Initial Stages of the Pyrolysis of Polyethylene - ACS Publications

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Initial Stages of the Pyrolysis of Polyethylene Konstantin V. Popov and Vadim D. Knyazev* Research Center for Chemical Kinetics, Department of Chemistry, The Catholic University, 620 Michigan Avenue Northeast, Washington, District of Columbia 20064, United States S Supporting Information *

ABSTRACT: An experimental study of the kinetics of the initial stages of the pyrolysis of high-density polyethylene (PE) was performed. Quantitative yields of gas-phase products (C1−C8 alkanes and alkenes) and functional groups within the remaining polyethylene melt (methyl, vinyl, vinylene, vinylidene, and branching sites) were obtained as a function of time (0−20 min) at five temperatures in the 400−440 °C range. Gas chromatography and NMR (1H and 13C) were used to detect the gas- and condensed-phase products, respectively. Modeling of polyethylene pyrolysis was performed, with the primary purpose of determining the rate constants of several critical reaction types important at the initial pyrolysis stages. Detailed chemical mechanisms were created (short and extended mechanisms) and used with both the steady-state approximation and numerical integration of the differential kinetic equations. Rate constants of critical elementary reactions (C−C backbone scission, two kinds of H-atom transfer, radical addition to the double bond, and beta-scission of tertiary alkyl radicals) were adjusted, resulting in an agreement between the model and the experiment. The values of adjusted rate constants are in general agreement with those of cognate reactions of small molecules in the gas phase, with the exception of the rate constants of the backbone C−C scission, which is found to be approximately 1−2 orders of magnitude lower. This observation provides tentative support to the hypothesis that congested PE melt molecular environment impedes the tumbling motions of separating fragments in C−C bond scission, thus resulting in less “loose” transition state and lower rate constant values. Sensitivity of the calculations to selected uncertainties in model properties was studied. Values and estimated uncertainties of four combinations of rate constants are reported as derived from the experimental results via modeling. The dependence of the diffusion-limited rate constant for radical recombination on the changing molecular mass of polyethylene was explicitly quantified and included in the extended kinetic mechanism, which appears critical for the agreement between modeling and experiment, particularly the agreement between the experimental and the calculated activation energies for product formation rates. Calculations were performed to estimate the contribution to the overall rate of radical recombination of the “reaction diffusion” phenomenon, where recombination is driven not by the actual motion of the recombining radical sites but rather by the migration of the radical site through PE melt due to rapid hydrogen transfer; this contribution was shown to be negligible for the conditions of the current work.

1. INTRODUCTION Combustion and flammability of polyethylene (PE) are important subjects of study directly related to such practical applications as fire safety and recycling of plastics. The process of pyrolysis is the initial step of combustion of the solid, with the gaseous products providing the fuel for the flame. Efforts have been directed onto elucidating the details of the chemical mechanism of PE pyrolysis (e.g., refs 1−5 and references therein). Nevertheless, the rates of the main elementary reactions that occur during pyrolysis and the roles of these reactions in the formation of the final products are still debatable due to the difficulty of experimental study of elementary reactions under the conditions of polymer melt. Early experimental studies of the pyrolysis of PE mostly utilized dynamic thermogravimetric analysis (TGA) experiments that can only provide effective kinetic rate constants of the overall process of mass loss and production of final gaseous products. A rather small number of publications present relative © XXXX American Chemical Society

abundances of products; absolute yields of different pyrolysis products as a function of time and/or temperature were determined only in a few studies.6−8 A very thorough analysis of the experimental and modeling studies of PE pyrolysis conducted by 2003 can be found in a review by Poutsma.3 In virtually all mechanistic models of PE pyrolysis most of the rate constants of elementary reactions are imported from the studies of similar reactions of smaller molecules investigated under the conditions of the gas phase, which, in general, is not justified without an experimental validation. In a number of modeling studies, good agreement was reached between experiment and modeling (e.g., refs 1, 2, and 4). However, when modeling efforts are directed at reproducing the results of TGA experiments or those where limited data on gas-phase Received: July 31, 2015 Revised: October 19, 2015

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DOI: 10.1021/acs.jpca.5b07440 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

chain length, which undergoes modification during the pyrolysis. Modeling with the extended mechanism was performed using integration of kinetic equations. The derived rate constant values were analyzed and compared with those of cognate reactions of small molecules in the gas phase and with the Arrhenius expressions used or suggested in earlier modeling studies of PE pyrolysis.

product relative abundances were obtained, the number of potentially adjustable parameters such as rate constants inevitably exceeded the number of observed dependences (such as residue weight fraction vs temperature for a particular heating rate). The resultant models provide adequate descriptions of the observables, but the veracity of individual rate constants used is not necessarily assured. The most detailed mechanistic study of the pyrolysis of PE is that of Levine and Broadbelt.4 The authors constructed a detailed model of high-density PE (HDPE) pyrolysis with rate constants of individual elementary reactions assigned based on the modeling studies of pyrolysis of polypropylene performed by the same group,9 recommendations from earlier studies, estimations performed using structure−activity relationships, and quantum chemical calculations.10 The predictions of the model were compared with the results of the experimental study of HDPE pyrolysis conducted earlier by the same group,8 where absolute yields of alkane and alkene products from methane to C34 compounds were determined at a temperature of 420 °C and reaction times between 30 and 240 min. The agreement between model predictions and experimental data was found to be excellent for condensable products and fair for gaseous products. The current study is primarily motivated by the interest in the values and temperature dependences of the rate constants of elementary reactions involved in PE pyrolysis as well as the applicability of gas-phase rate constant values to the conditions of PE melt. Therefore, it focuses on the initial stages of polyethylene (high-density polyethylene, HDPE) pyrolysis, where only several types of elementary reactions are expected to play major roles, and secondary processes involving the products of the initial stages are negligible. A novel aspect of the experimental part of this work is in the use of two techniques to probe both the gaseous products and the abundances of specific functional groups in the remaining condensed phase (polymer melt, which solidifies upon cooling). Nuclear magnetic resonance (NMR) spectroscopy was used to determine the kinetics of the formation of three types of carbon−carbon double bonds, terminal −CH3 groups, and chain branching points (CHY) in the liquid polymer melt phase. Gas chromatography (GC) was used to determine the yields of volatile products. Quantities of liquid-phase and gasphase products were measured as a function of time (5−20 min) in the 400−440 °C temperature range. Experimental conditions were selected to limit the molar fraction of the formed double bonds in the condensed phase to less than 0.5% and thus isolate for study only the initial stages of the overall process. The results were initially modeled with a short mechanism consisting of a limited number of reaction types: backbone C− C bond scission, beta-scission of alkyl radicals, hydrogen-atom transfer via abstraction, addition of alkyl radicals to double bonds, and radical recombination. Steady-state approximation was initially used to derive the relationships between the observable product yields and combination of rate constants of elementary reactions; analysis of the experimental data within the framework of the steady-state model provided initial approximations for the rate constants of critical reactions. The steady-state model was followed by modeling based on numerical integration of kinetic equations. Finally, the initial, short mechanism was extended to include reactions of secondary importance, as well as the dependence of the diffusion-limited radical recombination rate on the average PE

2. EXPERIMENTAL SECTION 2.1. Experimental Apparatus, Methods, and Materials. HDPE samples (Aldrich, density 0.9466 g cm−3, granules of ca. 0.02 g, melt index 43 g/10 min) were pyrolyzed in airtight reactors consisting of a 9 mL Pyrex test tube, a stainless steel Cajon junction that provided a vacuum-tight glass-to-metal connection, and a Nupro shut-off gas valve. An HDPE sample (0.040 ± 0.005 g) was placed into the test tube, and the junction and the valve were used to connect the reactor to a vacuum system to degas the sample (see below) and to fill the reactor with nitrogen in preparation for a pyrolysis experiment. The reactor was filled with nitrogen to a pressure exceeding atmospheric pressure by 5 Torr to avoid any potential leaks of the atmospheric air inside the reactor. During a typical pyrolysis experiment, the bottom of a sealed tube with an HDPE sample was immersed into a vessel filled with molten tin and kept there at constant temperature for a predefined time. The use of the molten tin as the thermostat enabled quick heating of the immersed reactor from room temperature to the desired temperature of pyrolysis. Due to the large mass of the molten tin (∼5 kg), the immersion of the reactor did not cause any significant change in the temperature of the thermostat. The temperature of the tin was measured using a type K (chromel− alumel) thermocouple. The temperature was controlled using an autotransformer regulating the voltage on the electric heating element and by a variable flow of nitrogen gas directed onto the surface of the molten tin. The combination of these two methods of temperature control kept the temperature of the tin at the desired value within a 1 °C range. A mechanical rotating rod with metal propeller-type blades was used to stir the molten tin in order to ensure homogeneity of the temperature inside the tin-containing vessel. Preliminary analysis showed that the HDPE used for the experiments contains non-negligible amounts of hydrocarbon gases trapped in the granules; the abundances of the trapped gases amounted to a few percent of the abundances of the gases released during pyrolysis experiments. Thus, in order to avoid distortion of the results of the pyrolysis experiments by mixing the gases trapped in the HDPE samples due to the manufacturing process and those produced during the pyrolysis experiment and also to remove any atmospheric air trapped in the samples, the latter were subjected to the procedure of degassing. The procedure consisted of evacuating the reactor with the sample while immersed in the vessel with molten tin at a temperature of 280 °C for at least 10 h. This temperature is below the onset of pyrolysis,3 and thus, the only process expected to occur in the molten HDPE was the escape of the trapped gases out of the melt. Pyrolysis experiments were performed using previously degassed samples over the temperature range 400−440 °C; the time of pyrolysis varied from 5 to 20 min. The lower limit of the temperature range was chosen so as to keep a substantial signal-to-noise ratio in the product analysis, which decreases with decreasing product yields at lower temperatures. The upper temperature limit was selected to keep the abundance of B


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Proton NMR spectroscopy was used to determine the concentrations of different types of double carbon−carbon bonds and −CH3 groups, whereas 13C NMR was used to analyze for the −CH3 and CHY (PE chain branching points) groups. In a typical NMR analysis, a piece of the solid residue of the HDPE pyrolysis was dissolved in a mixture consisting of 40 wt % of 1,2,4-trichlorobenzene and 60 wt % of deuterated benzene (C6D6); this solvent is similar to the that used by Qu et al.13 in their NMR study of cross-linking in low-density PE. Actual quantities of the solvents were chosen such that the mass of the dissolved HDPE residue in the NMR sample was equal to 10 wt %. HDPE dissolves in the used mixture of solvents only at elevated temperatures; therefore, the NMR tubes with the samples were flame sealed to avoid solvent evaporation. NMR analysis was performed using two spectrometer models: Bruker Ascend 400 FT NMR and JEOL JNMGSX270 FT NMR. Most of the experiments were performed using the Bruker instrument. Spectra were obtained at 120 °C, with 100 and 1024 scans for each 1H and 13C NMR spectrum, respectively. The experimental conditions for 1H and 13C experiments were chosen, respectively, as follows: pulse width, 10 and 14 μs; acquisition time, 1.47 and 0.655 s; number of points, 32 768 and 33 078; spectral width, 28 and 250 ppm; recycle delay, 1 and 2 s; broad-band decoupling14 was used for 13 C spectra to remove 1H spin−spin coupling with 13C nuclei. The 1H NMR analysis of the condensed-phase products produced the molar fractions (relative to −CH2−) of the following groups as a function of time and temperature: vinyl (RHCCH2, δ = 4.8−5.1 and 5.6−6.0 for the terminal hydrogens and the nonterminal vinylic hydrogen, respectively), vinylene (R1HCCHR2, δ = 5.2−5.6), vinylidene (R1R2C CH2, δ = 4.7−4.8), and −CH3 (δ = 0.8−1.0). Here, δ denotes chemical shift values in ppm, identified for particular groups by using the spectral assignments of ref 7 for PE and refs 15 and16 for long-chain alkanes and alkenes. The abundances of vinyl groups obtained from the two different kinds of H atoms were consistent with each other; the molar fractions used in the data analysis were obtained by averaging the values from the terminal and the nonterminal hydrogens. 13 C NMR spectroscopy was used to monitor the kinetics of two functional groups: −CH3 and CHY (i.e., a tertiary carbon or a chain branching point). Each of these groups is associated with multiple peaks on the spectrum as carbons adjacent to −CH3 and CHY have distinctly different chemical shifts. The following chemical shifts and types of C atoms were detected and quantified (associated labels are based on the terminology used in refs 13, 17, and 18 and others), the corresponding chemical shifts were assigned based on the data in these references: −CH3 (1S, δ = 13.6), next carbon relative to −CH3 (2S, δ = 22.4), the following carbon in the chain (3S, δ = 31.7), CHY (tertiary carbon, δ = 37.8), α (one of the three carbons next to the branching point, δ = 34.2), and β (C in the next position after α, δ = 26.9). The −CH2− midchain carbons produce a large peak with chemical shift δ = 30 ppm. No signal was observed at the chemical shift value (42.5 ppm13) corresponding to H links between PE chains. This was expected because H links are primarily products of termination (radical recombination) reactions, whereas all detected functional groups are produced in the reactions of chain propagation (vide infra). Broad-band decoupling used in the current study creates the nuclear Overhauser effect (NOE), which affects the peak

