a continuous variable. 5.1. PBE for thermal degradation Assuming linear dependence of the degradation reaction with the molecular weight, PBE for Equation (14) in the case of thermal degradation can be written as:
vpðx; tÞ ¼ k0 xpðx; tÞ þ 2k0 vt
Z∞
pðx0 ; tÞdx0
(17)
x
Applying the moment operation, i.e., multiplying by xn and integration in the semi-infinite interval, the following integrodifferential equation is obtained:
vpn ðx; tÞ vt
Z∞ ¼ k0 pnþ1 ðx; tÞ þ 2k0
2 pðx0 ; tÞ4
x
Zx 0
Rearranging and knowing that pn ðx; tÞ ¼
Z
3 xn dx5dx0 ∞
Fig. 9 shows the results of applying these equations to the thermal degradation at different temperatures. It can be seen that for 100 C and 60 C the slopes are negative and very close to zero. This behavior is related to that shown in Fig. 5. For this work, these constants can be considered zero and it can be assumed that the negative values of the slope are related with experimental errors. On the other hand for 120 C, Fig. 9 shows that the value of the slope is significant, then, the thermal degradation reaction should be taken into account with a value of k0 ¼ 2.79 1010 s1. 5.2. PBE for degradation with hydroperoxide For degradation reaction with the hydroperoxide, in Equation (15) and Equation (16) the hydrogen abstraction from primary radicals (CRO) and (CHO) could be considered as a single process of hydrogen abstraction from C(t) with a kinetic constant k23. Assuming the stoichiometric Kernel as the inverse function P(xjx0 ), ¼1/x0 and applying the moment operation to Equation (15) and Equation (16):
vpn ðx; tÞ ¼ k23 CðtÞ vt
(18)
Z∞
Z∞ xnþ1 pðx; tÞ k9b 0
Z∞ xnþ1 rðx; tÞ þ k4
x pðx; tÞdx:
0
0
n
vp ðx; tÞ ðn 1Þ ¼ k0 pnþ1 ðx; tÞ vt ðn þ 1Þ
nþ1
k0 p
p0 ðtÞ p0 ðt ¼ 0Þ ¼ k0 p1 ðtÞt p0 ðtÞ 1
p ðtÞ
p0 ðt ¼ 0Þ 1
p ðtÞ
¼ k0 t
1 1 ¼ k0 t Mn ðtÞ Mn0
x
0
2
pðx ; tÞ4 0
1 x0 rðx0 ; tÞ 0 dx x Zx
3 x dx5dx0 n
(21)
(22)
0
vpn ðx; tÞ ¼ k23 CðtÞpnþ1 ðx; tÞ k9b pnþ1 ðx; tÞ þ k9a r nþ1 ðx; tÞ vt k ðn 1Þ þ 4 r nþ1 ðtÞ k0 pnþ1 ðx; tÞ ðn þ 1Þ nþ1
(20)
Then Ns can be defined as:
Ns ¼
ðx; tÞ þ 2k0
(19)
For the zero moment, n ¼ 0 substituting in the Equation (19):
Z∞ xn
0
Z∞
n
vp ðx; tÞ 2k0 nþ1 0 ¼ k0 pnþ1 ðx; tÞ þ p ðx ; tÞ vt nþ1
xnþ1 pðx; tÞ 0
Z∞ þ k9a
n
165
(23) vr n ðtÞ ¼ k23 CðtÞpnþ1 ðx; tÞ þ k9b pnþ1 ðtÞ k9a r nþ1 ðtÞ vt n k r nþ1 ðtÞ nþ1 4
(24)
On the other hand, for the hydroperoxide decomposition, the concentration of hydroperoxide with time is given by:
CROOH ðtÞ ¼ CROOH ð0Þexpk1 t
(25)
The PBE expressed in terms of the primary radical concentration are given by:
vCRO$ ðtÞ ¼ k1 CROOH ðtÞ CRO$ ðtÞ vt
Z∞
vCHO$ ðtÞ ¼ k1 CROOH ðtÞ CHO$ ðtÞ vt
k2 ðxÞpðx0 ; tÞdx0
(26)
0
Z∞
k3 ðxÞpðx0 ; tÞdx0
(27)
0
Assuming that the initiation is much faster than the hydrogen abstraction, i.e., k1 >> k3, then Equation (26) and Equation (27) can be rearranged as:
Fig. 9. Calculation of the thermal degradation kinetic constants by PBE for three different temperatures.
