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Ind. Eng. Chem. Res. 1999, 38, 352-357
KINETICS, CATALYSIS, AND REACTION ENGINEERING Distribution Kinetics for Polymer Mixture Degradation Giridhar Madras† and Benjamin J. McCoy* Department of Chemical Engineering and Materials Science, University of California, Davis, California 95616
Most research on polymer degradation is for single polymers, even though the thermal decomposition of polymer mixtures is of interest both practically and theoretically. Polymer degradation rates depend on the mixture type, and adding a polymer can increase, decrease, or leave unchanged the degradation rate of the first polymer. We show how distribution kinetics theory, based on molecular weight distributions (MWDs), provides expressions for such degradation rates of binary polymer mixtures. The approach accounts for initiation, termination, hydrogen abstraction, and radical-chain scission in the governing equations for MWDs. Molecular weight moments yield expressions for molar and mass concentrations and rate coefficients for combinations of random-chain and chain-end scission. Experimental data show the concentration effect of poly(R-methylstyrene) (PAMS) on the degradation of polystyrene dissolved in mineral oil at 275 °C in a batch reactor. PAMS and polystyrene undergo mainly chain-end and randomchain scission, respectively. Samples analyzed by gel permeation chromatography yielded the time evolution of the MWD. The results indicated that, because of the interaction of mixed radicals with polymer by hydrogen abstraction, the polystyrene degradation rate decreases with increasing PAMS concentration. Introduction Thermochemical recycling of polymers has received growing attention in recent years as a potential alternative to disposal of plastic wastes. Understanding the thermal decomposition of polymers is also an important issue in polymer science and engineering.1 Most research has focused on degradation of single polymers. Waste streams, however, usually contain mixtures of polymers that are costly to separate prior to decomposition.2 Thus, it is important to study the degradation of polymer mixtures. The polymer degradation rate can be modified by the addition of conventional free-radical initiators, oxidizers, or hydrogen donors, but blending two polymers also alters the rate behavior.3 Degradation studies by pyrolysis of polymer mixtures have provided varied results contingent on experimental conditions. Some reports, for example, indicated a significant interaction between polystyrene and polyethylene,2,4,5 whereas others observed no interaction between these polymers.6,7 The polystyrene degradation rate during pyrolysis was found to increase3 in the presence of poly(methyl acrylate) and poly(butyl acrylate) but decreased8 with increasing molecular weight (MW) of added poly(R-methylstyrene) (PAMS). Usually in pyrolysis studies, only the volatile gases are monitored, and the MW change of the polymer is neglected. The reports suggest the need for a better understanding of the underlying reaction mechanisms. * To whom correspondence should be addressed. E-mail:
[email protected]. Phone: (530) 752-6923. Fax: (530) 7521031. † Present address: Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India.
Degradation of polymers in solution has been proposed to avoid problems encountered in multiphase pyrolysis processes.9 The degradation in solution of polystyrene,10-12 poly(styrene-allyl alcohol),13 poly(methyl methacrylate),14 and PAMS15 has been investigated. No studies on the degradation of polymer mixtures in solution, however, have been reported. In the current work, we examined the degradation of polymer mixtures by developing a detailed radical mechanism based on Rice-Herzfeld reactions.16,17 The interaction between different polymers occurs by radical abstraction of hydrogen atoms from either polymer.2 Expressions for the degradation rates for each polymer were obtained by applying distribution kinetics to the molecular weight distribution (MWD) of the reacting radicals and polymers. As polymers degrade by either chain-end scission or random-chain scission three cases are possible for the interactive degradation of two polymers. Both polymers can degrade by chain-end scission, or by random-chain scission, or one polymer can degrade by random-chain scission and the other polymer by chain-end scission. In addition to developing a new kinetics model, we present and interpret new experimental data for the concentration effect of PAMS on polystyrene degradation in solution at 275 °C. The present approach thus shows how the distribution kinetics framework can be applied to gain insight into the diverse observed behavior of degrading polymer mixtures. Experiments Analysis of MWDs. The high-performance liquid chromatograph HPLC; Hewlett-Packard 1050) system consists of a 100-µL sample loop, a gradient pump, and an
