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Ind. Eng. Chem. Res. 2001, 40, 1306-1311
KINETICS, CATALYSIS, AND REACTION ENGINEERING Degradation Kinetics for Polymer Mixtures in Solution Vishal Karmore and Giridhar Madras* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
The degradation rate of a polymer mixture depends on the particular polymer mixture, and the presence of a second polymer can increase, decrease, or not affect the degradation rate of the first polymer. The degradation rates of poly(vinyl acetate) in both the presence and absence of polystyrene at various temperatures (220-250 °C) were experimentally determined. The molecular weight distributions of the polymers were obtained by analyzing the samples by gel permeation chromatography. Continuous distribution kinetics were employed for determining the degradation rates of binary polymer mixtures. The results indicated that, because of the interaction of mixed radicals with polymer by hydrogen abstraction, the degradation rates of poly(vinyl acetate) are considerably enhanced in the presence of polystyrene, while the degradation rate of polystyrene decreases. Introduction There has been a lot of emphasis in the last few decades on the development of techniques for proper management of solid waste. With the advent of synthetic thermoplastics, polymers came to play an increasingly substantial role in society. Their use requires stability under severe conditions. This stability renders disposal very difficult. To manage such waste, polymeric material degradation is the focus of a great deal of current research. Direct combustion or landfilling has several associated environmental problems, and the cost of conventional disposal is likely to increase.1 Considerable research has focused on the reuse of solid plastic waste, and several methods, including pyrolysis, have been proposed for the recycling of plastics.2 Most of the research has addressed the degradation of single polymers. However, waste streams usually contain mixtures of polymers. Separation of these mixtures before recycling may be uneconomical.3 It is, therefore, necessary to understand the mechanisms of degradation of polymer mixtures and blends. Studies of the pyrolytic degradation of polymer mixtures have generated varied results. Some investigators3-5 have observed, during degradation, significant interaction between polystyrene and polyethylene, while others observed no interaction between these polymers.6,7 Investigators3,4 who have observed significant increases in the thermolytic degradation rate of polyethylene due to the presence of polystyrene attribute the interaction to hydrogen abstraction from polyethylene by polystyrene radicals. The degradation rate of polystyrene was enhanced 8-fold in the presence of poly(methyl acrylate) and poly(butyl acrylate) at 430 °C, while the degradation rates of poly(methyl acrylate) and poly(butyl acrylate) decreased 8-fold.8 It was also concluded that hydrogen abstraction plays an important * To whom correspondence should be addressed. Tel: 91080-3092321.Fax: 91-080-3600683.E-mail:
[email protected].
role in these polymer interactions. Photodegradation of a polystyrene-poly(vinyl acetate) blend is apparently influenced by interactions of radicals from both polymers resulting in an increased degradation rate for polystyrene.9 Studies have also been conducted on polypropylene and polyethylene mixtures.10 Degradation of polymers in solution has been proposed to rectify some of the problems faced in the commercial implementation of multiphase pyrolysis processes.11 The kinetics of degradation of pure polymers such as polystyrene,12 poly(styrene-allyl alcohol),13 poly(methyl methacrylate),14 poly(vinyl acetate),15 and polyethylene16 have been studied. The degradation in solution of a mixture of poly(R-methylstyrene) and polystyrene has been studied.17 In this system, the former polymer degrades by chain end scission while the latter degrades by random chain scission. The degradation of a mixture of polystyrene and poly(vinyl acetate), in which both the polymers degrade by random chain scission, will be described herein. A detailed radical mechanism based on Rice-Herzfeld reactions18,19 is postulated to explain the degradation kinetics of the polymer mixture. Because a polymer is a mixture of different size molecules, a molecular weight distribution (MWD) is needed to describe the polymer, and these distributions are characterized by the moments of the distribution of the molecules.20 A finite number of such molecules implies that the moments are the sums over the molecules in the polymer that constitute the distribution. These sums are more difficult to evaluate when compared to evaluating integrals. Therefore, continuous distribution kinetics is preferred to discrete kinetics for analyzing the time evolution of MWD of reacting polymers.21 The interaction between different polymers occurs by radicals abstracting hydrogen atoms from either polymer.3 The degradation rates for each polymer are then determined by applying distribution kinetics to the MWD. This new model is applied to determine the degradation rates for the polymers in the mixture.