vCRO$ ðtÞ vCHO$ ðtÞ ¼ ¼ k1 CROOH ðtÞ ¼ k1 CROOH ð0Þexpk1 t vt vt Solving Equation (28) leads to:
(28)
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
166
CRO$ ðtÞ ¼ CHO$ ðtÞ ¼ CðtÞ ¼ CROOH ð0Þ 1 expk1 t ¼ C0 1 expk1 t
(29)
Considering quasi-stationary state, the macroradical concentration with time does not change significantly, then:
vr n ðtÞ vr 0 ðtÞ vrðtÞ ¼ ¼ ¼0 vt vt vt
(30)
Replacing in Equation (24) and solving for r1(t):
r 1 ðtÞ ¼
k23 CðtÞ þ k9b 1 p ðtÞ k9a
(31)
and substituting Equation (31) in Equation (23) for zero moment (n ¼ 0):
vp0 ðx; tÞ p1 ðtÞk4 ¼ ðk23 CðtÞ þ k9b þ k0 Þ vt k9a
(32)
Now, because the initial conditions are known, and assuming that the first moment is invariable, i.e., the total mass does not change with time, Equation (32) can be solved by integration in time as:
Zt
Zt vp0 ðx; tÞ ¼
0
0
p1 ðtÞk4 ðk23 CðtÞ þ k9b þ k0 Þvt k9a
" k4 p1 ðtÞ k C ðk23 C0 þ k9b þ k0 Þt 23 0 p ðtÞ p ðt ¼ 0Þ ¼ k9a k1 # k t 1 k C e þ 23 0 k1 0
(33)
0
(34)
Assuming no thermal degradation, k0 might be neglected then Equation (34) can be expressed in terms of Ns as:
1 1 Mn ðtÞ Mn0 k4 k23 C0 k4 k9b k k C k k C t þ 4 23 0 ek1 t 4 23 0 ¼ þ k9a k9a k9a k1 k9a k1
NsðtÞ ¼
(35)
Denoting Ka ¼ k4k23/k9a and Kb ¼ k4k9b/k9a, the following relation is obtained:
NsðtÞ ¼ ðKa C0 þ Kb Þt þ
Ka C0 k1 t Ka C0 e k1 k1
Another simplification that hydrogen abstraction from the faster than macroradical chain Ka >> Kb, then the Equation (35)
NsðtÞ ¼
Fig. 10. Ka calculation for SBR degradation at 100 C and different concentrations of hydroperoxide from Equation (38), the dashed line is the fitting until 15000 s. The solid line represents the fit until 23000 s.
while the solid line, fitted the curve for longer times and leads to a Ka value of 3.88 104 with a R2 of 0.83235. Performing the same approximation but using Equation (36) directly, it is possible to calculate Ka and Kb at the same time, as shown in Fig. 11. Results from this approximation can be detailed in Table 2. At first glance, it can be confirmed that the order of magnitude of Ka is higher that Kb, in accordance to the approximation given by Equation (38). On the other hand, the difference observed in those fittings may arise from the rather large experimental errors involved, as shown in Fig. 3. For higher temperatures, i.e., 120 C, the kinetic constant for thermal degradation k0 should not be neglected, thus it should be counted as included in the Kb constant. A higher deviation from the theoretical approximation was observed for 120 C, as can be seen in the R2 values of the curve fittings. Fig. 12 shows the curve fittings for the SBR degradation at 120 C. Finally, Equation (36) was used to theoretically calculate the retardation of the reaction with addition of alkylbenzene as transfer agent. The aromatic group can stabilize a secondary radical in the a carbon of the aromatic ring through electron induction effect, involving another hydrogen abstraction step and additional
(36)
can be done is assuming that primary radicals is significantly transfer, that is, k23 >> k9b or is simplified to:
k4 k23 C0 k k C k k C t þ 4 23 0 ek1 t 4 23 0 k9a k9a k1 k9a k1
(37)
Considering that for the same temperature, all kinetic constants should be the same, then, changing the concentration of hydroperoxide allows calculating the Ka constant.