10.1021/ie9805335 CCC: $18.00 © 1999 American Chemical Society Published on Web 01/05/1999
Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999 353 Table 1. Distribution Kinetics of Primary Reactionsa reaction
primary reaction
transformation random-chain scission
R(x) f P(x) P(x′) f Q(x) + R(x′-x)
chain-end scission
P(x′) f Q(xs) + R(x′-xs)
addition reaction
xs is the MW of specific product P(x′) + Q(x-x′) f R(x)
a
rate expressions
moment expressions
∂p/∂t ) -∂r/∂t ) kr(x,t) ∂p/∂t ) -kp(x,t) ∂q/∂t ) k∫∞x p(x′,t) Ω(x,x′) dx′ ) ∂r/∂t ∂p/∂t ) -kp(x′,t) ∂q/∂t ) k∫∞x p(x′,t) Ω(xs,x′) dx′ ∂r/∂t ) k∫∞x p(x′,t) Ω(x-xs,x′) dx′ ∂p/∂t ) -kp(x,t)∫∞0 q(x′,t) dx′ ∂q/∂t ) -kq(x,t)∫∞0 p(x′,t) dx′ ∂r/∂t ) k∫x0p(x′,t) q(x-x′,t) dx′
) -dr(n)/dt ) kr(n)(t) dp(n)/dt ) -kp(n)(t) dq(n)/dt ) kp(n)(t)/(n + 1) ) dr(n)/dt dp(n)/dt ) -kp(n)(t) dq(n)/dt ) kxsnp(0)(t) n dr(n)/dt ) k∑j)0 (nj )xsj(-1)jp(n-j) dp(n)/dt ) -kp(n)(t) q(0)(t) dq(n)/dt ) -kq(n)(t) p(0)(t) n dr(n)/dt ) k∑j)0 (nj )p(j)(t) q(n-j)
dp(n)/dt
Stoichiometric kernels are Ω(x,x′) ) 1/x′, Ω(xs,x′) ) δ(x-xs), and Ω(x-xs,x′) ) δ[x-(x′-xs)]; rate coefficients are k(x) ) k.
on-line variable-wavelength ultraviolet (UV) detector. Three PL gel columns (Polymer Lab Inc.; 300 mm × 7.5 mm) packed with cross-linked poly(styrene-divinylbenzene) with pore sizes of 100, 500, and 104 Å are used in series. Tetrahydrofuran (HPLC grade; Fisher Chemicals) was pumped at a constant flow rate of 1.00 mL/ min. Narrow-MW polystyrene standards of MW 162 to 0.93 million (Polymer Lab and Aldrich Chemicals) were used to obtain the calibration curve of retention time versus MW, which was stable during the period of the experiments. The calibration curve, modeled as a secondorder polynomial, ensured higher accuracy in the measurement of lower MW polymers. Polystyrene Degradation. The thermal decomposition of polystyrene in mineral oil was conducted in a 100mL flask equipped with a reflux condenser to ensure condensation and retention of volatiles. To observe a significant effect of PAMS on the conversion, polystyrene of high MW was chosen. A total of 60 mL of mineral oil (Fisher Chemicals) was heated to 275 °C, and various amounts (0-0.60 g) of monodisperse PAMS (MW ) 11 000; Scientific Polymer Products) and 0.12 g of monodisperse polystyrene (MW ) 330 000; Aldrich Chemicals) were added. The temperature of the solution was measured with a type K thermocouple (Fisher Chemicals) and controlled within (1 °C by an Omega CN-2042 temperature controller. Samples of 1.0 mL were taken at 15-min intervals and dissolved in 1.0 mL of tetrahydrofuran (THF, HPLC grade; Fisher Chemicals). The chromatograph obtained by injecting 100 µL of this solution into the HPLC-GPC system was converted to MWD. The peaks of the reacted polystyrene, PAMS, and R-methylstyrene (AMS) were distinct, so that moments could be calculated by numerical integration. Because the solvent mineral oil is UV invisible, its MWD was determined with a refractive index (RI) detector. No change in the MWD of mineral oil (Figure 1) was observed when the oil was heated for 3 h at 275 °C without polystyrene. Theoretical Model According to the mechanism, polymers can transform without change in MW by hydrogen abstraction.16 Polymer radicals can also undergo chain scission to form lower MW products or undergo addition reactions to yield higher MW products. Chain scission can occur either at the chain end, yielding a specific product, or at a random position along the chain, yielding a range of lower MW products. Distribution kinetics is a straightforward and effective technique to represent macromolecular reactions. Continuous-distribution mass (population) balances are written for the various steps involved in the radical mechanism (Table 1). The rate coefficients are here
Figure 1. MWD of the inert solvent, mineral oil.
