Modeling the Evolution of the Full Polystyrene Molecular Weight

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Ind. Eng. Chem. Res. 2003, 42, 2722-2735

Modeling the Evolution of the Full Polystyrene Molecular Weight Distribution during Polystyrene Pyrolysis Todd M. Kruse, Hsi-Wu Wong, and Linda J. Broadbelt* Department of Chemical Engineering, Northwestern University, Evanston, Illinois 60208

The degradation of polystyrene was modeled at the mechanistic level using the method of moments to track structurally distinct polymer species. To keep the model size manageable, polymer species were lumped into groups, and within these groups, the necessary polymeric features for capturing the degradation chemistry were tracked. The pyrolysis reactions incorporated into the model included hydrogen abstraction, midchain β-scission, end-chain β-scission, 1,5-hydrogen transfer, 1,3-hydrogen transfer, radical addition, bond fission, radical recombination, and disproportionation. From the evolution of the zeroth, first, and second moments tracked for each dead species, polymer molecular weight distributions were constructed by summing the Schultz (Teymour, F.; Campbell, J. Macromolecules 1994, 27, 2460) and Wesslau (Pladis, P.; Kiparissides, C. Chem. Eng. Sci. 1998, 53 (18), 3315) distributions for the polymer groups. Model results were compared to experimental data collected in our laboratory, where polystyrene samples that differed in the shape and breadth of their initial distributions were pyrolyzed. The model was able to predict the formation of a bimodal distribution during the pyrolysis of polystyrene samples (molecular weight range of 10 000-500 000 g/mol) with narrow unimodal molecular weight distributions (polydispersity index < 1.1). This was accomplished by distinguishing the initial polymer from the polymer formed from midchain β-scission reactions within the model. At high conversions, all of the polystyrene samples investigated evolved to unimodal distributions, and these distributions were best captured by the Schultz distribution. 1. Introduction In the past decade, the pyrolysis of polymeric materials has grown as a resource recovery strategy.3-5 When polymeric materials are pyrolyzed, value is recovered by transforming polymers into fuels, chemicals, and monomer, where the monomer recovered can be used to produce new polymer.3 Unfortunately, the complete pyrolysis of polymers usually leads to a very diverse product distribution because of the high temperatures used and the complex free-radical reactions involved. Overall, the pyrolysis of polymeric materials has a high potential for growth in the future provided that economically feasible processes able to manage the diverse product distribution can be developed.6 The neat pyrolysis of polystyrene, a voluminous component of plastic waste, has been studied extensively.7-10 The main focus of these studies was to convert polystyrene to low molecular weight products. At low pressures ( 105) to produce a low molecular weight polymer.11 It was proposed that this low molecular weight polystyrene fraction could be used to make polystyrene products where low molecular weight polystyrene (average molecular weight ∼ 104) is desired.11 Extensive research has been performed on degrading polystyrene and other styrene-based polymers to produce a low molecular weight polymer.11-14 These works focused on degrading styrene-based polymers in various solvents to determine the effect of the solvents on degradation rates and the degradation mechanism. To develop a process that effectively degrades polystyrene to form a desired low molecular weight polymer, an understanding of how the molecular weight distribution (MWD) evolves under particular reaction conditions is required. Polystyrene properties are sensitive to the shape and breadth of the MWD,15-17 and the ability to predict the evolution of the MWD is needed. Results were shown for how the evolving MWDs of styrenebased polymers were affected by the presence of various solvents and different degradation temperatures,11-14 but no modeling work was performed. McCoy and co-workers have developed methods for modeling polymer degradation using the method of moments,18-27 where differential equations describing the evolution of the polymer moments are specified. While the moments characterize the MWD, the full shape and breadth of the MWD can only be reproduced by the method of moments if a form of the distribution

10.1021/ie020657o CCC: $25.00 © 2003 American Chemical Society Published on Web 04/25/2003

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Figure 1. Polystyrene pyrolysis reactions incorporated into the model. Reaction types are numbered in the preceding text.