10.1021/ie000820k CCC: $20.00 © 2001 American Chemical Society Published on Web 02/02/2001
Ind. Eng. Chem. Res., Vol. 40, No. 5, 2001 1307
Figure 1. MWDs for a polystyrene and poly(vinyl acetate) mixture.
Figure 2. Calibration curve based on polystyrene standards, log MW ) 9.57 - 0.0044 × retention time (in seconds).
Experiments
at 50 °C. The chromatograph was obtained by injecting (Rheodyne valve) 100 µL of the sample and monitoring the refractive index continuously with a differential refractometer (Waters R401). A calibration curve, obtained by injecting polystyrene standards (Figure 2), was used for converting the retention time to molecular weight. The peaks for both polystyrene and poly(vinyl acetate) from the reaction mixture were distinct, so that moments could be calculated by numerical integration.
Materials. Styrene and vinyl acetate monomers were obtained from Aldrich Chemical Co. The monomers were freed from inhibitor by washing with alkali and then distilling them under reduced pressure. Benzoyl peroxide was purified by dissolving in solvent (chloroform) and precipitating with a nonsolvent, methanol. Poly(vinyl acetate) and polystyrene were prepared by bulk polymerization with benzoyl peroxide as an initiator at 60 °C. Diphenyl ether and tetrahydrofuran (THF) were obtained from Merck and were distilled prior to use. Pretreatment of Polymers. Pretreatment of the polymers is necessary to obtain narrow fractions that are well separated by gel permeation chromatography (Figure 1). The fist peak in the figure corresponds to polystyrene. Polystyrene and poly(vinyl acetate) were dissolved in a solvent (acetone) and then reprecipitated with a nonsolvent (hexane). The obtained polymer was dried over calcium carbonate at room temperature until a constant weight was reached. Degradation Experiments. The thermal degradation of diphenyl ether solutions of a mixture of polystyrene [Mn ) 200 000 and polydispersity (PD) ) 1.3] and poly(vinyl acetate) (Mn ) 21 000 and PD ) 1.1) were carried out in three-neck, round-bottomed flask (equipped with a long condenser to ensure retention of volatiles) with continuous stirring. The study was done at various temperatures (220-250 °C). The temperature was controlled to (1 °C of the set point with a controller. The reactor was charged with 100 mL of diphenyl ether, and the temperature was raised to the desired reaction temperature. The desired amounts of poly(vinyl acetate) and polystyrene were then added to the reactor. The mass concentration of polystyrene was varied from 1 to 6 g/L, while the poly(vinyl acetate) mass concentration was kept constant at 1 g/L. Samples of 1 mL of the reaction mixture were taken at regular time intervals and dissolved in 1 mL of THF for subsequent analysis by HPLC-GPC. GPC Analysis. The samples are analyzed with an HPLC-GPC system (Waters). The eluent was THF at a flow rate of 1 mL/min. Three columns (Waters HR 4, HR 3, and HR 0.5) (300 × 7.5 mm), packed with crosslinked polystyrene-divinylbenzene, were used in series