assumed to be independent of MW, a reasonable assumption at low conversions.12 The integrodifferential equations obtained from the mass balances can be solved for MW moments. These moment equations are usually coupled ordinary differential equations that, in general, can be solved numerically. In the present treatment, two common assumptions simplify the governing equations. The long-chain approximation (LCA)18,19 postulates that the initiation and termination rates are negligible because such events are infrequent compared to hydrogen abstraction and propagationdepropagation chain reactions. The quasi-stationarystate approximation (QSSA) applies when radical concentrations are extremely small and their rates of change are negligible. The proposed scheme, based on the Rice-Herzfeld concept of chain reactions,16 includes the elementary steps of initiation, depropagation, hydrogen abstraction, and termination. The present approach is similar to our treatment of the hydrogen-donor effect on polymer degradation.17 Case I. One polymer undergoes random-chain scission, the other polymer undergoes chain-end scission, and both interact with each other. The degradation rate of polymer A undergoing random-chain scission is influenced by polymer B undergoing chain-end scission. We represent the reacting polymer A and its radicals as PA(x) and R•(x) and their MWDs as pA(x,t) and r(x,t), respectively, where x represents the continuous variable, MW. As the polymer reactants and random scission products are not distinguished in the distribution kinetics model,16 a single MWD, pA(x,t), represents the polymer in the mixture at any time t. The initiation-termination reactions are represented as kf
PA(x) S R•(x′) + R•(x-x′) kt
(1.1)
where S represents a reversible reaction. The reversible
354 Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999
∫x∞r(x′) Ω(x,x′) dx′ -
hydrogen abstraction process is
∂pA/∂t ) - khpA(x) + kHr(x) + kb
kh
PA(x) S R•(x) kH
∫x∞ri(x′) Ω(x,x′) dx′
kdpAre(0)(x) + kD
(1.2)
∂pB/∂t ) -khepB(x) + kHere(x) - kdpBr(0)(x) +
∫x∞ri(x′) Ω(x,x′) dx′
The irreversible depropagation chain reaction is kb
R•(x) 98 PA(x′) + R•(x-x′)
kD
∫x∞rs(x′) δ(x′-xs) dx′
∂qs/∂t ) kbs
(1.3)
∂rs/∂t ) kihre(x) - kiHrs(x) - kbsrs(x) Polymer B, the chain-end radical, the specific radical, and the specific product are represented as PB(x), Re•(x), Rs•(x), and Qs(xs), respectively, and their corresponding MWDs as pB(x,t), re(x,t), rs(x,t), and qs(xs,t). The formation of chain-end radicals by a reversible random scission initiation-termination reaction is kfs
PB(x) S Re•(x′) + Re•(x-x′) kts
(1.4)
Hydrogen abstraction by the chain-end radical is considered reversible, khe
PB(x) S
kHe
(1.9)
Re•(x)
(1.5)
Re•(x) S Rs•(x) kiH
(1.11) (1.12)
∂re/∂t ) khepB(x) - kHere(x) - kihre(x) + kiHrs(x) +
∫x∞rs(x′) δ[x-(x′-xs)] dx′ - kdpA(0)re(x) + ∞ kD∫x ri(x′) Ω(x,x′) dx′ (1.13)
kbs
∂r/∂t ) khpA(x) - kHr(x) - kbr(x) +
∫x∞r(x′) Ω(x,x′) dx′ - kdpB(0)r(x) + ∞ kD∫x ri(x′) Ω(x,x′) dx′
(1.14)
∫0xpA(x′) re(x-x′) dx′ + x kd∫0 pB(x′) r(x-x′) dx′ - 2kDri(x)
(1.15)
kb
∂ri/∂t ) kd
The moments for the polymer, radicals, and monomer, p(n), r(n), and q(n), respectively, are defined as
The chain-end radical may undergo radical isomerization via a cyclic transition state16 to form a specific radical, kih
(1.10)
(1.6)
p(n)(t) )
∫0∞xnp(x,t) dx
(1.16)
r(n)(t) )
∫0∞xnr(x,t) dx
(1.17)
q(n)(t) )
∫0∞xnq(x,t) dx
(1.18)