double carbon−carbon bonds in the condensed-phase pyrolysis products to less than 0.5% of the concentration of the −CH2− groups. The pyrolysis temperature in different sets of experiments was incremented by 10 °C, thus resulting in the following experimental temperatures: 400, 410, 420, 430, and 440 °C. At each combination of temperature and pyrolysis time, four experiments were performed; the abundances of the products obtained under each set of experimental conditions were averaged during the analysis. Calculations were performed to estimate the characteristic time of escape of the small molecule gaseous products from PE melt and the fraction of these products remaining dissolved in the melt. The diffusion coefficient and the solubility of C3H6 in PE melt11 were used in these calculations. The results demonstrate that the escape time (22 s at 400 °C) is much shorter than the interval between measurements (5 min). The solubility calculations show that less than 1% of small gaseous products are likely to remain trapped (dissolved) in liquid PE. Therefore, the measured abundances of the gaseous products reflect the real-time kinetics of these products’ formation in the pyrolysis process. The gaseous products of HDPE pyrolysis were analyzed using gas chromatography (GC) and gas chromatography/mass spectrometry (GC/MS). Although the GC/MS method enables identification of the chemical compounds responsible for individual chromatographic peaks, it could not be used routinely for analyzing the results of the experiments due to its lower sensitivity compared to the GC method used with a flame ionization detector (FID). Therefore, a single pyrolyzed HDPE sample was analyzed using the GC/MS method. This sample was pyrolyzed to a significant extent that produced large analyzable amounts of gaseous products. GC/MS analysis was used to identify chemical species for each observed chromatographic peak. The corresponding retention times of the products were used later for identification of species in the GC analysis, where the same chromatographic column and temperature program were used (see below). Pure samples of normal alkanes and alkenes from methane and up to pentane and pentene were also used to determine the retention times of these compounds, which were found to match those obtained from the GC/MS analysis. Both the GC/MS and the GC analyses used the same model of chromatograph (Hewlett-Packard 5890 Series II) and the same column (Agilent J&W GS-GasPro, i.d. 0.32 mm, 30 m length). The mass spectrometer part of the GC/MS setup was Hewlett-Packard 5989B, used with the 70 eV electron impact ion source. The temperature of the GC oven was kept at 50 °C for 5 min, with subsequent 20.0 °C min−1 increase to 200 °C, followed by 10 min at 200 °C. After each pyrolysis experiment, a 1 mL gas-phase sample of the contents of the reactor was used for GC analysis. In order to determine the abundances of gaseous products and account for fluctuations in the instrument’s sensitivity, a 0.5 mL injection of a calibration mixture was made before and after every injection of pyrolyzed samples. The calibration mixture consisted of methane (50% molar fraction), ethane (25% molar fraction), and ethylene (25% molar fraction). The abundances of the gaseous products were calculated under the assumption that the signal produced by a hydrocarbon in the FID detector is proportional to the number of carbon atoms in the hydrocarbon.12 Both 1H NMR and 13C NMR were used for analyzing the condensed-phase fraction of the products of HDPE pyrolysis. C


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The Journal of Physical Chemistry A intensities; the extent of the effect depends on the specific mechanisms through which nuclear spins relax after excitation.18 To enable quantitative analysis of 13C NMR spectra, generally, the recycle delay must be at least five times longer than the spin−lattice relaxation time (T1) in the sample.19 The pulse delay used in the current work (2 s) was shorter than the value of T1 for −CH3(1S), 2S, 3S, and −CH2− (3.3−12.4 s)20 and only somewhat longer than T1 for CHY, α, and β (1.3−1.5 s).20 However, according to Randall,18 in most cases 13C NMR analysis of polymers enables quantitative interpretation because most polymers exhibit the maximum possible value of NOE of 3, and thus, relative intensities of peaks do correspond to relative abundances of functional groups. A study by Qu et al.13 found that errors in quantitative mole fractions of PE functional groups obtained in 13C NMR experiments with pulse delays much shorter than five times T1 amounted to only a few percent of the peak intensity. Therefore, we use the abundances of −CH3 and CHY obtained in the current study as semiquantitative, i.e., the values are used in the analysis but with caution, and the effects of potential errors on the results of kinetic modeling are investigated, as described below. The intensities of different peaks associated with the same functional group (1S, 2S, and 3S for −CH3 and CHY, α, and β for CHY) were consistent with each other (detailed data on intensities vs time are presented in the Supporting Information). The values of molar fractions used in data analysis were thus obtained via averaging of these peak intensities, after correcting for different numbers of the corresponding carbons (3 α and β carbons for each CHY). 2.2. Results. The gas-phase products quantified via GC analysis were alkanes and alkenes ranging from methane (CH4) to octane (C8H18). Figure 1 shows examples of the kinetics of small-molecule gas-phase products formation at 420 °C. A complete set of data for the experimental yields of all detected gas-phase species is presented in the Supporting Information (Figures S1−S16, Tables S1−S5). Generally, the growth of gasphase product abundances is linear with time. Figure 2 shows relative abundances of gas-phase products, averaged on logarithmic scale over all experimental temperatures. As can be seen from the bar graph, the largest abundances are those of C3H8 and C3H6. Figures 3 and 4 show typical temporal profiles of the −CH3, CHY, vinylene, and vinylidene groups. The latter two groups are denoted as VE and VD, respectively, henceforth. All of these profiles display linear growth with time; thus, the main quantitative characteristics of these species used in further analysis are their respective rates of production, which are plotted in Figures 5, 6, and 7 in Arrhenius coordinates. The rate of formation of C2H4 is also plotted as a function of temperature in Figure 5. The signals of −CH3 obtained in individual 1H and 13C measurements differ to some extent (e.g., Figure 3); however, when production rates are plotted as a function of temperature (Figure 5), one can see that the results obtained using 1H and 13C NMR are in general agreement. The kinetics of vinyl groups (denoted as VL henceforth) is that of initial growth with a trend to saturation (Figure 8). Thus, data obtained at individual temperatures cannot be characterized by one quantity such as rate of formation; instead, two quantities are needed, e.g., the characteristic rise time and the abundance at saturation. The shapes of these curves as well as the abundances averaged over the period between 5 and 20 min (⟨[VL]⟩, Figure 6a) are thus used in further analysis, as described below. Table 1 lists the notations used for functional

Figure 1. Examples of the experimental abundances of small-molecule gas-phase products as a function of time. Error bars are one standard error of the mean. Lines are linear fits.

Figure 2. Relative abundances of experimentally detected gas-phase products of polyethylene pyrolysis. Values of abundances were averaged on logarithmic scale over all experimental temperatures.

groups and the values of the effective activation energies corresponding to the temperature dependences of the rates of formation of the detected functional groups, ethylene, and the average VL concentration.

3. KINETIC MODELING The analysis of experimental data was performed via kinetic modeling. As stated in the Introduction, the main motivation of the current study is the desire to determine the values and the temperature dependences of the rate constants of several classes of elementary reactions important at the initial stages of PE pyrolysis. Thus, modeling efforts were directed at this goal, as opposed to, e.g., creation of a comprehensive model capable D


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Figure 5. (Symbols) Experimental rates of formation of the −CH3 groups and ethylene as a function of temperature. (Lines) Calculations obtained using the steady-state model, the integrated short kinetic mechanism, and the final (extended) kinetic mechanism.

Figure 3. Examples of the experimental abundances of −CH3 and CHY groups as a function of time. Error bars are one standard error of the mean. Lines are linear fits.

Figure 6. (Symbols) (a) Experimental average molar fraction of vinyl groups (VL, averages over the 5−20 min period) and (b) rate of formation of the vinylene groups as a function of temperature. (Lines) Calculations obtained using the steady-state model, the integrated short kinetic mechanism, and the final (extended) kinetic mechanism.

Figure 4. Examples of the experimental abundances of vinylene (VE) and vinylidene (VD) groups as a function of time. Error bars are one standard error of the mean. Lines are linear fits.

of predicting all potentially experimentally observable quantities. First, a short mechanism was created, which includes all the important reaction types but at the same time is sufficiently E


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Figure 7. (Symbols) Experimental rates of formation of vinylidene (VD) and CHY groups as a function of temperature. (Lines) Calculations obtained using the steady-state model, the integrated short kinetic mechanism, and the final (extended) kinetic mechanism.

simple to enable analysis via both the steady-state approximation and the integration of the differential kinetic equations. Then the mechanism was extended to include other reactions, the importance of which is relatively minor compared to those in the short mechanism. The values and/or the ratios of rate constants were derived from fitting the calculated product formation rates and molar fraction profiles to the experimental data, as described below. 3.1. Short Mechanism. First, a short kinetic mechanism was created to describe the experimentally observed kinetics of product formation. The advantage of limiting the number of reactions to only the most important is that a steady-state approximation can be used for concentrations of radicals, which replaces differential equations with algebraic ones and thus greatly simplifies analysis of the relationships between observable quantities, such as product yields, and combinations of the rate constants of individual elementary reactions. 3.1.1. Included Elementary Reactions. The short mechanism is based on the Rice−Kossiakoff mechanism of the thermal cracking of alkanes,21,22 with added reactions of hydrogen-atom transfer via abstraction and corresponding formation of secondary alkyl radicals and beta-scission decomposition of these secondary radicals. The observed nonlinear behavior of the kinetics of vinyl groups (initial growth with a subsequent trend to saturation) indicates that reactions consuming olefins need to be accounted for. Thus, addition of radicals to olefins was included into the mechanism. Formation of vinylene groups is described via beta-scission of radicals formed by abstracting a hydrogen atom from a carbon next to an alkane branching site,3,23 as well as by abstraction of an allylic hydrogen next to a vinyl group followed by a hydrogen abstraction by the terminal carbon of the thus formed allylic radical.3 Formation of the vinylidene group is described

Figure 8. Experimental values of the molar fractions of vinyl groups as a function of time (symbols) and results of fitting with eq XXIII (lines).

Table 1. Effective Activation Energies (Ea) of the Rates of Formation of the Detected Functional Groups, Ethylene, and the Average VL Concentration functional group b

terminal vinyl vinylened vinylidenee terminal methyl chain branching ethylene

notation ⟨[VL]⟩ VE VD −CH3 CHY C2H4

c

Ea (kJ mol−1 a) 114 236 231 218 203 311

± ± ± ± ± ±

35 51 53 33 30 60

Uncertainties are one standard error of the fit. bRHCCH2 Abundances of VL averaged over the period between 5 and 20 min. dR1HCCHR2 eR1R2CCH2 a c

by the beta-scission of the tertiary alkyl radicals formed by the hydrogen abstraction at the alkane branching site. The following reactions are thus included in the short mechanism used for steady-state kinetic modeling. Here, reactants and products are denoted using terms emphasizing their functionality: −CH2− denotes a single C−C link in the polymer chain, CHY stands for a branching site, CHH is an H F


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The Journal of Physical Chemistry A link, which forms by a recombination of two secondary radicals, RP and RS denote primary and secondary radicals, and VL, VE, and VD stand for the vinyl, vinylene, and vinylidene groups, respectively. Rate constants for individual reactions are, again, described in terms of reaction types, e.g., kβ,S is for the betascission of a secondary radical, kHT,SP is for an H-atom transfer, i.e., abstraction by a secondary radical from a −CH3 group to form a primary radical, kAST denotes addition of a secondary alkyl radical to a VL group in the terminal position, etc. −CH 2− → 2R P R P → C2H4 + R P

kS

(2)

R P + −CH 2− → R S + −CH3

kHT,PS

(3)

R S + −CH3 → R P + −CH 2−

kHT,SP

(4)

R S → VL + R P

kβ ,S

R S + VL → R S + CHY

R S + R S → CHH

(5)

kAST

(6)

kRSS

(7)

Reactions 1−7 form the skeletal mechanism, in a sense that they largely determine the steady-state concentrations of the RP and the RS radicals and the yields of the main experimentally observable products: C2H4, VL, −CH3, and CHY. Initiation (reaction 1) forms primary radicals, which can decompose via beta-scission to form ethylene and another primary radical (reaction 2) or abstract a hydrogen atom and thus form a secondary radical and a −CH3 group (reaction 3). The secondary radicals can decompose to produce a vinyl group at the end of a long polymer chain (VL) and a primary radical (reaction 5) or revert to a primary radical via reaction 4 (the reverse of reaction 3). VL groups can be destroyed by terminal or nonterminal addition of secondary radicals. Only terminal addition (reaction 6) is included here because the rate constant for nonterminal addition is expected to be significantly lower (vide infra). Termination is described by the recombination of secondary radicals, reaction 7. As will be shown below, the concentrations of RS exceed those of RP by 1−2 orders of magnitude; therefore, recombination reactions involving RP are neglected. The following reactions that describe the formation of VE and VD can be considered as a perturbation of the skeletal mechanism R S + CHY → R SY + −CH 2−

3kHT,SS

(8)

R P + CHY → R SY + −CH3

3kHT,PS

(9)

R S + CHY → RT + −CH 2−

kHT,ST

(10)

R P + CHY → RT + −CH3 R SY → VE + R P

kβ ,S

R SY → VL + CHY + R P R SY + −CH 2− → CHY + R S

RT → VD + R P

kHT,PT

kHT,SS

kβ ,T

RT + −CH 2− → CHY + R S

(11) (12)