Figure 2. End-chain groups distinguished based on saturation and head/tail orientation.

is assumed. To this end, McCoy and co-workers constructed the full MWD of the polymer product from the zeroth, first, and second moments by assuming the distribution conforms to a Γ distribution.19,27 In a study on the degradation of poly(styrene-allyl alcohol), they were able to fit a Γ distribution to the initial polymer distribution and model the evolution of the polymer MWD over time by constructing a Γ distribution for the polymer product from the moments.19 They found that the Γ distribution was able to represent the evolving poly(styrene-allyl alcohol) distribution over all of the degradation time frames studied. However, in research they performed on the degradation of polystyrene,21,22,26 the evolution of the full polystyrene MWD was not

investigated. Mehta and Madras used the approaches developed by McCoy and co-workers to model the full MWDs of theoretical polymer systems with simple degradation mechanisms, but a comparison to experimental data was not performed.28 While limited work has been done to reproduce the full MWD for polymer degradation, more attention has been paid to predicting the MWD for polymerization chemistry. For polymerization processes, the Wesslau distribution and the Schultz distribution (a Schultz distribution is a Γ distribution on a weight basis) have been used to construct polymer MWDs from the moments.29,30 The Schultz distribution has been used to model polymer distributions during gelation,1 and the Wesslau distribution has been employed to model polymer distributions that contain branched polymer.2 The Schultz distribution is a more general form of the most probable distribution, and the Wesslau distribution is a log-normal distribution on a weight fraction basis.30 In previous work,31,32 we combined the approaches of McCoy and co-workers, Teymour and Campbell,1 and Pladis and Kiparissides2 for modeling polymer degradation and polymerization to develop a detailed mechanistic model for polystyrene degradation. This model incorporated chain-length-dependent rate parameters, tracked branched species, and used rate parameters primarily from the literature. For the model, unique

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Figure 4. Comparison of model results with experimental data for the evolution of the polymer Mn and polymer Mw for (a) 98 100 molecular weight polystyrene at 350 °C and (b) 98 100 molecular weight polystyrene at 420 °C.

Figure 3. Comparison of model results with experimental data for the evolution of the polymer Mn and polymer Mw for (a) 50 550 molecular weight polystyrene at 310 °C, (b) 50 550 molecular weight polystyrene at 350 °C, and (c) 42 500 molecular weight polystyrene at 380 °C.

In the present study, we extended this work by constructing full polymer MWDs from the zeroth, first, and second moments tracked for each dead species within our polystyrene model using the Schultz and Wesslau distributions. The polystyrene samples degraded differed in the shape and breadth of their initial distributions, and the effect of the initial MWD on the evolution of the MWD was investigated. The ability of the Schultz and Wesslau distributions to capture the experimental behavior as a function of time, temperature, and initial MWD was evaluated. A summary of the approach to model construction, the detailed chemistry included in the model, and the experimental procedure is given below. The complete model details are presented elsewhere.32 2. Model Development

polymer groups were devised that allowed the necessary polymeric features for capturing the degradation chemistry to be tracked while maintaining a manageable model size. The conversion among the species was described using typical free-radical reaction types, including hydrogen abstraction, midchain β-scission, end-chain β-scission, 1,5-hydrogen transfer, 1,3-hydrogen transfer, radical addition, bond fission, radical recombination, and disproportionation. The full model included over 2700 reactions and tracked 64 species. The model predictions for the evolution of Mn (numberaverage molecular weight) and Mw (weight-average molecular weight) and the yields of styrene, dimer, and trimer compared very well with experimental data obtained in our laboratory for the degradation of polystyrene over a large temperature range and with different initial molecular weights.32