Applying the moment operation, ∫∞0 [ ]x dx, to each balance equation yields n
The depropagation reaction yields the specific product and a chain-end radical from a specific radical,
Rs•(x)
kbs
98 Qs(xs) +
Re•(x-xs)
dpA(n)/dt ) -khpA(n) + kHr(n) + kbZn0r(n) (1.7)
where xs is the MW of the specific product. The interaction of the two degrading polymers is through hydrogen abstraction2 represented as a reversible disproportionation reaction. The end radical of polymer B combines with polymer A to form an intermediate radical complex that undergoes transformation to polymer B and a radical of polymer A, kd
kD
Re•(x) + PA(x′) S Ri•(x+x′) S PB(x) + R•(x′) kD
kd
kdpA(n)re(0) + kDZn0ri(n) (1.19) dpB(n)/dt ) -khepB(n) + kHere(n) kdpB(n)r(0) + kDZn0ri(n) (1.20) dqs(n)/dt ) kbsxsnrs(0)
(1.21)
drs(n)/dt ) kihre(n) - kiHrs(n) - kbsrs(n)
(1.22)
dre(n)/dt ) khepB(n) - kHere(n) - kihre(n) + kiHrs(n) + n
(1.8)
Including the intermediate complex, Ri•, facilitates the formulation of the population balance equations for the reversible disproportionation. If Ri• is ignored in eq 1.8, the forward and reverse rate coefficients would be kd and kD, respectively. Rate expressions and moments for reactions (1.1)(1.8) can be formulated with the aid of Table 1. Based on LCA, kf, kfs, kt, and kts are set to zero. The population balance equations16 for gain (+) and loss (-) terms in the reaction mechanism are
(nj )(xs)j(-1)jrs(n-j) - kdpA(0)re(n) + kDZn0ri(n) ∑ j)0
kbs
(1.23) dr(n)/dt ) khpA(n) - kHr(n) - kbr(n) + kbZn0r(n) kdpB(0)r(n) + kDZn0ri(n) (1.24) n
n
j)0
j)0
∑(nj )re(n-j)pA(j) + kd∑(nj )r(n-j)pB(j) -
dri(n)/dt ) kd
2kDri(n) (1.25) where Zn0 ) 1/(n + 1). The initial conditions for the
Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999 355
moments are
pB(n)(t)0) ) pB0(n), pA(n)(t)0) ) pA0(n), r(n)(t)0) ) 0 (1.26) When QSSA holds,
we represent the reacting polymers A and B as PA(x) and PB(x) and their radicals as RA•(x) and RB•(x). Their MWDs are pA(x,t), pB(x,t) and rA(x,t), rB(x,t), respectively. The initiation-termination reactions are kfA
dr(n)/dt ) drs(n)/dt ) dre(n)/dt ) dri(n)/dt ) 0
(1.27)
PA(x) S RA•(x′) + RA•(x-x′) ktA kfB
Expressions for the radical concentrations can be obtained by equating the expressions of the zeroth moments (n ) 0 in eqs 1.22-1.25) to zero. Equation 1.22 yields
rs(0) ) re(0)kih/(kbs + kiH)
PB(x) S RB•(x′) + RB•(x-x′) ktB
khA
(1.28)
PA(x) S RA•(x)
Equation 1.25 yields
khB
r(0) ) pA(0)(2kh + kdre(0))/(2kH + kdpB(0)) (1.30) re(0) ) pB(0)(2khekH + kd(khe + kh)pA(0))/(2kHekH + kd(kHepB(0)
+
kHpA(0)))
(1.31)
Because PAMS degrades only by chain-end scission, the zeroth moment (molar concentration) of the polymer is a constant,15
dpB(0)/dt
)0
(1.33)
Substituting eqs 1.28-1.31 into eq 1.19 yields
dpA(0)/dt ) kbr(0)
kHB
(2.3) (2.4)
The depropagation (chain-scission) reactions are kbA
RA•(x) 98 PA(x′) + RA•(x-x′) kbB
RB•(x) 98 PB(x′) + RB•(x-x′)
(2.5) (2.6)
The interaction between the polymers can be written as kd
kD
kD
kd
RB•(x) + PA(x′) S Ri•(x + x′) S PB(x) + RA•(x′) (2.7)
(1.32)
and therefore
pB(0) ) pB0(0)
PB(x) S RB•(x)
(1.29)
Equations 1.23 and 1.24 are solved simultaneously to obtain
(2.2)
The reversible hydrogen abstraction reactions are
kHA
ri(0) ) kd(re(0)pA(0) + r(0)pB(0))/(2kD)
(2.1)
(1.34)