1 kβ ,S 2

(13) (14) (15)

kHT,TS

kHT,SA

(17)

RA + −CH 2− → VL + R S

kHT,AS

(18)

RA + −CH 2− → VE + R S

kHT,AS

(19)

Reactions 8 and 9 are those of a hydrogen abstraction from one of the three carbons next to the alkane branching point (CHY), forming a secondary RSY radical. Reactions 10 and 11 abstract a tertiary H atom from CHY, thus forming a tertiary radical RT. RSY can decompose via scission of one of the two C−C bonds next to the branching point, thus producing a VE group and a primary radical (reaction 12), or via the scission of the C−C bond on the other side of the radical site, thus producing a VL group in the beta position relative to the CHY branching point (reaction 13). An alternative reaction for RSY is the hydrogen abstraction (reaction 14, the reverse of reaction 8). Similarly, the tertiary radical RT can react via beta-scission, producing VD and RP, or via abstraction, reverting to CHY + RS (reactions 15 and 16). Abstraction of an allylic hydrogen next to a VL group results in the formation of a delocalized allylic radical RA (reaction 17), which can further abstract a hydrogen atom from −CH2− by one of its two sites with radical character. In the case of the reacting terminal site, a VE group is formed in the beta position from the chain end (reaction 19). In the case of the nonterminal reacting site, the VL group and RS are regenerated (reaction 18). Reactions of abstraction from −CH3 groups are neglected here because of the low concentrations of the latter compared with those of the −CH2− groups. Reactions of radical isomerization (also sometimes referred to as backbiting) were not specifically distinguished in the model. These reactions affect distribution of small-molecule gas-phase products;1−3,24 however, modeling efforts performed in the current study did not attempt to reproduce the experimental yields of the latter. The detailed results on these experimental yields are nevertheless reported (Tables S1−S5 of the Supporting Information) so that any future modeling efforts directed at elucidation of the kinetics of specifically isomerization reactions can take advantage of these data. The reasons for abstaining from distinguishing radical isomerizations from the general class of abstraction reactions are as follows. If these reactions are included as different from reaction 2, rate constants for them must be assigned. Two options are possible. In one option, these rate constants can be treated as known and Arrhenius expressions would have to be assigned based on gasphase literature data. In the other option, one can attempt to derive these rate constants from fitting the model to the experimental data on gas-phase product yields. The first option is problematic because it cannot be unequivocally ascertained that the gas-phase rate constants are applicable. It is not clear to what extent PE melt environment may hinder the deformations of the chain end involved in these reactions. As described below (section 3.2.1), the results of modeling provide support to the hypothesis that PE melt impedes the tumbling motions of separating fragments in C−C bond scission; a similar phenomenon may affect isomerization reactions. If the second option is to be chosen, this would significantly complicate the reaction mechanism as individual reactive pathways specific to the radical site locations relative to the PE chain end will need to be tracked. In case of either option, the situation will also be complicated by the need to include not only isomerization reactions but also, potentially, sequences of H abstraction by RP

(1)

kβ ,P

R S + VL → RA + −CH 2−

(16) G


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approximation, the rate constant of the decomposition of the central bond in n-butane derived from the low-temperature experiments on the reverse reaction, that of recombination of ethyl radicals to form butane,33 and the known reaction thermochemistry34 were used for kS in the modeling of the current experimental study

from a nearby chain followed by another abstraction from the original chain, thus forming RS with radical site in positions close to the chain end. Such two-step sequences will form the same radical types as direct isomerization. Thus, a decision was made to put modeling of radical isomerizations and the resultant production of specific gasphase products other than ethylene beyond the scope if the current study. As discussed above, the main motivation of the current study is to determine the kinetic parameters of several major types of elementary reactions important at the initial stages of PE pyrolysis and not to create a comprehensive mechanism capable of describing every bit of observable data. It can be emphasized that in the modeling performed in the current work, particularly in the mechanism of reactions 1−19, isomerization reactions are not neglected but rather included in the broad category of H-transfer reactions. Other reactions that may occur but were not included in the model are those of molecular scission of PE chains (into alkane and alkene products) and C−C scission immediately followed by radical disproportionation, also resulting in alkane and alkene products. Molecular scission, although suggested in some earlier studies of PE pyrolysis, has a prohibitively large energy barrier. C−C scission followed by disproportionation is also not included because, as will follow from the results of modeling, the rates of production of −CH3 and VL groups in the reactions of propagation are orders of magnitude larger than the rate of C−C scission; therefore, if scission− disproportionation does occur, its effect is likely to be negligible compared to the overall rates of production of these groups. C−C scission at branching points, which can be expected to have a somewhat lower energy barrier compared to midchain scission, is also neglected. The justification for omitting these reactions will be clear from the results of modeling performed with the extended mechanism, in which reactions of decomposition of vinyl-terminated PE chains into RP and allyl radical are included; the latter reactions have even lower energy barriers than those of scission at branching points, and yet they are shown to have very little effect on the rates of product formation. Finally, in radical−radical reaction the disproportionation channels are ignored because of their relatively low expected branching fraction25,26 and the fact that, being a part of the set of termination reactions, they contribute much less to the rates of product formation than the reactions belonging to the chain propagation set. 3.1.2. Rate Constants of Individual Reactions. Initiation, kS. The rate constant of the C−C backbone scission has received substantial attention in the recent two decades. Relevant discussions are available in refs 3 and 27−32. In short, it has been suggested in a number of articles that backbone scission is likely to have lower activation energy compared to analogous reactions in the gas phase and thus higher values of the rate constants at the typical temperatures of the onset of polyethylene pyrolysis (300−400 °C). Our most recent reactive molecular dynamics (RMD) study of the condensed-phase effects of C−C bond scission32 demonstrated that the per-bond rate constant is lower in the liquid polymer melt by approximately an order of magnitude compared to the conditions of a single-polymer molecule in vacuum. Combined with the earlier RMD study of the effects of molecular chain length on the per-bond rate constant, this lead to the conclusion that per-bond rate constants of C−C scission of long alkanes in the condensed phase are likely to be similar to those of small alkanes in the gas phase.32 Thus, as the first

k S = 2.06 × 1017 exp( −43 611K /T ) s−1

(I)

In the skeletal chain reaction mechanism of reactions 1−7, the concentration of the chain carrier radicals and, thus, the overall reaction rate is determined by the ratio of kS and kRSS, the rate constants of recombination of the RS radicals. As will be seen below from the results of modeling, the use of the kS described by eq I to fit the experimental data of the current study would require values of kRSS in the range from 1 × 10−13 to 1 × 10−12 cm3 molecule−1 s−1, on average about an order of magnitude lower than Tsang’s recommendation for the gasphase recombination of the secondary i-C3H7 radicals.35 Such a high value of kRSS appears unrealistic for the conditions of the polymer melt, where the recombination of long-chain radicals is expected to be limited by the diffusion (see the discussion of kRSS below). Lowering the values of kRSS would require decreasing those of kS relative to eq I as well. It is instructive in this respect to recall that the RMD studies of refs 29 and 32 used the OPLS-AA force field36,37 that did not allow for the gradual decrease of the force constants of the bending and torsional terms associated with the C−C bond that is being stretched during C−C scission. It is this effect and the resultant increase of the partition functions of the tumbling motions of separating fragments in the “loose”38 transition state that is responsible for the large preexponential factors (1016−1017 s−1) of small alkane decomposition in the gas phase, compared to lower values of the preexponential factors (1013−1014 s−1) of reactions that do not proceed through particularly “loose” or “tight” transition states.38 It appears likely that the corresponding large-amplitude tumbling motions of the separating radical fragments will be severely impeded under the conditions of the polymer melt, which will likely result in a lower preexponential factor. Since the study of ref 32 did not describe the “loose” degrees of freedom explicitly, the conclusion of the similarity between the C−C scission rate constants of small molecules in the gas phase and those of the long chains in the PE melt reached in ref 32 does not contradict the notion of a lower preexponential factor in the polymer melt. Therefore, in the modeling process, first, the value of the radical recombination rate constants kRSS was selected on the basis of the estimated diffusion coefficient (∼10−15 cm3 molecule−1 s−1, about 4 orders of magnitude lower than expected from the gas-phase analogy), as described below, and then the ratios of kS/kRSS, and thus, the values of kS were obtained from fitting of the experimental data. Termination, kRSS. Under the conditions where diffusion affects the rate of encounters between radicals and thus the effective recombination rate, the latter can be calculated as kRSS = (kREC−1 + kDIFF−1)−1,39 where kREC and kDIFF are the (unhindered) recombination and the diffusion rate constants, respectively. The diffusion rate was estimated using the Smoluchowski equation40 kDIFF = 4πR 0D

(II)

Here, R0 is the critical distance at which the recombining radicals are sufficiently close for the reaction to occur without being hindered by diffusion. The value of R0 = 4.5 Å was H


Article


hand, since the rate constants of C−C scission in PE melt are expected to be equal to those of small analogous reactions of molecules in the gas phase (abstracting away from the issue of the “loose” degrees of freedom), the same can be expected of the reactions of beta-scission. Thus, the recommended expressions for the beta-scission of primary butyl radical50 and secondary hexyl radical51 were used. The latter rate constant was multiplied by 2 to reflect the increased reaction degeneracy in the case of the RS long-chain radical compared to the 2-C6H13 radical

estimated based on the distance between adjacent carbons belonging to different parallel chains in the lamellae of linear PE.41,42 When eq II is used for recombination of radicals in viscous solvents, a spin factor of 1/4 is usually added to the equation to reflect the fact that only one of the four possible spin combinations of the two approaching radicals will result in recombination on an attractive singlet potential energy surface. However, we do not include the spin factor in our estimate, under the assumption that in the case of slow diffusion the lifetime of a pair of radicals in a cage will likely be sufficiently long for the orientation of radical spins to randomly change and eventually provide a singlet configuration.43 The diffusion coefficient was estimated using the assumption that the segmental and the translational44 diffusion are characterized by the same values of mobility, as argued by Barner−Kowollik and Russell43 for polymer solutions with high concentrations of polymers (a PE melt is the limiting case of a solution with zero solvent concentration). The study of the Pearson et al.45 demonstrated that the PE diffusion coefficient at 175 °C depends on the weight-averaged molar mass via a power law D = 1.65 × Mw−1.98 cm2 s−1, in agreement with theory based on reptation46 dynamics of PE chains, which predicts the power of −2 dependence on Mw.46,47 Using this result and the activation energy of 26 kJ mol−1,48 we obtain the following expression for the diffusion coefficient and the rate constant D = 1770M w

−1.98

2 −1

exp(− 3127K /T ) cm s

kβ ,P = 1.05 × 1013 exp( −14 000K /T ) s−1 kβ ,S = 8.94 × 1011T 0.57 exp( − 14 114K /T ) s−1

(VII)

There are no experiment-based recommendations for the rate constant of gas-phase C−C beta-scission of a tertiary alkyl radical. Thus, kβ,T was estimated using expression for kβ,S, the difference between the energy barriers for the reactions of betascission of 2-butyl and 2-methyl-2-butyl radicals (3.5 kJ mol−1) obtained by Sayes et al.52 in their systematic quantum chemical study of the energy barriers of a large series of cognate reactions, and the 1.5 correction for the reaction path degeneracy kβ ,T = 1.34 × 1012T 0.57 exp( − 13 693K /T ) s−1

(III)

(VIII)

H-Atom Transfer, kHT,PS, kHT,SP, kHT,SS, kHT,PT, kHT,ST, kHT,TS, kHT,SA, and kHT,AS. The value of kHT,PS is linked to the experimentally observable quantity: the ratio of the rates of production of ethylene and vinyl groups (vide infra). Thus, kHT,PS was obtained from data fitting. The rest of the H-atom transfer rate constants were linked to kHT,PS via equilibrium constants and structure−activity relationships, as described below. kHT,SP is linked to kHT,PS via the equilibrium constant for the reaction

kDIFF = 1.00 × 10−3M w −1.98 exp( −3127K /T ) cm 3 molecule−1 s−1

(VI)

(IV)

The Mw value of the PE sample used in the experimental part of the current study was estimated as 67 700 Da based on the correlation between the melt index and Mw.49 Thus, we obtain kRSS = kDIFF = 2.72 × 10−13 exp(− 3127K /T ) cm 3 molecule−1 s−1

n‐C4 H 9 + n‐C4 H10 ⇄ s‐C4 H 9 + n‐C4 H10 kHT,PS KHT,PS = ΔH o 0 = −13.94 kJ mol−1 kHT,SP

(V)