2.1. Mechanistic Chemistry. The method of moments was used to develop differential equations describing the pyrolysis kinetics, and the mechanistic chemistry of interest was implemented by deriving the terms of the moment equations corresponding to each reaction type.32 The terms of the moment equations were derived for the following reactions: (1) chain fission, (2) radical recombination, (3) allyl chain fission, (4) hydrogen abstraction, (5) midchain β-scission, (6) radical addition, (7) end-chain β-scission, (8) 1,5hydrogen transfer (also 1,3-transfer), and (9) disproportionation. Examples of these reactions for polystyrene are pictured in Figure 1. As shown by our previous work, inclusion of both 1,5- and 1,3-hydrogen transfer was critical to describing the yields of trimer and dimer, respectively. Branching reactions were also included in the model, where end-chain radicals and unsaturated end-chain groups were allowed to combine with mid-


Figure 5. Comparison of Schultz and Wesslau distributions to experimental data for the degradation of 98 100 molecular weight polystyrene (PDI ) 1.14) at 350 °C at the following degradation times: (a) initial distribution, (b) 10 min, (c) 30 min, (d) 60 min, (e) 90 min, and (f) 180 min.

chain radicals via radical recombination and radical addition, respectively. However, the amount of branching predicted by the model is very mild; less than 10 wt % of the polymer chains are branched for the majority of the samples after all degradation times, and the number of branch points per chain in this branched group is approximately 1. In addition, the approach outlined by McCoy and Wang for describing random and proportioned chain fission and β-scission reactions18,19 was implemented into the moment equations. Chain fission and midchain β-scission of linear chains were assumed to occur at random points along polymer chains, while the fission and scission of branched species were assumed to have both proportional and random characteristics. 2.2. Specification of Rate Constants. Incorporation of these free-radical elementary step reactions in a model of polymer degradation required the specification of the rate parameters. The same approach used previously in our work to quantify rate constants was adopted.9 The rate parameters were dependent on not

only the reaction type but also the structural characteristics of the reactants and products. Assuming the validity of the Arrhenius relationship, a frequency factor and an activation energy for each reaction were specified. Each reaction of a given type (e.g., bond fission) shared the same frequency factor. The activation energy for each specific reaction was calculated using the Evans-Polanyi relationship,33 in which the activation energy is related linearly to the heat of reaction, i.e., E ) E0 + R∆Hr. The values of the heats of reaction were obtained from experimental polymerization data or based on analogous reactions of molecular mimics of the polymer structure as used previously.9,34,35 The frequency factors and the parameters E0 and R for each reaction type were primarily obtained from the literature.32 No optimization of the frequency factors or activation energies was carried out. The diffusion dependence of termination reactions was implemented using Smoluchowski’s equation36 for the rate constant of a diffusion-controlled reaction. The termination rate constant was assumed to be inversely

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Figure 6. Comparison of the Schultz distributions for the most significant polymer groups tracked within the model, the total Schultz distribution from the sum of the distributions of the groups, and the Schultz distribution calculated from the total polymer moments for the degradation of 98 100 molecular weight polystyrene (PDI ) 1.14) at 350 °C at the following degradation times: (a) 10 min, (b) 30 min, and (c) 60 min.

proportional to the chain length of the terminating radicals, and a termination rate constant was calculated for all termination reactions using the average length of all of the polymer radicals present in the polymer melt. In addition, hydrogen abstraction rate constants were assumed to be inversely proportional to the size of the abstracting radical. A complete summary of the rate parameters is provided in Table 2 of Kruse et al.32 2.3. Model Assembly and Solution. Because of the large number of species and reactions, a program written in the Perl programming language was developed to construct a list of the chemical reactions in traditional form from minimal user input. Another Perl program was then developed which could transform this