Theoretical Model Continuous distribution kinetics were applied to determine the rate coefficients for reactions involving the two polymers. The challenge in modeling degradation kinetics arises from the complexity of mixtures of molecules undergoing several elementary steps including bond fission, hydrogen transfer, and disproportionation. Quantitative modeling of these processes requires its formalism, and the elementary steps mentioned above can be organized into a general Rice-Herzfeld kinetics comprising initiation, propagation, and termination steps. Continuous-distribution mass balances were written for the various reactions based on the Rice-Herzfeld mechanism18,19 for degradation. The rate coefficients are assumed to be independent of molecular weight, a reasonable assumption at low levels of degradation.20 The integrodifferential equations obtained from the mass balances can be solved for molecular weight moments. The governing equations are then solved by making two assumptions:18,19 the long-chain approximation (LCA) that postulates that the initiation and termination rates are negligible and the quasisteady-state approximation (QSSA) that applies when the rate of change of radical concentration is extremely small. Polymers can undergo degradation by a wide variety of mechanisms, including chain end scission and random chain scission. A formulation has been derived to correspond to one polymer undergoing chain end scission and another polymer undergoing random chain scission.17 Polymers such as polyethylene, polypropylene, and polystyrene degrade predominantly by random chain scission. Both polystyrene and poly(vinyl acetate) un-
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dergo degradation predominantly by random chain scission.11,15 For this investigation, a kinetic model for the case when both polymers degrade by random chain scission and interact with each other has been developed. The degrading polymers, poly(vinyl acetate) (A) and polystyrene (B), are represented as PA(x) and PB(x) and their radicals as RA•(x) and RB•(x), respectively. Their MWDs are pA(x,t) and pB(x,t) and rA(x,t) and rB(x,t), respectively. The initiation-termination reactions are kfA
PA(x) y\ z RA•(x′) + RA•(x-x′) k
(1)
tA
∂ri/∂t ) kd
∫0xpA(x′) rB(x-x′) dx′ + x kd∫0 pB(x′) rA(x-x′) dx′ - 2kDri(x)
Applying the moment operation, ∫∞0 [ ]xn dx, to each mass balance equation yields
dpA(n)/dt ) -khApA(n) + kHArA(n) + kbArA(n)/(n + 1) kdpA(n)rB(0) + kDri(n)/(n + 1) (13) dpB(n)/dt ) -khBpB(n) + kHBrB(n) + kbBrB(n)/(n + 1) kdpB(n)rA(0) + kDri(n)/(n + 1) (14)
kfB
PB(x) y\ z RB•(x′) + RB•(x-x′) k
(2)
tB
drA(n)/dt ) khApA(n) - kHArA(n) - kbArA(n) + kbArA(n)/ (n + 1) - kdpB(0)rA(n) + kDri(n)/(n + 1) (15)
The reversible hydrogen abstraction reactions are khA
z RA•(x) PA(x) y\ k
(3)
HA
khB
PB(x) y\ z RB•(x) k
(4)
HB
drB(n)/dt ) khBpB(n) - kHBrA(n) - kbBrB(n) + kbBrB(n)/ (n + 1) - kdpA(0)rB(n) + kDri(n)/(n + 1) (16) n
dri(n)/dt ) kd
n
(nj )re(n-j)pA(j) + kd∑(nj )r(n-j)pB(j) ∑ j)0 j)0
The depropagation reactions are
2kDri(n) (17)
kbA
RA•(x) 98 PA(x′) + RA•(x-x′)
(5)
kbB
RB•(x) 98 PB(x′) + RB•(x-x′)
(6)
The interaction between the polymers can be written as kd
•
•
(12)
kD
•
RB (x) + PA(x′) y\ z Ri (x+x′) y\ z PB(x) + RA (x′) k k
On the basis of the QSSA assumption, expressions for the radical concentrations can be obtained by equating the expressions of the zeroth moments (n ) 0 in eqs 1517) to zero. Equation 17 yields
ri(0) ) kd(rB(0)pA(0) + rA(0)pB(0))/2kD
(18)
Equations 15 and 16 give
(7)
rA(0) ) (khApA(0) + kDri(0))/(kHA + kdpB(0))
(19)