The 420 °C estimated value of kDIFF is 3.0 × 10−15 cm3 molecule−1 s−1; comparing it with Tsang’s recommendation35 for the gas-phase recombination of the secondary i-C3H7 radicals kREC = 5.4 × 10−10 T−0.7 cm3 molecule−1 s−1 (5.5 × 10−12 cm3 molecule−1 s−1 at 420 °C), we can see that recombination of radicals is indeed in the diffusional limit and thus kRSS = kDIFF. Any reaction requiring an encounter of the reactive radical with a particular functional group in PE melt can be affected by diffusion as well. However, as will be shown below, the values of the rate constants of all such reactions in the chemical mechanism of PE pyrolysis used here are substantially lower than those of kDIFF and thus are not affected by diffusion. The reactions of abstraction of H atoms from −CH2− groups are not affected by diffusion regardless of their rate constant values because any radical in PE melt is always surrounded by −CH2− groups. Beta-Scission, kβ,P, kβ,S, and kβ,T. The gas-phase rate constants for the beta-scission of primary and secondary alkyl radicals were used. These reactions do not have either “loose” or especially “tight” transition states, and thus, the expectation of a lower preexponential factor in the PE melt (see the discussion of kS above) does not apply to them. On the other

Here and henceforth, KHT,XY denotes the equilibrium constant KHT,XY = kHT,XY/kHT,YX. The values of KHT,PS (8.5−10.0 at T = 400−440 °C) were calculated using the known heats of formation of the primary53 and the secondary54 butyl radicals and their molecular structures, vibrational frequencies, and torsional barriers from the models used in refs 50 and 55. The resultant ratio of the preexponential factors of kHT,SP(T) and kHT,PS(T) is 1.54, very close to the factor of 3/2 expected from the reaction path degeneracy. The values of kHT,SS were calculated from those of kHT,PS using the Evans−Polanyi relationship Ea = E0 + 0.5ΔHo0, where the value of 0.5 in the equation was selected because the reactions in question are nearly thermoneutral. Thus kHT,SS = kHT,PS × exp( −0.5 × 13.94 kJ mol−1/RT )

(IX)

kHT,TS was linked to kHT,SS using, again, the Evans−Polanyi relationship presented above, with ΔHo0 for the following reaction calculated using the literature data55−60 on the heats of formation, molecular structures, vibrational frequencies, and torsional barriers I


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The Journal of Physical Chemistry A kAPT = 3.3 × 10−21T 2.44 exp(− 2444K /T )

t ‐C4 H 9 + n‐C4 H10 ⇄ s‐C4 H 9 + i‐C4 H10

cm 3 molecule1 s−1

ΔH o 0 = +7.11 kJ mol−1

(X)

The ratio of the rates of nonterminal to terminal additions is calculated as the ratio of the rate constants of the reactions of addition of the methyl radical to the terminal and the nonterminal sites of the double bond of propene.50,55 The resultant ratio is

The resultant kHT,ST = 0.5 kHT,SS × exp(0.5 × 7.11 kJ mol−1/ RT), where the coefficient 0.5 is obtained from reaction degeneracy. Similarly, considering the thermochemistry of the above two reactions, we obtain kHT,PT = 0.5 kHT,PS × exp(0.5 × 7.11 kJ mol−1/RT). The values of kHT,TS were obtained from kHT,ST via the equilibrium constant KHT,TS (0.77−0.86 between 400 and 440 °C) calculated using the same thermochemical and molecular properties data. The rate constant for the abstraction of a hydrogen atom in an allylic position (kHT,SA) is rather uncertain. Using the difference in R−H bond energies at 0 K (41.8 ± 10.9 kJ mol−1)58 between secondary alkyl (s-C4H9) and allyl (C3H5) radicals with the Evans−Polanyi relationship given above, one obtains the activation energy difference of 21 kJ mol−1 and the corresponding factor of 38 difference in the values of kHT,SA and kHT,SS. On the other hand, if the coefficient of 0.2561 is used in the Evans−Polanyi relationship instead of 0.5 (abstraction of an allylic hydrogen is exothermic, compared to almost thermoneutral in the case of secondary hydrogen), the activation energy difference becomes 10.5 kJ mol−1 and the corresponding difference in the values of kHT,SA and kHT,SS is only a factor of 6. As will be demonstrated below, the production of vinylene groups is sensitive to the value of kHT,SA. Therefore, kHT,SA was used as a fitting parameter in analyzing the experimental kinetics of product formation. The value of kHT,AS is linked to that of kHT,SA through the equilibrium constant KHT,SA (34−51 at T = 400−440 °C) calculated using the known58 enthalpy of reaction

kASN/kAST = kAPN/kAPT = 0.0779T 0.29 exp( − 560K /T ) (XI)

Therefore, kASN and kAPN are calculated from the values of kAST and kAPT and eq XI. Summary of Rate Constants. The following rate constants are set as equal to those of cognate reactions in the gas phase: kβ,P and kβ,S, with a reaction path degeneracy correction for the latter. kR,SS is estimated using existing data on PE diffusivity. This leaves four rate constants that were obtained from the fitting of the experimental kinetic data: kS, kHT,PS, kHT,SA, and kAST. The rest of the rate constants used in modeling are linked to those listed above via factors based on quantum chemical studies of differences in reaction energy barriers, Evans− Polanyi relationship, or equilibrium constants. One exception is kAPT, which is estimated from the rate constant of a similar gasphase reaction and quantum chemical data on energy barrier differences. As will be shown below, the predicted kinetics of product formation under the experimental conditions of the current study is insensitive to kAPT. 3.1.3. Kinetic Modeling: Short Mechanism, Steady State. Application of the steady-state approximation method to the kinetics of PE pyrolysis within the framework of the short mechanism of reactions 1−19 results in simple equations for RP, RS, RSY, RT, and RA, linking the concentrations of these radicals and the experimentally observable quantities with combinations of the rate constants of key reactions. These equations are presented below, accompanied by a discussion of the expected values of the involved rate constants. A derivation of these steady-state equations is given in the Supporting Information.

s‐C4 H 9 + C3H6 ⇄ C3H5 + n‐C4 H10 ΔH o 0 = −41.84 kJ mol−1

and molecular properties of the involved molecules from the models used in refs 55, 60, 62, and 63. Kinetic modeling is not sensitive to the values of kHT,AS because the only two reactions of RA (eqs 18 and 19) are assumed to have equal rate constants. Radical Addition to Double Bonds, kAST, kASN, kAPT, and kAPN. As will be shown below, the predicted temporal profiles of vinyl groups (VL) are very sensitive to the value of kAST. The reason for this sensitivity is that the main channel of consumption of VL is reaction 6, terminal addition of the secondary radical. Thus, the values of kAST were obtained in the fitting of experimental data. Nonterminal addition, having noticeably lower values of rate constants,50,55 is neglected in the short mechanism of reactions 1−19 used for the steady-state analysis. Also neglected in the short mechanism are the reactions of addition of primary radicals, due to the much lower concentrations of RP compared with RS (vide infra). However, all of these originally neglected reactions are included in the final, extended mechanism of PE pyrolysis, which was used in modeling performed via integration of kinetic differential equations. Therefore, the corresponding rate constants are also discussed here. The estimate for kAPT is based on the rate constant of the ethyl radical addition to ethylene,50 with 2.1 kJ mol−1 reduction of the energy barrier due to alkyl substitution on the nonterminal end of the double bond (based on the quantum chemistry study of Sayes et al.52) and the reaction path degeneracy factor of 0.5

[R S] =

k S[−CH 2−] kRSS

[R P] = [R S]

[R SY] =

[RT] =

(XII)

kHT,SP[−CH3] + kβ S kHT,PS[−CH 2−]

(XIII)

3kHT,SS[R S] + kHT,PS[R P] 3 k 2 βS

+ kHT,SS[−CH 2−]

kHT,ST[R S] + kHT,PT[R P] kβ T + kHT,TS[−CH 2−]

[RA] = [R S]

kHT,SA[VL] 2kHT,AS[−CH 2−]

=

[CHY ] (XIV)

[CHY ]

(XV)

[R ][VL] 1 KHT,SA S 2 [−CH 2−] (XVI)

After several times the induction period τ= J

1 2 k SkRSS[−CH 2−]

(XVII) DOI: 10.1021/acs.jpca.5b07440 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article


Equations for the rates of production and concentrations of CHY, VE, and VD are presented in the Supporting Information. The experimental part of the current study yielded experimentally observable time dependences of molar fractions (relative to −CH2−) of the following species or functional groups: C2H4, −CH3, VL, VE, and VD. The molar fraction of CHY groups was also obtained as a function of time; however, the reliability of [CHY] measurements is questionable because of the short recycle delay time in the 13C analysis of the samples, as described above. The experimental temporal profiles of C2H4, −CH3, CHY, VE, and VD appear linear, and thus, the associated growth rates can be used as approximately time-independent experimental quantities to which the properties of the model must be fitted. The temporal profiles of VL are those of growth to saturation. Each can be characterized by two quantities: the rate of the initial growth and the saturation limit. Thus, since the CHY data were not targeted for modeling, six observable quantities at each temperature had to be reproduced with the model. As mentioned above (end of section 3.1.2), four rate constants (kS, kHT,PS, kHT,SA, and kAST) were treated as unknown and thus were fitted to reproduce the experimental data. As follows from eqs XVIII and XIX, [CH3] is directly linked to [RS] and, through eq XII, to the value of kS. Equation XXIII shows that the two-parameter profile of [VL] is sensitive to kVL (kVL = kAST + 1/2kHT,SA) and [RS]; the latter is determined by kS. The equation for [C2H4] (eq XXI) consists of three terms, two of which grow linearly with time and the third is quadratic. As shown in the Supporting Information, the first term is predominant, and thus, the rate of formation of ethylene is primarily determined by the following ratio

the concentration of RS remains constant, whereas those of other radicals increase in time because of the growth of the concentrations of the −CH3, CHY, and VL groups. The concentration of the −CH3 groups is expected to undergo a linear growth rate( −CH3) ≡

d[−CH3] = kβ S[R S] dt

(XVIII)

[−CH3] = [−CH3]0 + rate( −CH3) × t = [−CH3]0 + kβ S[R S]t

(XIX)

Ethylene is produced only in reaction 2; therefore, the rate of its production is proportional to the concentration of RP, which slowly increases in time. Even though ethylene escapes the polymer melt into the gas phase, its quantity is expressed here in terms of concentration recalculated for the PE melt conditions for the sake of convenience. d[C2H4] = kβ P[R P] dt ⎛ ⎞ kβ S [−CH3] ⎟⎟ = kβ P[R S]⎜⎜ + KHT,PS[−CH 2−] ⎠ ⎝ kHT,PS[−CH 2−]

rate(C2H4) ≡

(XX)

⎛ ⎞ kβS [− CH3]0 ⎟⎟t [C2H4] = kβ P[R S]⎜⎜ + KHT,PS[− CH 2−] ⎠ ⎝ kHT,PS[− CH 2−] kβ PkβS[R S]2 t 2

+

2KHT,PS[− CH 2−]

(XXI)

Under the experimental conditions of the current study, the linear term in eq XXI is greater than the quadratic term, although the latter is not negligible. Vinyl groups (VL) are produced primarily in reaction 5, with a minor contribution from reaction 13, and consumed in reactions 6 and 17; one-half of VL lost in reaction 7 is restored in reaction 18, and the other half is transformed into vinylidene groups via reaction 19. The steady-state approximation, after removing negligible terms (see the derivations in Supporting Information), yields the following differential and integrated equations for [VL]

kβ Pkβ S[R S] kHT,PS[−CH 2−]

With fixed values of kβ,P and kβ,S and [RS] determined from fitting the profiles of either [−CH3] or [VL], experimental rates of C2H4 yield thus can be fitted to obtain the values of kHT,PS. The steady-state analysis presented in the Supporting Information demonstrates that, with known [RS], the rates of production of VE and VD are most sensitive to kHT,SA and kβ,T, respectively. Considering these sensitivities, two fitting methods can be used. In the first method, [VL] profiles are used to extract the values of [RS] (and thus kS) and kVL (Figure 8, solid lines). The rates of C2H4 production then are fitted to obtain kHT,PS. Finally, [VE] formation rates are fitted to obtain kHT,SA, which, combined with kVL will yield kAST. In the second method, [−CH3] production rates are used to determine [RS] (and thus kS). Then [VL] profiles are fitted using just one fitting parameter, kVL (Figure 8, dashed lines), with [RS] obtained from the preceding step. The rest of the fitting procedure is the same as in the first method. Both methods of data fitting were used. Initial concentrations of CHY, needed in both methods, were taken from the experimental data obtained in 13C NMR analysis, even though these data are characterized as semiquantitative, as described above. The sensitivity of the modeling results to [CHY]0 was studied as a part of the analysis. Varying the initial molar fraction CHY by a factor of 2 resulted in the changes of the fitted values of kHT,SA by less than 22% in the case of increasing [CHY]0 and by less than 10% in the case of decreasing [CHY]0. The fitted values of kAST were affected to the extent of less than

⎛ ⎞ d[VL] 1 = kβ S[R S] − ⎜kAST + kHT,SA ⎟[R S][VL] ⎝ ⎠ dt 2 = kβ S[R S] − k VL[R S][VL]

[VL] =

kβ S k VL

⎛ kβ S ⎞ −k [R ]t + ⎜[VL]0 − ⎟e VL S k VL ⎠ ⎝

(XXII)

(XXIII)

Here k VL = kAST +

1 kHT,SA 2

(XXIV)

accounts for the processes consuming VL. Experimental data indicate that the production rate of VL is greater than that of C2H4 by 1−2 orders of magnitude. Comparing the corresponding terms in eqs XX and XXII and recalling that rate constants of beta-scission of primary and secondary radicals can be expected to be similar in magnitude, one comes to the conclusion that the concentration of RS exceeds that of RP by at least an order of magnitude. K


Article

The Journal of Physical Chemistry A 4%. The rest of the fitted rate constants were not affected, as follows from eqs XVIII, XX, and XXII. The resultant values of the fitted rate constants and concentrations of RS and RP are given in Tables SX−SXX in the Supporting Information. The temperature dependences of the fitted rate constants are shown in Figures 9 and 10, together

Figure 10. Temperature dependences of the values of rate constants of reactions 17 (kHT,SA) and 6 (kAST). (Symbols) Values obtained via fitting of experimental data at individual temperatures with equations S-XXXI (Supporting Information) and XXIII derived using the steadystate approximation. (Lines) (a) Arrhenius expressions derived using the Evans−Polanyi method with coefficients 0.25 (lower dotted line) and 0.5 (upper dotted line) and those used in the steady-state model (dashed line) and in the final extended kinetic mechanism (solid line). (b) Rate constant of the gas-phase terminal addition of the C2H5 radical to C3H664 (dotted line), and the Arrhenius expressions for kAST used in the steady-state model (dashed line) and in the final extended kinetic mechanism (solid line).