list into moment rate terms. These terms were then assembled into a set of ordinary differential equations that included three (zeroth, first, and second) moment equations for each unique species. The resultant set of stiff differential equations was then solved using DASSL.37 To choose what polymer species were to be tracked, a lumping scheme was devised that lumped polymer species into groups based on their end-chain structure. The different types of end-chain groups possible for polystyrene are shown in Figure 2. For polystyrene, the phenyl group on the head end of a monomer unit provides resonance stabilization for any radical, and therefore its reactivity must be distinguished from tail ends. In addition, unsaturated groups have different reactivities and participate in different reactions (such as radical addition) compared to saturated end-chain groups, so they also must be tracked accordingly. Therefore, to model polystyrene, both head and tail ends, saturated and unsaturated, were differentiated in the model developed. To include branching, all branched species were lumped into one branched group, where the branched group was then described by several properties defining the structure of the branched species (i.e., end-chain types, average number of branches, and fraction of branched species with one branch). A total of 11 dead polymer groups were tracked in this polystyrene degradation model. The model also tracked structural irregularities within polymer chains. All atypical bonds (tail-tail and headhead) were tracked explicitly, and using the concentrations of the different types of bonds, probabilities were incorporated into the model to partition fission reactions between possible breaking bonds. In addition, different heats of reaction were used to characterize each possible bond fission reaction. 2.4. Determining the Polymer MWD. The model follows the zeroth, first, and second moments for each unique species tracked. To determine the full polymer MWD from the zeroth, first, and second moments, a form of the polymer MWD needs to be assumed. In this study, the validity of two different distributions, the Schultz and Wesslau distributions, was examined. For the Schultz model distribution, eqs 1-5 detail how the full polymer MWD (on a weight basis) is obtained from the polymer moments.1 The model was solved using the Saidel-Katz approximation38 (eq 5) for the third moments, which is consistent with the Schultz distribution and is needed to obtain a closure on the differential equations for the moments of dead polymeric species. For each ith polymer group tracked, the number-average chain length (X h in) and the weight-average chain i length (X h w) for the group are calculated by dividing the first moment by the zeroth moment and the second moment by the first moment, respectively. These values are then used to calculate the parameters zi and yi that describe the Schultz distribution S(x,i) for each group, where x represents the chain length in number of monomer units. The total Schultz distribution [Stotal(x)] is then determined using eq 4, where the contribution each group makes to the total distribution is weighted based on the group’s mass relative to the total polymer mass. The variables µi1 and µt1 represent the first


Figure 7. Comparison of Schultz and Wesslau distributions to experimental data for the degradation of 98 100 molecular weight polystyrene (PDI ) 1.14) at 420 °C at the following degradation times: (a) initial distribution, (b) 1 min, (c) 2 min, and (d) 3 min.

moments of the ith polymer group and all polymers, respectively.

zi )

yi(xyi) e

Γ(zi + 1) X h in

X h wi - X h in

[

W(x,i) ) (2πx2σi2)-1/2 exp -

zi -xyi

S(x,i) )

accordingly, as shown in eq 9.

(1)

(2)

]

(ln x - ln xmi)2 2σi2

(6)

xmi ) (µi2/µi0)1/2

(7)

σi2 ) ln[µi0µi2/(µi1)2]

(8)

N

yi )

Wtotal(x) )

zi + 1

(3)

X h iw

µi3 ) µi0(µi2/µi1)3

N

Stotal(x) )

µi3 )

S(x,i) (µi1/µt1) ∑ i)1

2µi2µi2 µi1

-

W(x,i) (µi1/µt1) ∑ i)1

µi2µi1 µi0

(4)

(5)

For the Wesslau model distribution, eqs 6-10 detail how the full polymer MWD (on a weight basis) is obtained from the polymer moments.2 The model was solved using an approximation for the third moments (eq 10) that is consistent with the Wesslau distribution. The parameters σi2 and xmi, which characterize the Wesslau distribution for the ith polymer group, are calculated using the zeroth, first, and second moments (µi0, µi1, and µi2) of each group. The total Wesslau distribution [Wtotal(x)] is then calculated by summing the distributions of the polymer groups and weighting them

(9) (10)