where r(0) and re(0) are given by eqs 1.30 and 1.31 with pB(0) ) pB0(0). The differential equation (1.34) can be expressed as a first-order rate, as we will show, when reasonable assumptions are made. The summation of first moments (n ) 1) for polymer and radicals (total mass concentration) must be constant when there are no losses from the reactor. For n ) 1, eqs 1.19-1.25 are added to obtain
d[p(1) + r(1) + rs(1) + re(1) + ri(1)]/dt ) 0 (1.35) thus confirming the mass balance. The proposed mechanism represents the interaction of radicals of two reacting polymers and shows how the polymer undergoing chain-end scission affects the degradation rate of the polymer undergoing random-chain scission. The degradation rate coefficient is a function of the added polymer concentration and the solvent and also depends on the temperature and pressure. This can explain the varied results found in experiments for degradation rates in polymer mixtures. Case II. Both polymers undergo random-chain scission and interact with each other. When polymers degrade by random-chain scission and interact with each other,
The population balance equations and their moments are readily formulated with the aid of Table 1. Case III. Both polymers undergo chain-end scission and interact with each other. Certain polymers such as PAMS undergo only chain-end scission. In cases where both polymers A and B degrade by chain-end scission and interact with each other, the radical reactions for polymers can be written for the mechanism. Polymers A and B, the chain-end radical, the specific radical, and the specific product are represented as PA(x), ReA•(x), RsA•(x), QsA(xsA) and PB(x), ReB•(x), RsB•(x), QsB(xsB), respectively. The formation of chain-end radicals by a reversible random-scission initiation-termination reaction is kfsA
PA(x) S ReA•(x′) + ReA•(x-x′) ktsA kfsB
PB(x) S ReB•(x′) + ReB•(x-x′) ktsB
(3.1) (3.2)
Hydrogen abstraction by the chain-end radical is considered reversible, kheA
PA(x) S ReA•(x) kHeA kheB
PB(x) S ReB•(x) kHeB
(3.3) (3.4)
The chain-end radical can undergo radical isomerization via a cyclic transition state to form a specific radical,
356 Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999 kihA
ReA•(x) S RsA•(x)
(3.5)
kiHA kihB
ReB•(x) S RsB•(x)
(3.6)
kiHB
The depropagation reactions yield the specific products and chain-end radicals from specific radicals, kbsA
RsA•(x) 98 QsA(xs) + ReA•(x-xs) kbsB
RsB•(x) 98 QsB(xs) + ReB•(x-xs)
(3.7) (3.8)
where xsA and xsB are the MWs of the specific products of polymers A and B, respectively. The interaction of the two polymers is given by the disproportionation reaction
ReB•(x)
kd
•
kD
+ PA(x′) S Ri (x+x′) S PB(x) +
concentration on polystyrene degradation at 275 °C. The mass of specific products11,12 formed by polystyrene chain-end scission at 275 °C for 10 h is less than 2%. In our experiments lasting 3 h or less, the molar production rate of such specific products is then negligible relative to the random scission rate. Since polystyrene therefore degrades mainly by random-chain scission and PAMS by chain-end scission, the polystyrene degradation rate is given by eq 1.34. If the rates of the elementary steps were known, this equation could be solved numerically. However, reasonable assumptions allow an analytical solution for the polystyrene degradation rate. The polystyrene concentration is very small, pA(0) < 6 × 10-6 mol/L, so that it can be neglected in eq 1.31; thus,
re(0) ) pB(0) × 2khekH/(2kHekH + kdkHepB(0)) (4.1) Substituting this into eq 1.30 yields after rearrangement
ReA•(x′)
(3.9)
r(0) ) pA(0) × 2[(2 + β)kh/kH + βkhe/kHe]/(2 + β)2 (4.2)