Figure 9. Temperature dependences of the values of rate constants of reactions 1 (kS) and 9 (kHT,PS). (Symbols) Values obtained via fitting of experimental data at individual temperatures with eqs XVIII and XX derived using the steady-state approximation. (Lines) (a) Rate constant of the gas-phase decomposition of n-C4H10 to two C2H5 radicals (see text, dotted line), and the Arrhenius expressions used in the steady-state model (dashed line) and in the final extended kinetic mechanism (solid line). (b) Rate constant of the gas-phase abstraction of the secondary H atom from C3H8 by the C2H5 radical35 (dotted line), and the Arrhenius expressions used in the steady-state model (dashed line) and in the final extended kinetic mechanism (solid line).

kHT,PS, kHT,SA, and kAST are given in Table 2. Also in Table 2 are given Arrhenius or modified Arrhenius parameters for other reactions (1−19) in the short mechanism, calculated from kS, kHT,PS, kHT,SA, and kAST as described in section 3.2.1. Temporal profiles of the concentrations of −CH3, C2H4, VL, VE, VD, and CHY products or groups were then calculated using the Arrhenius or the modified Arrhenius parameters given in Table 2 and the steady-state model of eqs XII−XXIV and SXXVIII, S-XXXII, and S-XXXV (Supporting Information). For the linear and approximately linear dependences, effective rates of product formation were calculated via linear fits through the calculated values over the 0−20 min time range. Figures 5, 6, and 7 display the modeled temperature dependences of the product rates of formation (dotted lines) together with the experimental rates. For vinyl groups (VL), two plots illustrate the agreement of the model with the experiment: the actual [VL] vs time profiles for individual temperatures (Figure 11, dotted lines) and the temperature dependence of the averaged VL concentrations (modeled and experimental) over the time period of 5−20 min (Figure 6a). As can be seen from the plots, the short kinetic mechanism used with the steady-state model reasonably well describes the rates of formation of −CH3, C2H4, and VE and the profiles of VL. This is not surprising because the critical rate constants were fitted to reproduce these experimental data. However, as was pointed out above, four fitting parameters (kS, kHT,PS, kHT,SA, and kAST) were used to describe a set of five

with the Arrhenius dependences of the rate constants of selected cognate gas-phase reactions of small molecules.35,64 The fitted [VL] profiles are shown in Figure 8 together with the experimental data. The data-fitting process described above resulted in the values of the rate constants at individual temperatures; two methods of data fitting resulted in two sets of rate constant values. As can be seen from Figures 9 and 10, rate constants derived for specific temperatures show substantial amounts of scattering on Arrhenius plots. To create a model that would describe the experimental kinetics of product formation at all temperatures, Arrhenius expressions for kS, kHT,PS, kHT,SA, and kAST were developed from the fitted values presented in Figures 9 and 10. The rather narrow temperature range of the experimental study and the amount of data scattering do not allow for the determination of the activation energies. Therefore, activation energies of cognate reactions in the gas phase were used, and the preexponential factor for each rate constant was selected to minimize deviations between the Arrhenius expression and the individual fitted values on the logarithmic scale. The resultant Arrhenius parameters for kS, L


Article


Table 2. Parameters of the Rate Constant Expressionsa Used in the Steady-State Model (short kinetic mechanism) and in the Extended Kinetic Mechanism of Polyethylene Pyrolysis over the Temperature Range of 400−440 °C SS model rate constant

a

relevant reactions

kS kS,A

1 20

kβ,P kβ,S kβ,T

2, 23, 24, 34 5, 12, 13, 28, 29, 30 15

kHT,PS kHT,SP kHT,SS kHT,ST kHT,PT kHT,TS kHT,SA kHT,AS

3, 9, 25, 27 4 8, 14, 26, 31, 35 10 11 16 17 18, 19,21

kAST kASN kAPT kAPN

6, 39 22, 37 32, 38 33, 36

kRSS kRb

7, 40, 41 7a, 40a, 41a

A

full model

n

Ea

C−C backbone scission 1.60 × 10+15 0.00 43 611 beta-scission of radicals 1.05 × 10+13 0.00 14 000 8.94 × 10+11 0.57 14 114 1.34 × 10+12 0.57 13 693 H-atom transfer (abstraction) 6.85 × 10−13 0.00 6157 1.06 × 10−12 0.00 7998 6.85 × 10−13 0.00 6995 3.42 × 10−13 0.00 6567 3.42 × 10−13 0.00 5729 1.89 × 10−12 0.00 7896 6.85 × 10−13 0.00 5561 2.27 × 10−11 0.00 10 566 radical addition to double bonds 7.03 × 10−13 0.00 4666

2.72 × 10−13

radical recombination 0.00

A

n

Ea

3.11 × 10+15 2.02 × 10+14

0.00 0.00

43 611 38 711

1.05 × 10+13 8.94 × 10+11 1.34 × 10+12

0.00 0.57 0.57

14 000 14 114 12 833

5.57 8.62 5.57 2.78 2.78 1.54 5.57 1.85

× × × × × × × ×

10−13 10−13 10−13 10−13 10−13 10−12 10−13 10−11

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

6157 7998 6995 6290 5729 8172 4908 9913

4.64 5.48 3.30 2.57

× × × ×

10−13 10−14 10−21 10−22

0.00 0.29 2.44 2.73

4666 5226 2444 3004

2.46 × 10−53

0.00

3127

3127

The rate constant expressions used have the form k = AT exp(−Ea/T). The base units are cm, molecule, s, and K. bRate constant is for the fourthorder reactions, as described in text. n

experimental quantities: the rates of formation of −CH3, C2H4, and VE and two parameters of VL profiles. Thus, the agreement also provides support for the adequacy of the kinetic mechanism as a whole. The agreement between the modeled and the experimental rate of VD formation is approximate (Figure 7a). No data fitting was performed specifically for VD, which also provides support for the adequacy of the mechanism. For CHY, the temperature dependence of the rate of formation is not reproduced: the two lowest temperature data points show good agreement, but at higher temperatures the calculated rates exceed the experimental ones. 3.1.4. Kinetic Modeling: Integration of Kinetic Equations. The kinetics of PE pyrolysis was also simulated using the mechanism of reactions 1−19 and the kinetics integration program Kinsolve.65 The results are displayed in Figures 5, 6, 7, and 11 with the solid lines. In the cases of all observable functional groups or species with the exception of VL the lines on the plots corresponding to the steady state and the integration calculations are barely distinguishable (and not distinguishable in the case of −CH3). In the case of VL temporal profiles (Figure 11), up to 15% difference is observed at long reaction times (15−20 min, more pronounced at higher temperatures), and also deviations are visible at shorter reaction times at lower temperatures. The short reaction time deviations are due to the nonzero induction period, which is neglected in the steady-state model. Equation VI gives the values of τ = 160, 96, 59, 36, and 23 s for the experimental temperatures of 400, 410, 420, 430, and 440 °C, respectively. At the hightemperature end of the experimental range the effect of the induction period disappears, but at the low-temperature end it

is non-negligible. The long-time minor deviations are due to the neglect in the steady-state model of several minor terms in the kinetic equations (detailed derivations of the steady-state equations and estimates of minor terms are given in the Supporting Information). 3.2. Extended Mechanism, Integration of Kinetic Equations. The main purpose of the steady-state kinetic analysis described above was to emphasize the relationships between the rate constants of the most important elementary reactions and to establish the rate constant combinations that are linked to the observable experimental data. After the steadystate analysis and its confirmation by kinetic simulation via integration of kinetic equations, an extended kinetic mechanism was created. This mechanism included relatively minor reactions, such as nonterminal additions to the double bond of VL, reactions of primary radicals, and others. Inclusion of these reactions in the steady-state model would render it impractical by preventing derivation of the algebraic eqs XII−XXIV and S-XXVIII, S-XXXII, and S-XXXV (Supporting Information). One more benefit of using a full kinetic mechanism unconstrained by the demands of the steady-state approximation is the inclusion of the dependence of the effective rate constant of the diffusion-limited recombination of radicals (termination) on the average PE chain length, which undergoes significant change during the course of the reaction. The method of including this dependence and its effects on modeling are described below. 3.2.1. Additional Elementary Reactions. The following reactions were included in the extended mechanism M


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abstraction of a secondary H atom leading to the formation of RS and the R1−CH(−CH3)−CH(−R2)−R3 group, reflected by the formal reaction products −CH3 and CHH (denoting the presence of an H link). RS or RP can abstract one of the six secondary hydrogens in beta position to the H link in CHH, thus producing a secondary radical of the R1−CH(−CH3)−CH(−R2)−CH(•)−R3 type, denoted as RSH (reactions 26 and 27). The latter’s likely channels of beta-scission include reactions 28−30 (depending on which of the carbon−carbon bonds gets broken) and an abstraction of a secondary hydrogen from a PE chain (reaction 31, the reverse of reaction 26). Abstraction from the −CH3 group of CHH is neglected because of a higher energy barrier. 3kHT,SS

R S + CHH → R SH + −CH 2−

3kHT,PS

R P + CHH → R SH + −CH3

(27)

1 kβ ,S 2

R SH → R P + VE + CHY

R SH → R S + VE

(26)

(28)

1 kβ ,S 2

(29)

1 kβ ,S 2

R SH → R P + VL + CHH

(30)

kHT,SS

R SH + −CH 2− → R S + CHH

(31)

Terminal addition of RP to VL (reaction 32) results in formation of RS. Nonterminal addition (reaction 33) forms a primary radical with the radical site on a side methylene group attached to the PE chain (R1−CH(CH2•)−R2), denoted here as RPY. This radical can undergo a beta-scission to VL + RP (reaction 34, reverse of reaction 33) or abstract an H atom from a nearby PE chain (reaction 35), thus forming RS and the −CH3 and CHY functional groups R P + VL → R S

Figure 11. Experimental values of the molar fractions of vinyl groups as a function of time (symbols) and calculated [VL] vs time dependences obtained using the steady-state model, the integrated short kinetic mechanism, and the final (extended) kinetic mechanism.

VL → R P + C3H5

k S,A

(20)

C3H5 + −CH 2− → R S + C3H6

kHT,AS

(22)

R PH → R S + VL

kβ ,P

(23)

R PH → R P + VL + CHY

kβ ,P

R PH + −CH 2− → R S + CHH + −CH3

kAPN

(33)

R PY → R P + VL

2kβ ,P

(34)

R P + VE → R SY

kHT,SS

(35)

kAPN

(36)

Addition of RS to VE forms an H-linked radical with the radical site in the beta position relative to the H-link. This type of radical is lumped together with RSH, because most of its subsequent reactions will be of the same types as reactions of RSH (reactions 28−31) R S + VE → R SH

(24)

kHT,PS

R P + VL → R PY

Addition of RP to VE results in formation of RSY

Here, vinyl groups can decompose forming a primary radical and the small allyl (C3H5) radical. The most likely subsequent reaction for the latter is abstraction of an H atom to form propene and RS kASN

(32)

R PY + −CH 2− → R S + −CH3 + CHY

(21)

R S + VL → R PH

kAPT

kASN

(37)

Terminal addition of either RP or RS to VD forms a tertiary radical (RT); in the case of RS an additional CHY group is also formed

(25)

Nonterminal addition of RS to VL (reaction 22) leads to the formation of a primary-type radical R1−CH(−CH2•)−CH(−R2)−R3, labeled “RPH” here to emphasize the presence of an H link between two carbon chains. Its subsequent reactions include the reverse decomposition (reaction 23), beta-scission (reaction 24) to RP and a long chain with an attached side vinyl group (hence the formal products CHY and VL), and

R P + VD → RT

kAPT

R S + VD → RT + CHY

(38)

kAST

(39)

Nonterminal addition of radicals to VD is neglected because of the low abundance of VD and lower values of the rate constants for nonterminal (compared with terminal) addition. N