3. Experimental Section The experimental data to which the model results were compared have all been reported previously.9,32,39 Briefly, isothermal batch pyrolysis experiments in argon were carried out on polystyrene samples with numberaverage molecular weights of 98 100 [polydispersity index (PDI) ) 1.14 ) Mw/Mn], 50 550 (PDI ) 1.14), and 42 500 (PDI ) 1.02) at 310, 350, 380, and 420 °C. The 98 100 and 50 550 molecular weight polystyrene samples were anionically polymerized as previously described.9,34 The 42 500 molecular weight polystyrene was a narrow polystyrene standard obtained from Scientific Polymer Products Inc. Reaction times ranged from 1 to 180 min. To analyze the full molecular weight range of products observed from the polymer pyrolysis, three complementary analytical techniques were used:9,34 gel permeation chromatography (GPC), gas chromatography, and gas chromatography-mass spectroscopy. The error bars shown in the figures in this paper represent the

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Figure 8. Comparison of Schultz and Wesslau distributions to experimental data for the degradation of 42 500 molecular weight polystyrene (PDI ) 1.02) at 380 °C at the following degradation times: (a) initial distribution, (b) 5 min, (c) 10 min, (d) 15 min, and (e) 20 min.

standard deviations of experiments, which were all performed in triplicate. Experimental MWDs were determined from GPC data for the experiment that gave an Mn closest to the average Mn of the three experiments performed at each condition. 4. Polystyrene Pyrolysis Modeling In the model developed for polystyrene, only midchain head radicals were allowed because of the resonance stabilization the phenyl substituent provides. Specific midchain radicals formed from backbiting reactions were tracked separately from other midchain radicals. Branching was regulated by only allowing linear chains to add as branches to branched species. As the branched species themselves degraded, the average branch length dropped to only a few monomer units long, and it was assumed that these short branches would interfere with the addition of branches from other branched species. Based on the anionic polymerization procedure9,34 followed to make most of the polystyrene samples de-

graded, the initial polystyrene structure was assumed to consist of linear chains with saturated tail ends. This type of polymer is shown as the polymer chain undergoing bond fission in reaction (1) in Figure 1, where it is labeled P_tt to represent a polystyrene chain with two saturated tail (t) ends. Because the model developed tracked polymer groups based on end-chain features, all of the polymer mass was placed into the P_tt group initially. However, during model runs polymer chains with end-chain features different from that of the initial polymer are produced, and the polymer mass becomes distributed between several polymer groups. For example, the midchain β-scission reaction in Figure 1 produces chains with unsaturated tail ends, and head end-chain radicals formed during the pyrolysis can abstract hydrogen to form saturated head ends. Examples of chains with these features are labeled P_Tt and P_th in Figure 1, where the labels represent chains with saturated tail (t) ends and unsaturated tail (T) or



saturated head (h) ends. A total of 11 dead polymer groups were tracked within the model. To provide confirmation that the model accurately captures the moments of the MWDs, comparisons of the model results and experimental data for the evolution of the polymer Mn and the polymer Mw for different polymers and reaction temperatures are shown in Figures 3 and 4. For the modeling results at 420 °C in Figure 4b, a warm-up time of 55 s was estimated by extrapolating the total yield of low molecular weight products to zero conversion. This warm-up period was incorporated into the model results at 420 °C because it was a significant fraction of the degradation time frame of interest. The full model results compared to experimental data for the evolution of the total Mn and