NsðtÞ 1 1 ¼ Ka t þ ek1 t C0 k1 k1
(38)
Fig. 10 shows the results for this approximation. It can be seen that deviation of the theoretical behavior occurs after 15000 s. For the dashed line, the value of Ka is 6.65 104 with a R2 of 0.90616,
Fig. 11. Ka and Kb calculation for SBR degradation at 100 C and different concentrations of hydroperoxide from Equation (36).
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna Table 2 Curve fitting for kinetic constants calculation. Ka ª 100 C until 25000 s ª100 C until 15000 s b 100 C 0.28% Equation (36) b 100 C 0.32% b 100 C 0.50% b 120 C 0.20% b 120 C 0.60% b 100 C 10% Alkylbenzene b 100 C 20% Alkylbenzene a b
3.88 6.65 1.67 6.32 1.06 1.16 7.65 6.67 7.10
104 104 104 106 106 105 105 1025 106
Kb
R2
0 0 1.23 1.09 4.06 1.05 1.17 7.10 1.00
0.83235 0.9062 0.9906 0.9804 0.9334 0.7427 0.8735 0.9617 0.9273
107 109 1015 109 109 1011 1022
Calculated from Equation (38). Calculated from Equation (36).
167
magnitude of the shift. The larger shifting effect occurs for Kb, which means that the reaction of Equation (12) is the most affected by the addition of the transfer agent. From this set of experiments, the experimental error or the raw polymer heterogeneities were too high; then, the calculation of the kinetic constant has been compromised. Nevertheless, some results from this experimental work can be highlighted as follows: No thermal degradation was observed below 100 C, but for 120 C, the effect of temperature starts to be significant. Addition of an initiator like cumene hydroperoxide leads to a significant increase in polymer degradation. In the concentration range of hydroperoxide studied, no significant differences were found for the induced degradation Addition of a transfer agent, such as an akylbenzene in the solvent, retards significantly the degradation process. 6. Conclusions
Fig. 12. PBE approximation for SBR degradation at 120 C.
intermediate radical species to the general reaction. Radicals in solvent molecules are more stable than on the polymer chain, and this effect retards the reactions of Equation (10), Equation (11) and Equation (12) shifting both k23 and k4 to lower values. Similar results have been obtained for polystyrene degradation [24] where hydrogen donor solvents decreased the polymer degradation. For Fig. 13 shows the fitting from Equation (36) and the results are summarized in Table 2. Despite the fact that no clear trend can be drawn from the kinetic constant, it is important to point out the
In this work, the characterization of the degradation reaction of SBR in solution was studied by the changes in molecular weight and its distribution, despite the fact that chemical changes were not detectable by spectroscopic techniques. The population balance equation analysis is an important tool to quantify the degradation process from the changes in molecular weight distribution. Despite large experimental errors, it can be concluded that the degradation of SBR in solution in the absence of oxygen occurs mainly by random chain scission events which are dependent on the molecular weight. The primary radicals from the hydroperoxide decomposition lead to the formation of macro-radicals in the main chain, which may generate new smaller chains from their scission. The results confirm the hypothesis of the degradation mechanism expected for SBR. A simple transfer agent as an aromatic solvent can be used during specific applications to retard SBR degradation, either to retard the SBR degradation at high temperatures or to control the degradation kinetics. Acknowledgments The authors would like to thank to PDVSA Intevep, for funding through the ID-6085-0000 project in the Department of Well Productivity. Also FONACIT (Science and Technology Ministry of the Bolivarian Republic of Venezuela) and Ministry of Foreign Affairs (France) for funding received through project: PCP 2011001409 (Postgraduate Cooperation Project). References
Fig. 13. Curve fitting for the SBR degradation with 0.50% of hydroperoxide at 100 C and different concentrations of alkylbenzene in the solvent.