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The rates of formation of −CH3, C2H4, VE, and VD decreased, on average, by factors of 1.25, 1.5, 2.5, and 10; profiles of VL were affected only to a minor extent. The effect was more pronounced at higher temperatures, where −CH3 groups are formed faster, which resulted in significant reduction of the calculated effective activation energies on the Arrhenius plots of product formation rates. As a result, some of the rate constants of the important reactions had to be adjusted for the calculated results to match the experimental data, as described below. To preserve the agreement between experiment and calculations for VL, the effective overall rate constant of RS reaction with VL (kVL) had to be preserved. After the inclusion of reaction 22

Finally, recombinations of RP and cross-combination of RP and RS are included R P + R P → −CH 2−

kRSS

(40)

2kRSS

R P + R S → CHY

(41)

3.2.2. Rate Constants. The rate constants of additional reactions 20−41 included in the extended model were assigned based on the reaction type and adjusted for reaction path degeneracies, as needed. The assignment of rate constants is indicated above, next to the chemical eqs 20−41. The rate constant kS,A for the decomposition of a PE chain with a vinyl group (VL) producing an allyl radical and RP (reaction 20) was estimated by multiplying the value of kS by the ratio of the rate constants for decomposition of 1-C4H8 (into CH3 + C3H5)66 and C3H8.34 The reactions of radical recombination (termination) were assumed to have the same rate constants for primary and secondary radicals, limited by diffusion. According to eq IV, kDIFF and thus kRSS are inversely proportional to the square of the weight-averaged molecular mass of the PE chain, Mw (to Mw−1.98, to be exact). Assuming that Mw is inversely proportional to [−CH3] and using eq V for the estimated Mw = 67 700 Da of the PE sample used in the current work, we can obtain for the diffusion-limited kRSS dependence on [−CH3] kRSS = 2.72 × 10−13

[−CH3]2 [−CH3]02

k VL = kAST +

(XXV)

Since [−CH3] depends on time, kRSS also increases with time. For practical purposes, to include the quadratic dependence of eq XXV on [−CH3], the recombination reactions 7, 40, and 41 were replaced in the mechanism with formal reactions of the fourth order kR

(7a)

kR

R P + R P + 2 −CH3 → −CH 2 − + 2 −CH3

(40a)

R P + R S + 2 −CH3 → CHY + 2 −CH3

2kR

(41a)

where kR = 2.72 × 10−13

1 e−3127K / T [−CH3]20

= 2.46 × 10−53e−3127K / T cm9 molecule−3 s−1

(XXVII)

(in the steady-state model, the third term was not present, eq XXIV). The values of kASN are linked to those of kAST via eq XI; the average ratio kAST/(kAST + kASN) calculated using eq XI over the experimental temperature range is 0.812. Thus, as the first step of model adjustment, the new values of kAST were calculated by multiplying the old preexponential factor of kAST by this factor. Further adjustments of the rate constants made to improve the agreement between experiment and calculations for each experimentally observable quantity are described in the Supporting Information. The resultant calculated product formation rates and VL profiles and average abundances are shown as a function of temperature together with the experimental data and the results of steady-state analysis in Figures 5, 6, 7, and 11 (solid lines). The Arrhenius and the modified Arrhenius parameters of the final selected rate constants are given in Table 2; the corresponding temperature dependences of kS, kHT,PS, kAST, and kHT,SA are shown in Figures 9 and 10 (solid lines) in comparison with the data for cognate gas-phase reactions of small species. 3.3. Sensitivity of the Models to Rate Constants and Uncertainty in [CHY]0. As discussed above (section 2.1), the results of 13C NMR analysis of the −CH3 and CHY products of PE pyrolysis are treated as semiquantitative, due to the uncertainty in to what extent the NMR signal is proportional to the molar fraction of the particular group. Although the −CH3 formation rates obtained using 13C and 1H NMR spectroscopy agree (Figure 5), the 13C NMR data for CHY groups cannot be independently verified. Therefore, to estimate whether the associated uncertainty has any substantial effect on the analysis, the sensitivity of the final kinetic model to the initial concentration of CHY groups was performed. Kinetic modeling was repeated twice, with [CHY]0 increased and decreased by a factor of 2. The results indicate that the predicted rates of formation of VE, −CH3, CHY, and C2H4 and the concentrations of VL change by less than 7% as a result of [CHY]0 variations. However, the predicted rates of VD formation change by factors of 1.3−2.1, depending on the temperature, with lower initial concentrations of CH Y corresponding to lower predicted rates of VD formation and vice versa. This is an expected result because VD is formed exclusively via the thermal decomposition of RT (reaction 15), which, in turn, is formed by an abstraction of a tertiary H atom from CHY (reaction 10). If the initial concentration of CHY changed compared to the value used in the final model, the rate constant of reactions 15 (kβ,T) and/or 10 (kHT,ST) needs to be

e−3127K / T cm 3 molecule−1 s−1

R S + R S + 2 −CH3 → CHH + 2 −CH3

1 kHT,SA + kASN 2

(XXVI)

3.2.3. Kinetic Modeling (integration). The kinetics of reactions 1−41 (with formal reactions 7a, 40a, and 41a substituted for the original ones to account for the evolution of the diffusional rate constant) was modeled using the integration program Kinsolve.65 The results indicated that inclusion of reactions 20−41 in the model had a relatively minor effect on the calculated product formation rates and the profiles of VL. The largest effect was the increase in the calculated rate of formation of −CH3 at the highest experimental temperature, 440 °C, by 45%, primarily due to the introduction of the new type of primary radical, RPY, which leads to −CH3 formation via H-atom abstraction. However, inclusion of the dependence of the radical recombination rate on [−CH3] had a dramatic effect on the results. As recombination accelerated in time due to the increasing diffusion coefficients, the overall rate of pyrolysis decreased. O


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than 16% (mostly at 400 °C due to the increased induction period). The rate of C2H4 formation decreased by 7−21% (agreement here can be improved by adjusting kHT,PS), and that of VD formation decreased by 63−73%, i.e., approximately by a factor of 3. The large effect observed in the case of VD is due to the factor of 3 decrease in [RS] (see equation S-XXXIV in the Supporting Information). The agreement with experiment can be restored by adjusting kβ,T and/or kHT,SA, in a way similar to that described in section 3.2.3. Second, kβ,S was decreased by a factor of 3. Similarly, kS was increased by a factor of 9, and kAST, kHT,SA, and kASN were decreased by a factor of 3. The results were similar to those caused by the increase in kβ,S but opposite in the directions of change. Again, adjusting kHT,PS, kβ,T, and/or kHT,SA would restore agreement with experiment. Finally, the effects of varying kβ,P were investigated by varying it by a factor of 3. The model is also inherently sensitive to kβ,P; thus, the values of kHT,PS had also to be changed by the same factor to keep the ratio of the rate constants the same and thus counterbalance the effect on the rate of formation of C2H4 (see eq XX, where the first term in parentheses is dominant). The resultant changes in the rates of formation of all products and [VL] concentrations did not exceed 5%. It can thus be concluded that the ability of the kinetic model to describe the experimental observations is independent of the particular choice, within reasonable limits, of the rate constants for the reactions of beta-scission of the primary and the secondary alkyl radicals, provided that the rate constants of other key reactions are adjusted accordingly if the rates of betascission are changed.

adjusted further to reproduce the experimental VD formation rates. As follows from eq XII, the steady-state concentration of RS, which largely determines the overall rate of pyrolysis, is affected primarily by the ratio of kS and kRSS (reactions 1 and 7, initiation and termination). This conclusion holds for the final, extended mechanism as well. Increasing both kS and kRSS by a factor of 100 resulted in essentially the same values of [RS] and produced less than 7% effect on the calculated rates of formation of all products and VL concentrations. Decreasing kS and kRSS, however, can have a more significant effect on the model predictions because of the resultant increase in the incubation period (see eq XVII). When both kS and kRSS were decreased by a factor of 10, significant changes in the rates of product formation were observed, particularly at the lowtemperature end of the experimental range: between 38% and 63% at 400 °C but less than that at higher temperatures. The predicted profiles of [VL] at 400, 410, and 420 °C became visibly distorted in comparison with the experimental data, with delayed growth due to the long induction periods. Decreasing kS and kRSS by a lower factor of 3 results in a less pronounced effect due to the increased induction time, with the rates of product formation changing by only 6−17% at 400 °C and 0.3−6% at 440 °C. There was still a visible delay in the predicted VL growth, but in this case, the predicted [VL] profile was comparable to the experimental data within estimated uncertainties and scatter of the latter. One can therefore conclude that the model is insensitive to simultaneous changes in kS and kRSS, although in the case of decreasing these rate constants the insensitivity extends only to variations of approximately by a factor of 3. In the development of the model of PE pyrolysis in the current work, temperature dependences of two rate constants were taken as equal to those of cognate reactions of small radicals in the gas phase. These are the reactions of betascission of the secondary and the primary alkyl radicals (kβ,S and kβ,T, used for reactions 5 and 2 and other related reactions in the mechanism). It is therefore instructive to investigate to what extent variations in these rate constants would affect the results of the modeling. Equations XII−XXIV and S-XXVIII, SXXXII, and S-XXXV (Supporting Information) derived from the steady-state approximation are helpful in that respect. As can be seen from these equations, kβ,S appears (in combinations with other rate constants) in the expressions for the rates of formation and concentrations of all observable products. Thus, the model is very sensitive to the values of kβ,S, and this sensitivity is an inherent property of the model. However, as demonstrated by the analysis presented below, if kβ,S is changed, other rate constants can be changed as well in such a way that the ability of the model to reproduce the experimental observations is preserved. To investigate the effects of variations in kβ,S and mitigating changes in other rate constants, kinetic modeling was repeated with the following modifications to the mechanism. First, kβ,S was increased by a factor of 3. In order to keep the predicted −CH3 rates, the value of kS was decreased by a factor of 32 = 9 to keep the product kβ,S[RS] the same. To maintain the same asymptotic values of [VL] and the characteristic VL rise times, kVL also had to be increased by a factor of 3; this was achieved by multiplying kAST, kHT,SA, and kASN by 3 (see the discussion of the three components of kVL in section 3.2.3 and eq XXVII). The resultant predicted rates of formation of −CH3, VE, and CHY and the values of [VL] changed by less

4. DISCUSSION 4.1. Experimental Rates of Product Formation and Product Concentrations. The relative abundances of gasphase products (Figure 2) are in general qualitative agreement with the results of earlier studies where gas-phase products were quantified (refs 6 and 8 and a review in ref 3) in that the largest abundances are those of C3 products: propane and propene. Modeling of PE pyrolysis performed in the current study purposely abstained from targeting rates of formation of gas-phase products other than ethylene. As described above (section 3.1.1), modeling the rates of formation of these gaseous products would involve explicitly describing radical site-specific reactions of radical isomerization, which, in turn, would seriously complicate the analysis and extend this work well beyond its stated goals. Therefore, of all gas-phase products, only the rates of formation of C2H4 are used in modeling and thus discussed below. Earlier quantitative studies of the kinetics and product abundances of the initial stages of the pyrolysis of high-density polyethylene (HDPE) include those of Tsuchiya and Sumi,6 Kuroki et al.,7 and De Witt and Broadbelt.8 Kuroki et al. used 1 H and 13C NMR to study the abundances of the same functional groups monitored in the current work (−CH3, CHY, VL, VE, and VD) in the PE samples left over after lowtemperature (350−390 K) pyrolysis in the atmosphere of nitrogen. The pyrolysis times ranged from 30 to 300 min. Tsuchiya and Sumi used gas chromatography to study the gaseous products of HDPE pyrolysis in the 375−425 °C temperature range. The pyrolysis time was 20 min in all experiments; the rates of product formation for comparison with the results of the current study were calculated here assuming linear dependence of product yield on time. De Witt P


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The Journal of Physical Chemistry A and Broadbelt studied HDPE pyrolysis at the single temperature of 420 °C and pyrolysis times of 30−240 min. Both gaseous and liquid products, up to C34 in size, were analyzed by gas chromatography. Product yields were reported as molar ratios relative to the initial numbers of moles of HDPE. To enable comparison with the results of the current study and that of Tsuchiya and Sumi, where results are expressed relative to the concentration of the −CH2− groups, the value of number-averaged molecular mass (Mn) of the HDPE samples used by De Witt and Broadbelt is needed. This value was not reported in ref 8, although that of the weight-averaged molar mass (Mw = 125 000) was given; however, in a later article by Levine and Broadbelt,4 in which kinetic modeling of the results of ref 8 was performed, the value of Mn = 62 500 was used, obtained by assuming the polydispersity value of two. In the current work, comparison of the current experimental data with those of De Witt and Broadbelt is done under the assumption that the same value of Mn was used in ref 8 to calculate the numbers of moles of the HDPE samples. Figures 12, 13, and 14 compare the experimental results of the current study with those of refs 6−8. Also, kinetic modeling

Figure 13. Comparison of the results of the current study with those of Kuroki et al.7 (Symbols) Experimental rates of formation of VE and VD groups as a functions of temperature (error bars are one standard error of the mean). (Lines) Calculations performed with the final (extended) kinetic mechanism, with conditions corresponding to those of the respective experiments.