the total Mw and the yields of styrene, dimer, and trimer are shown elsewhere.32 The ability of the model to predict the evolution of Mn and Mw and the yields of styrene, dimer, and trimer over a 110 °C temperature range and for different initial polymer molecular weights was shown. The yield of polymer for the results shown in Figures 3 and 4 ranged from 90 wt % at 310 °C after 360 min to 30 wt % at 350 °C after 180 min. To examine the model results for the polymeric species more closely and to test the ability of different assumed distributions to capture the experimental data, MWDs have been constructed by summing the Schultz and Wesslau distributions for the 11 dead polymer groups within the model. Figure 5 compares the experimental polymer MWDs with the model Schultz and Wesslau distributions for the degradation of 98 100 molecular weight polystyrene (PDI ) 1.14) at 350 °C for a series of reaction times. These distributions are normalized such that the area under each distribution profile on a mass fraction basis in Figure 5 equals 1 (equivalent to normalizing by the first moment of the distribution). While the initial distribution does not conform to either model distribution, it is similar enough to them that at 10 min both model distributions compare very well with the experimental distribution. For the remaining degradation times shown, both model distributions are similar to the experimental distributions, with the Schultz distribution giving slightly better results at the longest degradation times. To analyze the evolution of the model distributions in Figure 5 more closely, Figure 6 compares the full Schultz model distribution (determined by summing the distributions of the polymer groups tracked) to the Schultz distributions for the major polymer groups within the model at three different degradation times. All of the polymer mass was placed into the P_tt group initially. As is evident from Figure 6, the major polymer groups that form during the course of the degradation are those with unsaturated tail ends (P_Tt and P_Th). This indicates that the major reaction pathway influencing the evolution of the MWD is midchain β-scission. From work performed by Madras and McCoy,40 it has been shown that the MWD for polymers undergoing random chain scission evolves to a special case of the Schultz distribution where the parameter zi ) 1 (corresponding to a PDI ) 2). In this study, the degradation of the polystyrene sample in Figure 5 leads to the formation of a polymer fraction with a PDI close to 2 after 60 min, and from 60 to 180 min the polystyrene MWD is best modeled using the Schultz distribution. Thus, random midchain β-scission appears to be the dominant degradation pathway, and because of the predominance of this pathway, the polystyrene MWD is best modeled using the Schultz distribution at long degradation times. Figure 6 also compares the full Schultz model distributions to the full Schultz distributions calculated from the total polymer moments. The full distribution calculated from the total polymer moments would be h tw for the overall simply obtained by using X h tn and X polymer in eqs 1-3. Figure 6a shows that there are significant differences between the full Schultz model distribution calculated from the sum of the distributions of all of the polymer groups tracked (see eq 4) and the Schultz distribution determined from the total moments of all of the dead polymer species. However, the differences in Figure 6 are relatively minor, and at longer

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Figure 10. Comparison of Schultz and Wesslau distributions to experimental data for the degradation of 50 550 molecular weight polystyrene (PDI ) 1.14) at 350 °C at the following degradation times: (a) initial distribution, (b) 10 min, (c) 30 min, (d) 60 min, (e) 120 min, and (f) 180 min.

degradation times, the differences between these two distributions become negligible. For this example, calculating a full Schultz distribution by either summing the group distributions or using the total polymer moments will both give a good approximation to the experimental distribution at all degradation times. In Figure 7, the model MWDs are compared to experimental data for the degradation of the same initial polymer shown in Figure 5 but at a higher temperature of 420 °C. Similar to the results in Figure 5, both model distributions compare well with the experimental distributions once some polymer has degraded, with the Schultz distribution giving better results at the longest degradation time. In addition, the experimental distribution in Figure 7c (2 min of degradation at 420 °C) has nearly the same Mn, Mw, and shape as the experimental distribution in Figure 5d (60 min of degradation at 350 °C). Based on this result, the evolution of the shape and breadth of the polymer MWD appears to depend very little on the degradation temperature, and the model was able to somewhat predict

this result. The greater differences between the model and experimental results in Figure 7c compared to Figure 5d are primarily due to the greater differences between the model and experimental molecular weights shown in Figure 4 for the degradation of this polystyrene sample at 420 °C. Figure 8 compares the model results to experimental data for the evolution of the polymer MWD for the degradation of 42 500 molecular weight polystyrene (PDI ) 1.02) at 380 °C. The initial distribution for this polystyrene sample is extremely narrow, and this distribution is represented well by either a Schultz or a Wesslau distribution. However, the experimental distribution evolves to a multimodal distribution at 5 and 10 min before becoming more of a unimodal distribution again at 15 min. Both model distributions predict a bimodal distribution at 5 and 10 min, and both evolve to a unimodal distribution at 15 min. While the agreement between the model distributions and the experimental data is not perfect, the model distributions capture the general breadth and magnitude of the data.