The intermediate complex, Ri•, facilitates the formulation of the population balance equations for the reversible disproportionation. On the basis of the LCA assumption, the initiation and termination steps are neglected and the population balance equations17are
rB(0) ) (khBpB(0) + kDri(0))/(kHB + kdpA(0))
(20)
D
d
∫x∞rA(x′)
∂pA/∂t ) -khApA(x) + kHArA(x) + kbA
∫x∞ri(x′) Ω(x,x′) dx′
Ω(x,x′) dx′ - kdpArB(0) + kD
(8)
∫x∞rB(x′)
∂pB/∂t ) -khBpB(x) + kHBrB(x) + kbB
∫x∞ri(x′) Ω(x,x′) dx′
Ω(x,x′) dx′ - kdpBrA(0) + kD
(9)
∂rA/∂t ) khApA(x) - kHArA(x) - kbArA(x) +
∫x rA(x′) Ω(x,x′) dx′ - kdpB rA(x) + ∞ kD∫x ri(x′) Ω(x,x′) dx′
kbA
∞
dpA(0)/dt ) kbArA(0)
(21)
dpB(0)/dt ) kbBrB(0)
(22)
Expressions for rA(0) and rB(0) can be obtained by solving eqs 19 and 20 simultaneously with eq 18. The result is substituted in eqs 21 and 22 to obtain the time variation of the polymer molar concentration. The first moments can also be determined in a similar fashion.
dpA(1)/dt ) dpB(1)/dt ) 0
(23)
Equation 23 indicates that the polymer mass remains constant.
(0)
(10)
∂rB/∂t ) khBpB(x) - kHBrB(x) - kbBrB(x) +
∫x∞rB(x′) Ω(x,x′) dx′ - kdpA(0)rB(x) + ∞ kD∫x ri(x′) Ω(x,x′) dx′
Substituting eqs 18-20 in eqs 13 and 14 gives
kbB
(11)
Results and Discussion The hypothesized interaction of degrading polymers is through generated radicals and their rates of hydrogen abstraction. When kd ) kD ) 0, the two polymers react independently and the moment eqs 13-17 are identical to those derived for a single polymer undergoing random chain scission.17 Equations 21 and 22 show the dependence of the degradation rate of a
Ind. Eng. Chem. Res., Vol. 40, No. 5, 2001 1309
Figure 3. Semilog plot of Mn0A/MnA versus reaction time, t, for poly(vinyl acetate) at different temperatures to obtain the rate coefficient kA (by eq 26). Legend: 1, 250 °C; 2, 240 °C; b, 230 °C; 9, 220 °C.
polymer on the concentration of the other polymer in the mixture. The degradation rate equations for binary mixtures show that, depending on the particular polymer, the presence of another polymer in the mixture can increase, decrease, or have no effect on the degradation rate. Thus, the rate coefficient for random chain scission of poly(vinyl acetate) is a function of the concentration of polystyrene in the mixture through the fundamental radical rate parameters, which depend on the experimental conditions. In this study, degradation rates for poly(vinyl acetate) (A) and polystyrene (B) have been investigated at various temperatures (220-250 °C) as individual polymers and as mixtures in solution. The influence of the ratio of the mass concentration of polystyrene on the degradation rate of poly(vinyl acetate) at 220 °C has also been determined. The degradation rate of poly(vinyl acetate) in the absence of polystyrene can be readily determined by substituting eq 19 in eq 21 with rB(0) ) pB(0) ) 0, which yields
dpA(0)/dt ) kApA(0)
(24)
where kA ) kbAkhA/kHA. Equation 24 can be solved with the initial condition, pA(0) ) pA0(0), to yield
ln(pA(0)/pA0(0)) ) kAt
(25)
Because, by eq 23, the polymer mass (pA(1) ) pA0(1)) is constant and the number-average molecular weight is the ratio of the first moment to the zeroth moment (MnA ) pA(1)/pA(0)), eq 25 gives
ln(Mn0A/MnA) ) kAt
(26)
The expression for the degradation rate of polystyrene can be derived similarly.