and those of VD cannot be unequivocally characterized as linear due to the large data scatter. These data were nevertheless fitted to linear dependences for the sake of comparison with the results of the current work. The error bars in Figure 13 reflect the larger amounts of data scatter for VE and VD compared with −CH3 and CHY (Figure 12). The profiles of VL in ref 7 are not well defined. These are not expected to be linear but rather have a pattern of growth to saturation. Thus, to enable comparison, average abundances over the experimental time periods were calculated for both the experimental data7 and the modeled profiles. Although the temperature ranges of ref 7 and the current work do not overlap, trends observed in Figures 12, 13, and 14a indicate that the rates of formation and the abundances of all products observed by Kuroki et al. are higher than in the current study. A likely more meaningful comparison is between the experimental data of ref 7 and the lines calculated using the model developed based on the results of the current work. These lines do not form continuous dependences with those calculated to model the results of the current study because modeling results depend on the initial concentrations of various functional groups. The HDPE sample of Kuroki et al. had Mw = 1.5 × 105 Da and Mn = 1.4 × 104 Da (polydispersity of 10.7), with linear chains containing a VL group at one end of each chain. Figure 14b displays the rates of ethylene formation obtained in the current work (solid circles), in the study of Tsuchiya and Sumi (open squares), and in that of De Witt and Broadbelt (open triangle). Modeling (solid lines) was performed for the conditions of all three experimental studies. The authors of ref 6 did not provide any information related to Mw or Mn; therefore, calculations were performed assuming that their

Figure 12. Comparison of the results of the current study with those of Kuroki et al.7 (Symbols) Experimental rates of formation of −CH3 and CHY groups as a function of temperature (error bars are one standard error of the mean). (Lines) Calculations performed with the final (extended) kinetic mechanism, with conditions corresponding to those of the respective experiments. Filled and open circles in the upper graph correspond to the rates of −CH3 formation obtained using 1H and 13C NMR spectroscopy, respectively.

calculations using the final extended model of HDPE pyrolysis were performed for the experimental conditions of these studies. The results of modeling are shown in Figures 12, 13, and 14 with solid lines. In the study of Kuroki et al.,7 the profiles of −CH3 and CHY show linear growth with time and the rates of formation are thus well defined. The profiles of VE have more relative scatter, Q


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the pyrolysis experiments. It is known that the presence of oxygen accelerates polymer pyrolysis;67 however, it is unknown whether the abundance of dissolved or trapped atmospheric oxygen in the PE samples of refs 6 and 7 was sufficient to cause the observed differences. 4.2. Major Quantitative Rate Constant Results Derived from Experiment. As discussed in section 3.3, the ability of the developed kinetic mechanism to reproduce the experimental data is not so much sensitive to the values of individual rate constants as to the ratios of rate constants. The relationships between the observable quantities such as product formation rates and rate constants ratios are best analyzed using eqs XII−XXIV and S-XXVIII, S-XXXII, and S-XXXV (Supporting Information) derived by applying the steadystate approximation to the short mechanism of reactions 1−19. In the following discussion, corresponding combinations of rate constants that can be referred to as derived from experiment are listed for the major observable quantities. These combinations were calculated using the rate constant temperature dependences from the final, extended kinetic mechanism, and their uncertainty factors were estimated based on the scattering of the experimental data. The rate of formation of −CH3 groups is sensitive to φ1 = kβ S

Figure 14. Comparison of the results of the current study with those of Kuroki et al.,7 Tsuchiya and Sumi,6 and De Witt and Broadbelt.8 (a) (Symbols) Experimental average molar fraction of vinyl groups (VL). Current work: averages over the 5−20 min period. Kuroki et al.: averages at each temperature taken over different time periods within the 30−300 min range (error bars are one standard error of the mean). (b) (Symbols) Experimental rates of formation of C2H4 as a function of temperature. (Lines) Calculations performed with the final (extended) kinetic mechanism, with conditions corresponding to those of the respective experimental data sets.

kS kR

(XXVIII)

which varies with temperature from 2.0 × 1025 to 3.6 × 1026 cm6 molecule−2 s−1 in the case of the short mechanism and steady-state analysis and from 2.8 × 1025 to 5.1 × 1026 (7.4 × 1026 exp(−34 751K/T) cm6 molecule−2 s−1 in the case of the extended kinetic mechanism. We recommend the latter set of values and the associated Arrhenius expression, because of the higher quality of the extended kinetic mechanism. The associated uncertainty can be estimated as a factor of 1.5 from the envelope of uncertainties surrounding the temperature dependence of the −CH3 rate of formation (Figure 5a). Here, kR represents the fourth-order pseudo-reaction (7a) introduced to account for the change in the diffusion-limited radical recombination rate constant kRSS due to that in the average length on the PE chains as reaction progresses; in the case of the steady-state model, which did not include the dependence of kRSS on PE chain length, kR was calculated as kR = kRSS[CH3]0−2. The steady-state approximation rate of formation of the C2H4 product is given by eq XX. The second term in parentheses is minor, between 3% and 4% of the first one. Thus, by using eqs XII, XVIII, and XX and removing constant terms, we obtain that the ratio of the rates of formation of −CH3 and C2H4 is determined by the following quantity

sample of PE was similar to that of Kuroki et al., except for the absence of the terminal VL groups. As can be seen from the plot, the rates of ethylene production from Tsuchiya and Sumi is noticeably higher than those obtained in the current study, whereas that of De Witt and Broadbelt is lower. Similar effects were observed for other gas-phase products: refs 6 and 8 reported higher and lower yields, respectively, compared to the current study. Modeling fails to reproduce the higher yield of C2H4 in ref 6 but is in a reasonable agreement with the results of De Witt and Broadbelt. The calculated line passes lower for the conditions of De Witt and Broadbelt, compared to those of the current study, primarily because of the different values of kRSS = kDIFF. The value of kDIFF is higher for the conditions of ref 8 because of the low assumed value of polydispersity in their PE sample. As discussed below (section 4.2), the way in which estimation of kDIFF was introduced in the current study results in its dependence on polydispersity, which may (or may not) be an artifact of the model. Therefore, it is impossible to unambiguously conclude whether the results of the current work are in agreement with those of De Witt and Broadbelt until more is known about diffusion-limited reactions under the conditions of PE melt. The reasons for deviations between the experimental and the calculated results of the current work and the experimental data of Kuroki et al.7 and Tsuchiya and Sumi6 are unknown. Both earlier studies show higher product yields at all temperatures. One tentative hypothesis that can be offered is based on the fact that only in the current experimental study PE samples were thoroughly degassed in the molten state before

φ2 =

kHT,PS kβ P

(XXIX) −21

φ2 decreases with temperature from 7.5 × 10 to 3.9 × 10−21 cm3 molecule−1 in the case of steady-state analysis and from 6.1 × 10−21 to 3.2 × 10−21 (5.3 × 10−26 exp(+7843K/T)) cm3 molecule−1 in the case of the extended kinetic mechanism. Again, the latter dependence is recommended, with the uncertainty factor of 1.75 estimated from the envelope of uncertainties surrounding the temperature dependence of the ratio of −CH3 and C2H4 rates of formation. The asymptotic concentrations of VL is most sensitive to the ratio R


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The Journal of Physical Chemistry A φ3 =

kβ S k VL

between the values of kβ,T and the rate constants of the gasphase reactions of decomposition of tertiary radicals is not conclusive. Two reactions of the latter type have been studied experimentally, those of decomposition of the 2-methyl-2pentyl68 and 2-methyl-2-hexyl69 radicals. Their reported rate constants (multiplied by 3 to account for the reaction path degeneracy) are compared with the values of kβ,T in the Supporting Information (Figure S27). The values of the multiplied by 3 experimental rate constants are higher68 or lower69 than kβ,T, with an order of magnitude difference between the two experimental studies. The rate constant of the backbone C−C scission (kS) adopted in the extended kinetic model is compared with that of the decomposition of n-butane into two ethyl radicals in Figure 9a. As can be seen from the plot, kS is almost 2 orders of magnitude (a factor of 66) lower that the analogous gas-phase rate constant. The observation appears to provide support to the hypothesis that congested PE melt molecular environment impedes the tumbling motions of separating fragments in C−C bond scission, thus resulting in less “loose” transition state and lower rate constant values. This conclusion, however, should be treated with caution because of the large uncertainties associated with the rate constant for radical−radical recombination, kRSS. All combined uncertainties in the dependences of the PE self-diffusion coefficient on temperature and molar mass (both Mn and Mw), those in the Mw value of the PE sample used in the experimental part of the current work, and those due to the details of the use of the Smoluchowski equation (such as in the value of R0) can amount to an order of magnitude uncertainty in kRSS or even larger. The uncertainty in the thermochemistry of n-butane decomposition (3.7 kJ mol−1)34,56 provides another factor of 2. Apparently, more research is needed in the field of PE diffusion and diffusionlimited recombination of long-chain radicals. Figures S28−S33 (Supporting Information) compare the temperature dependences of the rate constants obtained from data fitting (kS, kHT,PS, and kAST) as well as those selected at the beginning of model development (kRSS, kβ,P, and kβ,S) with the Arrhenius expressions used in earlier PE pyrolysis models developed by Faravelli et al.,2 Poutsma,3 and Levine and Broadbelt.4 The values of kHT,PS and kAST used in earlier models differ from those derived here by not more than a factor of 3. This result appears reasonable considering that earlier studies used estimates based on gas-phase rates and the results of the current work are in agreement with gas-phase based expectations, as discussed above. The values of kβ,P agree within a factor of 2.5, all of the expressions used being based on gasphase experimental dependences. Those of kβ,S used in the current study agree with the values of refs 2 and 3; however, the large preexponential factor estimated in ref 4 makes this rate constant approximately an order of magnitude larger. Greatest disparities between the values used in different studies are observed for kRSS and kS. Reference 4 uses the largest kRSS, between 3.6 × 10−11 and 4.0 × 10−11 cm3 molecule−1 s−1 between 400 and 440 °C; such values are expected of recombination of small radicals in the gas phase. Reference 2 uses approximately 40 times lower values, ca. 10−12 cm3 molecule−1 s−1, and the diffusion-derived values of kRSS used in the current study are approximately 4 orders of magnitude lower than the gas-phase rate constants. Finally, the values of kS used in refs 2 and 4 are lower than those of the current study by average factors of 6.5 and 2.8, respectively, whereas the estimate

(XXX)

(eq XXIII), where kVL = kAST + kASN + 1/2kNT,SA (eq XXVII) is the effective rate constant of the overall reaction of RS with VL (addition at both sites and H abstraction in the allylic position). The values of φ3 increase with temperature from 3.3 × 1019 to 7.4 × 1019 molecules cm−3 in the case of steady-state analysis and from 3.6 × 1019 to 8.0 × 1019 (5.9 × 1025 exp(−9635K/T) molecules cm−3 in the case of the extended kinetic mechanism. The uncertainty factor of 1.3 is estimated from the envelope of uncertainties of the temperature dependence of the average experimental [VL]. Finally, the steady-state equation for the rate of VE production is given by equation S-XXXI (Supporting Information), in which the first term is relatively minor (7− 8%). Using only the second term, we obtain (d[VE]/dt) = 1/ 2kHT,SA[RS][VL]. Combining this equation with the rate of −CH3 production in eq XVIII and using φ3 for the asymptotic concentration of [VL] (eq XXIII), we obtain that the ratio of the rates of formation of VE and −CH3 is approximately equal to 1/2kHT,SA/kVL, and considering that kVL = kAST + kASN + 1/ 2kNT,SA (kVL = kAST + 1/2kNT,SA in the case of steady-state approximation), the ratio of rates is most sensitive to φ4 =

kHT,SA kAST + kASN

(XXXI)

The values of φ4 are approximately independent of temperature: 0.26−0.28 in the case of steady-state analysis and 0.63 in the case of the extended kinetic model. The latter is recommended, and the uncertainty factor is estimated as 2 from a combination of data scattering of the experimental rates ratio and the minor dependence of [VE] on the rates of betascission (section 3.3). Although experimental dependences of [CHY] and [VD] on time and pyrolysis temperature are also sensitive to combinations of rate constants (equations S-XXVIII and SXXXV), we do not present the corresponding combinations here as directly derived from the experimental results because of the potential uncertainties in [CHY] related to the quantitative use of 13C NMR spectra. The experimentally derived Arrhenius expressions for φ1−φ4 are compiled in Table 3, together with the corresponding estimated uncertainty factors. Table 3. Experimentally Derived Arrhenius Parameters for φ1−φ4 (eqs XXVIII−XXXI) and the Estimated Associated Uncertainty Factors φ

A

E/K

uncertainty factor

φ1 φ2 φ3 φ4

7.4 × 1026 cm6 molecule−2 s−1 5.3 × 10−26 cm3 molecule−1 5.9 × 1025 molecules cm−3 0.63

34751 −7843 9635 0

1.5 1.75 1.3 2

Figures 9 and 10 provide comparison between the values of the rate constants adjusted during data fitting and those of analogous reactions of small molecules in the gas phase, where such data are available. The values of kHT,PS and kAST are in good agreement with the gas-phase rate constants. No data are available for kHT,SA; however, its value is within the limits of uncertainty of the Evans−Polanyi estimation. Comparison S