Most importantly, however, they illustrate the ability of the model to capture the formation of a complex distribution (asymmetric and multimodal) from a uniform unimodal initial distribution. To analyze the evolution of the model distributions in Figure 8 more closely, Figure 9 compares the full Schultz model distribution to the Schultz distributions of the major polymer groups within the model at three different degradation times. All of the polymer mass was placed into the P_tt group initially, and the major polymer groups that evolve during the course of the degradation are those produced during random midchain β-scission (P_Tt, P_Th, and P_th). The degradation of the polystyrene sample in Figure 8 leads to the formation of a polymer fraction with a PDI close to 2 after 15 min, and from 15 to 20 min the polystyrene MWD is best modeled using the Schultz distribution. This supports the conclusion drawn earlier from the analysis of Figure 6 that random midchain β-scission is the dominant degradation pathway. Because of the predominance of this pathway, the polystyrene MWD is best modeled using the Schultz distribution at high conversions. Figure 9 also compares the full Schultz model distributions to the full Schultz distributions calculated from the total polymer moments. For this case, Figure 9a shows that there are major differences between the full Schultz model distribution calculated from the sum of the distributions of all of the polymer groups tracked and the Schultz distribution determined from the total moments of all of the dead polymer species tracked. The difference between these two distributions in Figure 9 does not become minor until after 15 min of degradation. For this example, calculating a full Schultz distribution by summing the group distributions captures the experimental distributions much better. Calculating the full Schultz distribution from the total polymer moments only gives a unimodal distribution at all degradation times, which is a very poor representation of the experimental data. Next, a comparison of the model results to experimental data for the degradation of 50 550 molecular weight polystyrene (PDI ) 1.14) at 350 °C is shown in Figure 10. The initial distribution for this polystyrene sample has some bimodal character, where a narrow high molecular weight polymer peak has a low molecular weight shoulder. The model results for both distributions are very poor for the initial distribution, 10 min results, and 30 min results, where the experimental distributions have bimodal characteristics. However, the model results for the Schultz and Wesslau distributions improve over time as the experimental distribution evolves to a unimodal distribution, with the Schultz distribution once again giving better results at the longest degradation time. Using a very poor approximation for the initial distribution appears to have greatly affected the results at low conversions. To obtain a better approximation for the initial distribution in Figure 10a, the distribution was modeled as a combination of two separate distributions. The original strategy of putting all of the polymer mass into the P_tt does not allow the initial distribution for this sample to be captured. Figure 11 shows the results of fitting two Schultz distributions and two Wesslau distributions to the initial distribution in Figure 10a. The fitted distributions consist of a narrow high molecular weight portion (Mn ) 62 500; PDI ) 1.02; mass

Figure 11. Approximating initial distribution of 50 550 molecular weight polystyrene as the sum of (a) two Schultz distributions and (b) two Wesslau distributions (labeled P_th and P_tt).

fraction ) 0.62) and a broad low molecular weight fraction (Mn ) 36 000; PDI ) 1.22; mass fraction ) 0.38). Within the model, the high molecular weight fraction was lumped into the polymer group consisting of linear chains with tail ends (P_tt), while the low molecular weight portion was put into the polymer group consisting of linear chains with a head and a tail end (P_th). Putting all of the initial polymer into either the P_tt or P_th group gives nearly identical model results, indicating that starting with either tail (t) or head (h) ends on the initial polymer has a negligible effect on the model results. It was thus unimportant which molecular weight portion was put into which polymer group, i.e., P_tt or P_th. However, it was critical that the initial polymer was divided into two polymer groups because this greatly affects the evolution of the full MWD. Figure 12 compares the model results to the same experimental data shown in Figure 10, except that now the model distribution in Figure 11 was used to approximate the initial distribution. Using this distribution, the model results predicted the formation of a bimodal distribution at low conversions. For this case, capturing the shape of the initial distribution was critical to being able to accurately predict how the distribution evolves. It is only at long degradation times that the Schultz and Wesslau distributions depicted in Figure 10 gave nearly the same results as their counterpart in Figure 12. The model distributions appear to be independent of the shape and breadth of the initial distribution after 60 min of degradation, but to capture the short time behavior, the shape of the initial distribution is key.