Table 1 facilitates writing of the mass balance equationsand their moment forms. Analytical solutions may be possible for special values of the rate parameters. Otherwise, the coupled ordinary differential equations can be solved numerically. The equations show how the degradation rate of a polymer depends on the concentration of the other polymer in the mixture. Of course, the degradation rate coefficient is also a function of temperature and pressure and the nature of the solvent.
where β ) pB(0)kd/kH. Equation 4.2 combined with eq 1.34 yields the rate expression
kD
kd
dpA(0)/dt ) kr pA(0)
(4.3)
where
kr ) 2kb[(2 + β)kh/kH + βkhe/kHe]/(2 + β)2 (4.4) Because pB(0) is constant, the solution to eq 4.3 is
Interpretation of Experimental Results Polymers thermolytically degrade by chain-end scission and/or random-chain scission. In polymer mixtures the interaction is complex and is influenced by the experimental conditions and polymer type. For example, Roy et al.6 observed no interaction between polystyrene and polyethylene at pressures higher than 200 mmHg but observed interaction at lower pressures. McCaffrey et al.2 observed a significant increase in the thermolytic degradation rate of polyethylene due to the presence of polystyrene. They proposed that the interaction was due to hydrogen abstraction from polyethylene by polystyrene radicals. Gardner et al.3 observed an 8-fold increase in the pyrolytic chain-end scission rate of polystyrene at 430 °C in the presence of poly(methyl acrylate) (PMA) or poly(butyl acrylate) (PBA), whereas the chain-end degradation rates of PMA and PBA decreased 8-fold. However, PBA or PMA did not affect the PAMS degradation rate. They also concluded that hydrogen abstraction plays the key role in polymer interactions. We have proposed degradation mechanisms for three common types of polymer degradation with interactions by a disproportionation reaction for hydrogen abstraction. The degradation rate equations (for example, eq 1.34 with eqs 1.30 and 1.31) for binary mixtures show that, depending on the particular polymer, another polymer in the mixture can increase, decrease, or have no effect on the degradation rate. As shown by these equations, the interaction would depend on the fundamental radical rate parameters (and thus temperature and pressure) and the concentrations of the polymers in the mixture. As an example of degradation kinetics in a polymer mixture, we have measured the influence of PAMS mass
pA(0)(t) ) pA0(0) exp(krt)
(4.5)
and a plot of ln(p(0)/p0(0)) should be linear in time with slope kr. The random-scission degradation rate coefficient, kr, was determined from the experimental data by analyzing the time dependence of the polymer mixture MWDs. Polystyrene degrades rapidly at low reaction times because of weak links in the polymer chain caused by side-group asymmetry or chain-branching.12,20 The weak and strong links in polystyrene can be represented by additive distributions,12 so that the total molar concentration, pAtot(0), of the polymer is the sum of the molar concentrations of the weak, pAw(0), and strong, pA(0), links. Because the weak link concentration is approximately 2 orders of magnitude smaller than the strong link concentration, only the rate coefficients for strong link scission are examined in this study. The initial molar concentration of the strong links in polystyrene, pA0(0), is