ln(Mn0B/MnB) ) kBt
(26a)
Figures 3 and 4 show the variation of the numberaverage molecular weight of poly(vinyl acetate) and
Figure 4. Plot of ln(Mn0B/MnB) versus reaction time, t, for polystyrene at different temperatures to determine the rate coefficient of polystyrene kB (by eq 26a). Legend: 1, 250 °C; 2, 240 °C; b, 230 °C; 9, 220 °C.
polystyrene with time, respectively. The degradation rate coefficients of poly(vinyl acetate) and polystyrene (kA and kB, respectively) were obtained by linear regression from the slopes of plots of polymer molecular weight versus time. The degradation rate coefficients at 220, 230, 240, and 250 °C are 1.5 × 10-5, 4.3 × 10-5, 6.2 × 10-5, and 1 × 10-4 s-1 for poly(vinyl acetate) and 6.8 × 10-6, 9.0 × 10-6, 1.5 × 10-5, and 2.2 × 10-5 s-1 for polystyrene. These degradation rates correspond to the rates when the other polymer is not present in the mixture. Equations 21 and 22 have to be solved simultaneously to obtain the degradation rates of the individual polymers in the polymer mixture. It is necessary to obtain ri(0) in terms of the polymer concentrations by substituting eqs 19 and 20 in eq 18. This yields
ri(0) ) kdpA(0)pB(0)(khBkHAk1 + khAkHBk2)/ [kD(2kHAkHBk1k2 - kdkHBk2pB(0) - kdkHAk1pA(0))] (27) where k1 ) 1 + kdpB(0)/kHA and k2 ) 1 + kdpA(0)/kHB. Equation 27 is substituted into eq 19 to obtain rA(0), and this is substituted in eq 21 to obtain
dpA(0)/dt ) kApA(0)(1 + kdpB(0)k3/khA)/k1
(28)
where k3 ) (khBkHAk1 + khAkHBk2)/[kHBkHA(k1 + k2)]. This is the rate equation which governs the degradation rate of poly(vinyl acetate) in the presence of polystyrene. If the rates of the elementary steps are known, eq 28 can be directly solved. Reasonable assumptions described below, however, permit solutions for the poly(vinyl acetate) degradation rate in the polymer mixture. It is observed experimentally (in this study) that the degradation of polystyrene is negligible in the presence of poly(vinyl acetate) and eq 22 can be written as dpB(0)/ dt ) kbBrB(0) ) 0. Solving this with the initial conditions provides pB(0) ) pB0(0). On the basis of the experimental data for the thermal degradation of the pure polymers,12,15 it is also reasonable to assume kdpB(0)/kHA , 1 and kdpA(0)/kHB , 1; i.e., k1 = 1 and k2 = 1. These assumptions reduce eq 28 to
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dpA(0)/dt ) kApA(0)(1 + pB0(0)kdkhB/2kHBkhA) (29) Equation 29 can be solved to obtain
ln(pA(0)/pA0(0)) ) ln(Mn0A/MnA) ) kA(1 + k4pB0(1))t ) kABt (30) where k4 ) kdkhB/2Mn0BkHBkhA and kAB, as indicated by eq 30, is the degradation rate coefficient of poly(vinyl acetate) when polystyrene is present in the system. Figure 5 shows the variation of the molecular weight of poly(vinyl acetate) in the presence of polystyrene. It is apparent from Figure 5 that the degradation of poly(vinyl acetate) is highly enhanced in the presence of polystyrene. The degradation rate coefficient for poly(vinyl acetate) in the polymer mixture, kAB, is determined from the slope of the lines in Figure 5. The regressed values of kAB at 220, 230, 240, and 250 °C, when the mass concentration of polystyrene in the mixture is 1 g/L, are 3 × 10-5, 2.4 × 10-4, 4.3 × 10-4, and 6.9 × 10-4 s-1, respectively. These degradation rate coefficients are significantly higher than the rate coefficients observed in the absence of polystyrene in the mixture. For example, the degradation rate coefficient at 250 °C for poly(vinyl acetate) in the presence of polystyrene is nearly 7 times greater than the degradation rate of pure poly(vinyl acetate). The increase in the degradation rate is directly reflected by k4. From eq 30,
kAB/kA - 1 ) k4pB0(1)
Figure 5. Plot of ln(Mn0A/MnA) versus reaction time, t, for poly(vinyl acetate) in the polymer mixture to determine kAB (by eq 30) at various temperatures with pB0(1) ) 1 g/L and at various mass concentrations of polystyrene at 220 °C. Legend: (a) at (1) 250 °C, (2) 240 °C, (b) 230 °C, and (9) 220 °C with pB0(1) ) 1 g/L; (b) at 220 °C with ([) pB0(1) ) 3 g/L and (f) pB0(1) ) 6 g/L.