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in the experiments of De Witt and Broadbelt8 and the experimental data may well be artifactual, as it relies on the polydispersity of 2 estimated by Levine and Broadbelt4 for the PE sample used in ref 8. At later stages of PE pyrolysis where the average molar mass decreases to approximately 103 Da, a transition from reptation46 dynamics of PE chains to Rouse70 behavior is expected, and the power law of kR ∝ Mw−2 is expected to change. A review of relevant literature can be found in ref 3. The values of Mn and Mw of the PE samples used in the current study are highly uncertain. The molar fractions of the −CH3 and the CHY groups obtained in the NMR analysis are too close to derive the value of Mn (the uncertainty limits of [−CH3]0 and [CHY]0 overlap, regardless of whether 1H or 13C NMR analysis was used for −CH3). The value of Mw needed to evaluate the diffusion-controlled rate constant of radical recombination was estimated based on the correlation with the melt index (section 3.1.2). As PE pyrolysis progresses, the concentrations of −CH3 and the CHY groups diverge and it becomes possible to estimate the value of Mn from the differences in [−CH3]0 and [CHY]. Figure S34 in the Supporting Information demonstrates the dependence of the thus estimated Mn on time together with the corresponding results of kinetic modeling using the extended model. Under the conditions of very long PE chains and thus slow diffusion, an additional term for kRSS may become important. This term will be due to the “reaction diffusion” phenomenon, where recombination is driven not by the actual motion of the recombining radical sites but rather by the migration of the radical site through PE melt due to rapid hydrogen transfer. The term “reaction diffusion” was introduced by Schulz in his 1956 study of polymerization71 to describe chain-end diffusion by propagational growth, and the concept was further developed by other researchers. A review can be found in ref 43. For the PE pyrolysis system, we estimate the rate constants for the “reaction diffusion”, kRD, as follows. The effective diffusion coefficient is estimated here as D = 1/ 3λ⟨v⟩, where λ is the average distance “travelled” by the radical site during an individual act of H transfer and ⟨v⟩ is the average velocity of this travel, ⟨v⟩ = λ/τ. Here, τ is the characteristic time needed for H transfer, which can be expressed through kHT,SS: τ = 1/(N × kHT,SS × [−CH2−]/2). N is the number of reaction paths (reaction path degeneracy) equal to the number of adjacent carbons suitable for reaction (only one of the two hydrogens of each of these carbons has the right orientation for reaction), and the coefficient 2 appears due to the fact that kHT,SS is the rate constant per −CH2− group with two hydrogens. The value of N = 4 was selected based on the relative positions of carbons within a PE lamella, and λ = 4.25 Å was taken as the distance between adjacent carbons belonging to different chains or chain folds within a lamella.41,42 Using the Smoluchowski equation and estimating R0 as equal to λ, we obtain

of ref 3 is 2 orders of magnitude greater and essentially coincides with the expected gas-phase rate constant. When comparing rate expressions used or obtained in the current work with those of earlier studies, one should keep in mind that each study developed its model for a specific purpose. For example, modeling performed in refs 2−4 targeted distributions of pyrolysis products as a function of molecular chain length. If only relative abundances are compared with experimental data2,3 or if comparison is made not at specific times but at specific degrees of conversion (e.g., 100% conversion2), the agreement is likely to be insensitive to the rates of initiation and termination. As explained above, the current work did not set a goal of comprehensive description of all processes occurring in PE pyrolysis; rather, it focused on the values of the rate constants of elementary reactions involved at the initial stages of polyethylene pyrolysis. 4.3. Rate Constants for Radical Recombination (chain termination). The diffusion-limited rate constant for radical recombination was estimated in section 3.1.2 using the Smoluchowski equation, and then the dependence on the PE chain length changing during the pyrolysis process was introduced in section 3.2.2 by replacing the second-order reactions in eqs 7, 40, and 41 with the formally fourth-order reactions in eqs 7a, 40a, and 41a. This appears to be the most uncertain part of the mechanism and the model of PE pyrolysis in terms of the rate constant values. Uncertainties in the critical distance R0, PE diffusion coefficients, their dependence on PE chain length, and the value of Mw of the PE sample all contribute to the overall uncertainty in the chain termination rate constant. The dependence of kRSS = kDIFF on the PE chain length was expressed in section 3.1.2 via the weight-averaged molar mass of PE, Mw, using the relationships derived in refs 45 and 47. Later, when the effect of decreasing during pyrolysis PE chain length was introduced in section 3.2.2, it was expressed in terms of kRSS dependence on [−CH3], which for linear PE is linked to the number-averaged molar mass, Mn. As a result, the expression used for the rate constant kR of the formal fourthorder termination reactions in eqs 7a and 40a depends on both Mw and Mn or, rather, on their ratio, or polydispersity (a detailed explanation is given in the Supporting Information) 2 1.00 × 10−3 ⎛ M n(0) ⎞ −3127K / T 1 ∼ kR = 2 ⎟e 2⎜ 28 [−CH 2−] ⎝ M w (0) ⎠ PDI2

(XXXII)

where PDI is the polydispersity index. Here, the units of kRSS and kR are cm3 molecule−1 s−1 and cm9 molecule−3 s−1, respectively. The resultant dependence of kR on polydispersity is, most likely, an artifact of the model. It appeared due to the fact that the dependence on the diffusion coefficient on PE chain length is expressed via Mw and not Mn in ref 45. At the same time, there is a lack of certainty in the question of whether the diffusion coefficient of molten PE has any dependence on polydispersity. This further emphasizes the uncertainty in the recombination rate constant values. Clearly, diffusion of radicals in PE melt and the related rates of radical recombination need to be studied further in order to achieve predictive ability; without such knowledge extrapolation of the values of kR selected for the current model to other PE systems is problematic. A consequence of this uncertainty is that the agreement between the calculated rates of ethylene formation

kRD =

4πλ 3N kHT,SS[−CH2−] 6

(XXXIII)

Substituting the values of λ and N, we obtain kRD = 10.4kHT,SS

(XXXIV)

For the conditions of the current work, thus estimated kRD ranges from 1.8 × 10−16 to 3.2 × 10−16 cm3 molecule−1 s−1 or between 6% and 9% of the kRSS under the initial pyrolysis T


Article

The Journal of Physical Chemistry A conditions; as pyrolysis progresses, kRSS increases and kRD becomes even less important. Therefore, the potential effects of reaction diffusion were neglected in the current study. However, they may become non-negligible for very long PE chains and/or high temperatures, because the effective activation energy of kHT,SS and thus kRD is higher than that of kRSS. 4.4. Effects of k DIFF Dependence on Pyrolysis Progress. The increase in the diffusion-limited rate of radical recombination with the decreasing during pyrolysis average molecular mass of PE was first proposed in 1954 by Wall et al.72 as a potential explanation of the experimentally observed phenomenon of initial fast pyrolysis followed by a slower stage. An increase in the effective rate constants of radical recombination decreases the kinetic chain length and slows down the overall pyrolysis. The dependence of the diffusionlimited radical recombination rate constant on PE molar mass was subsequently discussed by Mita and Horie73,74 and others; a review can be found in ref 3. The current study is the first modeling work that we are aware of where the dependence of kDIFF on the changing molecular mass of PE was explicitly quantified and included in a kinetic model. This inclusion resulted in significant improvement of the agreement between modeling and experiment. In particular, the phenomenological activation energies for the formation of major products decreased compared with the models where such dependence was not included; the calculated slopes of the temperature dependences of product rates of formation in Arrhenius coordinates became significantly closer to the experimental ones (Figures 5, 6b, 7). Faster overall pyrolysis at higher temperatures means faster decrease in the average PE molar mass and thus faster increase in the rate of chain termination. One consequence of the effect of kRSS = kDIFF decreasing during pyrolysis is that, for sufficiently long initial PE chains, the rates of appearance of major products eventually converge to the same values, regardless of the initial PE chain length. This is accompanied by the convergence of the time-dependent −CH3 concentration and, thus, Mn. Figure 15 demonstrates predicted time dependences for −CH3 and C2H4 abundances and formation rates for hypothetical linear PE samples with the numbers of C atoms N of 1000, 10 000, and 100 000 at 400 °C. The same value of kR (reactions 7a and 40a) is used as in the model used to describe the experiments performed in the current study. The plots in Figure 15 demonstrate an initial fast increase in pyrolysis rate followed by a relaxation to a lower value, with the same asymptotic behavior for the three samples of different molar masses. Similar behavior is observed in modeling performed for lower (360 °C) and higher (440 °C) temperatures, except for higher and lower rates, respectively, and that at lower temperature the induction period is visible on the time scale of the plots (Figures S35−S37 in the Supporting Information). Once again, the demonstrated effects of pyrolysis progress on the rate of diffusion-controlled termination rates emphasizes the importance of the knowledge of the dependences of such rates on molar mass of PE and, potentially, its distribution as well as other conditions. Such knowledge is currently not of the quality that would enable predictions of pyrolysis kinetics of different PE samples. Clearly, more research is needed in this field.

Figure 15. Predicted time dependences for −CH3 and C2H4 abundances and formation rates for hypothetical linear PE samples of different chain lengths (N) at 400 °C. All three samples demonstrate the same asymptotic behavior for rates of product formation at long pyrolysis times.

5. SUMMARY An experimental study of the kinetics of the initial stages of the pyrolysis of high-density polyethylene (PE) was performed. Quantitative yields of gas-phase products (C1−C8 alkanes and alkenes) and functional groups within the remaining polyethylene melt (methyl, vinyl, vinylene, vinylidene, and branching sites) were obtained as a function of time (0−20 min) at five temperatures in the 400−440 °C range. Gas chromatography and NMR (1H and 13C) were used to detect the gaseous and the condensed-phase products, respectively. The temporal profiles of all products except the end-chain vinyl groups display linear growth with time; the kinetics of vinyl groups is that of initial growth with a trend to saturation. The largest abundances among the gaseous products are those of species with the C3 composition, propane and propene, in agreement with the results of earlier studies where gas-phase products have been quantified. Comparison of the experimental data with existing literature on the rates of production of gaseous products6,8 and functional groups in the condensed phase7 demonstrate divergence of up to an order of magnitude between the results of the current and earlier studies as well as larger differences between the results of the latter.6,8 Two chemical kinetic mechanisms of PE pyrolysis were created. The short mechanism (reactions 1−19) was used with the steady-state approximation to derive relationships between the observable quantities such as product formation rates and the ratios of the rate constants of the most important elementary reactions (eqs XII−XXIV and S-XXVIII, S-XXXII, and S-XXXV (Supporting Information)). Both the short and the extended (reactions 1−41) mechanisms were used to model the kinetics of PE pyrolysis via integration of differential kinetic equations. Rate constants of critical elementary reactions (C−C backbone scission, two kinds of H-atom transfer, radical addition to the double bond, and beta-scission of tertiary alkyl radicals) were adjusted using the steady-state U



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analysis equations as a guide, resulting in an agreement between the model and the experiment. The number of experimentally observable quantities such as rates of product formation (7) exceeded that of adjustable parameters of the model (5). The values of adjusted rate constants are in general agreement with those of cognate reactions of small molecules in the gas phase, with the exception of the rate constants of the backbone C−C scission, which is found to be approximately 1−2 orders of magnitude lower. This observation provides tentative support to the hypothesis that congested PE melt molecular environment impedes the tumbling motions of separating fragments in C−C bond scission, thus resulting in less “loose” transition state and lower rate constant values. Sensitivity of the calculations to selected uncertainties in model properties was studied. Values and estimated uncertainties of four combinations of rate constants (φ1−φ4, eqs XXVII−XXXI) are reported as derived from the experimental results via modeling (Table 3). The dependence of the diffusion-limited rate constant for radical recombination (kRSS = kDIFF) on the changing molecular mass of PE was explicitly quantified and included in the extended kinetic model. Accounting for the dependence of kDIFF on PE molar mass resulted in significant improvement of the agreement between modeling and experiment, particularly the agreement between the experimental and the calculated activation energies for product formation rates. Calculations performed for model PE chains of different lengths (103−105 carbon atoms) demonstrate a pattern of initial fast pyrolysis followed by a slow down caused by shortening of PE chains and associated increase in the rate of kinetic chain termination via recombination of radicals. For sufficiently long initial PE chains, rates of appearance of major products eventually converge to the same values, regardless of the initial PE chain length. Calculations were performed to estimate the contribution to the overall rate of radical recombination of the “reaction diffusion” phenomenon, where recombination is driven not by the actual motion of the recombining radical sites but rather by the migration of the radical site through PE melt due to rapid hydrogen transfer. For the conditions of the current work, the contribution of reaction diffusion is negligible but may become more important for very long PE chains and/or high temperatures. Kinetic modeling of PE pyrolysis performed in the current study demonstrates the critical role of diffusion-controlled rates of kinetic chain termination in the overall kinetics of pyrolysis. The results emphasize the importance of the knowledge of the dependences of such rates on molar mass of PE and, potentially, its distribution as well as other conditions. Clearly, more research is needed in this field.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b07440. Detailed data on the experimental yields of all detected gas-phase species and condensed-phase functional groups, derivation of the steady-state equations, fitted rate constants and concentrations of RS and RP, predicted time dependences for −CH3 and C2H4 abundances, and formation rates as well as RS and RP concentrations for hypothetical linear PE samples with different numbers of C atoms (PDF) V


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