determined from the intercept of the regressed line of the pAtot(0)/pAtot0(0) data for t g 45 min. The slopes, corresponding to the rate coefficient for random scission, kr, are determined from the plot of ln(pA(0)/pA0(0)) versus time, as given by eq 4.5 (Figure 2). The initial mass concentration of PAMS, pB0(1) ) pB0(0)MB0, is expressed in terms of the number-average molecular weight, MB0. The line for zero concentration of PAMS is curved slightly due to a MW-dependent rate. However, generally, the linearity of the data in Figure 2 supports the hypothesized mechanism. Madras et al.11,12 show how data are analyzed when the polystyrene degradation rate coefficient is a function of MW, i.e., kr(x). Equation 1.34 explains how the polystyrene degradation rate coefficient depends on the PAMS
Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999 357
Acknowledgment The financial support of NSF Grant CTS-9810194 and Pittsburgh Energy Technology Center Grant DOE DEFG22-94PC94204 is gratefully acknowledged. The authors thank Professor J. M. Smith for helpful discussions and Dr. Yoichi Kodera for analyzing the mineral oil sample. Literature Cited
Figure 2. Plot of ln(pA(0)/pA0(0)) versus time for polystyrene degradation at 275 °C for four PAMS concentrations: (b) polystyrene (2 g/L) only; (2) polystyrene (2 g/L) + PAMS (2 g/L); ([) polystyrene (2 g/L) + PAMS (5 g/L); (9) polystyrene (2 g/L) + PAMS (10 g/L).
Figure 3. Effect of PAMS mass concentration, pB0(1), on the rate coefficient for random-chain scission, kr, of polystyrene at 275 °C. The curve indicates the fit to determine the rate coefficient groups, kd/kH and kbkhe/kHe.
concentration. Only zeroth moments (molar concentrations) are required to derive this key relationship. The polystyrene degradation rate coefficient, kr, given by eq 4.4 decreases with increasing PAMS mass (or molar) concentration, as shown by a plot of kr versus pB0(1) (Figure 3). Error bars in Figure 3 indicate the average error in kr of (0.001/min. Of the three parameter groups appearing in the expression for kr, the group kbkh/kH ) 0.013/min was determined as the intercept (pB(1) ) 0). The other two groups, kd/kH ) 9500 L/mol and kbkhe/kHe ) 0.001/min, were evaluated by fitting the data with eq 4.4 by least-squares minimization. Although the data cannot provide individual values of the separate rate constants, evaluation of the parameter groups corroborates the validity of the kinetics model. The hypothesized interaction of the degrading polymers is through the free radicals and their rates of hydrogen abstraction. When kd ) kD ) 0, the two polymers react independently and the moment equations (1.19)-(1.25) are identical to those derived for a single polymer undergoing chain-end scission or randomchain scission.16 The rate coefficient for random-chain scission of polystyrene is a function of the PAMS mixture concentration through the fundamental radical rate parameters, kb, kh, khe, kd, and kH, and the initial number-average molecular weight of PAMS. The addition of PAMS inhibits the random-chain scission of polystyrene, similar to the effect of hydrogen donors on the degradation of polystyrene.17 These results suggest that the interaction between polymers involves hydrogen abstraction.
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Received for review August 10, 1998 Revised manuscript received November 17, 1998 Accepted November 19, 1998 IE9805335