(31)
To determine the validity of this equation at various polystyrene concentrations, the degradation rate, kAB, was determined at polystyrene concentrations of 1, 3, and 6 g/L and the rate coefficient was found to be 3 × 10-5, 6.1 × 10-5, and 10.6 × 10-5 s-1, respectively (Figure 5). A plot of kAB/kA - 1 versus the mass concentration of polystyrene (pB0(1)) is linear, as shown in Figure 6, and the value of k4 (determined from the slope) at 220 °C is 1.0. The values of k4 at other temperatures can be calculated directly from eq 31. This indicates the validity of eqs 30 and 31 for degradation at the polymer concentrations used. This equation, however, may fail at very high polystyrene concentrations because the assumption of kdpB(0)/kHA , 1 used in reducing eq 28 to eq 30 will be invalid. In such cases, eq 28 has to be used to determine the degradation rate coefficients. The activation energies of the degradation rate coefficients were also determined. Figure 7 shows an Arrhenius plot for the degradation of poly(vinyl acetate) (kA), polystyrene (kB), poly(vinyl acetate) in the mixture (kAB), and k4. The activation energies are 120, 80, 200, and 112 kJ/mol, respectively. These activation energies are comparable to the values reported for the random scission degradation rates of other polymers in solution.13,15 The interaction of the two polymers degrading by random chain scission is through hydrogen abstraction2,3 and is represented by a reversible disproportionation reaction, eq 7. The increase in the degradation rate of poly(vinyl acetate) in the presence of polystyrene can be quantified by eq 30 and is directly dependent on the rate coefficient, k4, and thus on kd. Because kd, the rate coefficient for the forward reaction (eq 7), is large, this indicates that the polystyrene radical, RB, interacts with the polymer, poly(vinyl acetate), PA, resulting in an
Figure 6. Plot of kAB/kA - 1 versus the mass concentration of polystyrene to determine k4 at 220 °C.
increased degradation of poly(vinyl acetate). Thus, this paper presents the theoretical development for determining the degradation rates of a binary polymer mixture and a demonstration of how the presence of another polymer may significantly affect the degradation rate for a polymer. The theoretical model is validated by experimental data for the degradation of a mixture containing both poly(vinyl acetate) and polystyrene. Nomenclature kA ) overall degradation rate coefficient of poly(vinyl acetate) (A) kb ) rate coefficient for depropagation kB ) overall degradation rate coefficient of polystyrene (B) kAB ) overall degradation rate coefficient of poly(vinyl acetate) (A) in the presence of polystyrene (B) kd ) rate coefficient for the interaction of PB and RA• kD ) rate coefficient for the interaction of RB• and PA kf ) rate coefficient for initiation
Ind. Eng. Chem. Res., Vol. 40, No. 5, 2001 1311
Figure 7. Arrhenius plot of kA, kB, kAB, and k4 versus temperature for the determination of activation energies.