[1] Chiantore O, Tripodi S, Sarmoria C, Valles E. Mechanism and molecular weight model for thermal oxidation of linear ethyleneebutene copolymer. Polymer 2001;42:3981e7. [2] Jiang DD, Levchik GF, Levchik SV, Wilkie CA. Thermal decomposition of crosslinked polybutadiene and its copolymers. Polym Degrad Stab 1999;65: 387e94. [3] Gordon Cameron G, Meyer J, McWalter I. Thermal degradation of polystyrene. 3. A Reappraisal. Macromolecules 1978:696e700. [4] Kim Y-C, McCoy BJ. Degradation kinetics enhancement of polystyrene by peroxide addition. Ind Eng Chem Res 2000;39:2811e6. [5] Madras G, Chattopadhyay S. Optimum temperature for oxidative degradation of poly(vinyl acetate) in solution. Chem Eng Sci 2001;56:5085e9. [6] Ramkrishna D. Population balances: theory and applications to particulate systems in engineering. San Diego, CA. USA: Academic Press; 2000. p. 355. [7] Tobita H. Simulation model for the modification of polymers via crosslinking and degradation. Polymer 1995;36:2585e96. [8] Chodak I, Zimanyova E. The effect of temperature on peroxide initiated crosslinking of polypropylene. Eur Polym J 1984;20:81e4. [9] Karmore V, Madras G. Kinetics for thermal degradation of polystyrene in presence of p-toluene sulfonic acid. 2002. p. 117e21.
168
ez et al. / Polymer Degradation and Stability 111 (2015) 159e168 O. Verna
[10] McCoy Benjamin J, Wang M. Continuous-mixture fragmentation kinetics: particle size reduction and molecular cracking. Chem Eng Sci 1994;49: 3773e85. [11] Schnabel W. Polymer degradation, principles and practical applications. 1981. p. 192e3. New York. [12] Mccoy B. Continuous kinetics of cracking reactions: thermolysis and pyrolysis. Chem Eng Sci 1996;5:2903e8. [13] Madras G, Smith J, McCoy B. Thermal degradation of poly (a-methylstyrene) in solution. Polym Degrad Stab 1996;52. [14] McCoy BJ, Madras G. Degradation kinetics of polymers in solution: dynamics of molecular weight distributions. AIChE J 1997;43:802e10. [15] Kodera Y, McCoy BJ. Distribution kinetics of radical mechanisms: reversible polymer decomposition. AIChE J 1997;43:3205e14. [16] Madras G, McCoy BJ. Oxidative degradation kinetics of polystyrene in solution. Chem Eng Sci 1997;52:2707e13. [17] Madras G, Smith JM, McCoy BJ. Thermal degradation kinetics of polystyrene in solution. Polym Degrad Stab 1997;58:131e8.
[18] Madras G, McCoy BJ. Time evolution to similarity solutions for polymer degradation. AIChE J 1998;44:647e55. [19] Sterling WJ, McCoy BJ. Distribution kinetics of thermolytic macromolecular reactions. AIChE J 2001;47:2289e303. [20] Karmore V, Madras G. Thermal degradation of polystyrene by Lewis acids in solution. Ind Eng Chem Res 2002;41:657e60. [21] Smagala TG, Mccoy BJ. Mechanisms and approximations in macromolecular reactions: reversible initiation, chain scission, and hydrogen abstraction. 2003. p. 2461e9. [22] Madras G, McCoy BJ. Kinetics and dynamics of gelation reactions. Chem Eng Sci 2007;62:5257e63. [23] Madras G, Chung GY, Smith JM, Mccoy BJ. Molecular weight effect on the dynamics of polystyrene degradation. 1997. p. 2019e24. [24] Sato S, Murakata T, Baba S. Solvent effect on thermal degradation of polystyrene 1990;40:2065e71.