kh ) rate coefficient for hydrogen abstraction kH ) rate coefficient for the reverse of the hydrogen abstraction reaction kt ) rate coefficient for termination P ) reacting polymer p(x,t) ) MWD of the polymer R• ) reacting radical Ri• ) intermediate radical in reaction 7 r(x,t) ) MWD of the radical Subscripts A ) poly(vinyl acetate) B ) polystyrene Superscripts (n) ) nth moment of the distribution
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(5) Koo, J. K.; Kim, S. W. Reaction Kinetic Model for Optimal Pyrolysis of Plastic Waste Mixtures. Waste Manage. Res. 1993, 11, 515. (6) Roy, M.; Rollin, A. L.; Schreiber, H. P. Value Recovery from Polymer Wastes by Pyrolysis. Polym. Eng. Sci. 1978, 18, 721. (7) Wu, C. H.; Chang, C. Y.; Hor, J. L.; Shih, S. M.; Chen, L. W.; Chang, F. W. On the Thermal Treatment of Plastic Mixtures: Pyrolysis Kinetics. Waste Manage. 1993, 13, 221. (8) Gardner, P.; Lehrle, R.; Turner, D. Polymer Degradation Modified by Blending with Polymers Chosen on the Basis of Their Φ-Factors. J. Anal. Appl. Pyrolysis 1993, 25, 11. (9) Kaczmarek, H. Photodegradation of Polystyrene and Poly(Vinyl Acetate) BlendssI. Irridation of PS/PVAc Blends by Polychromatic Light. Eur. Polym. J. 1995, 31, 1037. (10) Teh, J. W.; Rudin, A.; Yuen, S. Y.; Keung, J.n C.; Pauk, D. M. LLDPE/PP Blends in Tubular Film Extrusion: Recycling of Mixed Films. J. Plast. Film Sheeting 1994, 10, 288. (11) Sato, S.; Murakata, T.; Baba, S.; Saito, Y.; Watanabe, S. Solvent Effect on Thermal Degradation of Polystyrene. J. Appl. Polym. Sci. 1990, 40, 2065. (12) Madras, G.; Smith, J. M.; McCoy, B. J. Thermal Degradation Kinetics of Polystyrene in Solution. Polym. Degrad. Stab. 1996, 58, 131. (13) Madras, G.; Smith, J. M.; McCoy, B. J. Effect of Tetralin on the Degradation of Polymer in Solution. Ind. Eng. Chem. Res. 1995, 34, 4222. (14) Madras, G.; Smith, J. M.; McCoy, B. J. Degradation of Poly(Methyl Methacrylate) in Solution. Ind. Eng. Chem. Res. 1996, 35, 1795. (15) Madras, G.; Chattopadhyay, S. Effect of Hydrogen Donors on the Degradation of Poly(Vinyl Acetate) in Solution. J. Appl. Polym. Sci. 2000, in press. (16) Guy, L.; Fixari, B. Waxy Polyethylenes from Solution Thermolysis of HDPE: Inert and H-Donor Solvent Dilution Effect. Polymer 1999, 40, 2845. (17) Madras, G.; McCoy, B. J. Continuous Distribution Kinetics for Polymer Mixtures Degradation. Ind. Eng. Chem. Res. 1999, 38, 352. (18) Nigam, A.; Fake, D. M.; Klein, M. T. Simple Approximate Rate Law for Both Short Chain and Long Chain Rice Herzfeld Kinetics. AIChE J. 1994, 40, 908. (19) Gavalas, G. R. The Long Chain Approximation in Free Radical Reaction Systems. Chem. Eng. Sci. 1966, 21, 133. (20) Madras, G.; Chung, G. Y.; Smith, J. M.; McCoy, B. J. Molecular Weight Effect on the Dynamics of Polystyrene Degradation. Ind. Eng. Chem. Res. 1997, 36, 2019. (21) Madras, G.; McCoy, B. J. Discrete and Continuous Models for Polymerization and Depolymerization. Chem. Eng. Sci. 2000, in press.
Received for review September 15, 2000 Revised manuscript received December 18, 2000 Accepted December 19, 2000 IE000820K