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Figure 12. Comparison of Schultz and Wesslau distributions to experimental data for the degradation of 50 550 molecular weight polystyrene (PDI ) 1.14) at 350 °C at the following degradation times: (a) initial distribution, (b) 10 min, (c) 30 min, (d) 60 min, (e) 120 min, and (f) 180 min. The initial distribution was approximated as the sum of two distributions as shown in Figure 11.

To analyze the evolution of the model distributions in Figure 12 more closely, Figure 13 compares the full Schultz model distribution to the Schultz distributions of the major polymer groups within the model at three degradation times. Similar to Figures 6 and 9, the major polymer groups that form during the course of the degradation are those produced during random midchain β-scission (P_Tt and P_Th). Even though this polymer sample has a bimodal initial distribution, the degradation of the polystyrene sample in Figure 12 leads to the formation of a polymer fraction with a PDI close to 2 after 60 min, and from 60 to 180 min the polystyrene MWD is best modeled using the Schultz distribution. This supports the conclusion drawn earlier from the analysis of Figures 6 and 9 that random midchain β-scission is the dominant degradation pathway, and thus the polystyrene MWD is best modeled using the Schultz distribution at high conversions. In addition, this finding is in agreement with work performed by Madras and McCoy,40 where they found that

the MWD for polymers undergoing random chain scission evolves to a special case of the Schultz distribution (corresponding to a PDI ) 2) irrespective of the initial distribution. Figure 13 also compares the full Schultz model distributions to the full Schultz distributions calculated from the total polymer moments. For this case, Figure 13a shows that there are major differences between the full Schultz model distributions and the Schultz distributions determined from the total moments. The differences in Figure 13 only become minor after 60 min of degradation. Similar to the example in Figure 9, calculating a full Schultz distribution by summing the group distributions is a much better model of the experimental MWDs because calculating the full Schultz distribution from the total polymer moments only gives a unimodal distribution at all degradation times. The evolution of the distribution for the degradation of the same polymer examined in Figures 10 and 12 is compared to the model distributions in Figure 14 for a



reaction temperature of 310 °C. For this case, capturing the shape of the initial distribution was critical once again, and the initial distribution was modeled using the two Schultz distributions shown in Figure 11. At this lower temperature, the formation of a bimodal distribution is clearly observed, with the main initial polymer peak shrinking over time as a secondary low molecular weight peak grows. Both model distributions are able to predict the bimodal nature of this evolving distribution. The source of the bimodal distributions within the model results can be examined by comparing the origin of the unimodal distribution for the 98 100 molecular weight sample (shown in Figure 6) to the origin of the bimodal distributions for the 42 500 and 50 000 molecular weight polystyrene samples (shown in Figures 9 and 13). For the 98 100 molecular weight sample, the distribution of the primary polymer types formed from midchain β-scission (i.e., P_Tt and P_Th) overlaps the

original polymer distribution (P_tt) significantly, resulting in a total distribution that remains unimodal. For the 50 550 and 42 500 molecular weight polystyrene samples, however, the distributions for the primary polymer types formed from midchain β-scission are separated enough from the original distribution that the peaks of these distributions do not overlap the P_tt distribution. For the 42 500 molecular weight polystyrene, this separation arises because the initial polymer distribution is very narrow (PDI ) 1.02). Similarly, the main polymer peak for the 50 550 molecular weight polystyrene is very narrow (PDI ) 1.02), and as a result a bimodal distribution forms. The model was able to predict the formation of a bimodal distribution from an initial unimodal distribution by distinguishing the initial polymer from the polymer formed from midchain β-scission reactions. This was achieved by tracking multiple polymer groups (11 total dead polymer groups), where polymeric species with different molecular weights and different end-chain features were separated. Tracking multiple polymer groups also allowed an initial bimodal distribution to be captured (which was demonstrated with our 50 550 molecular weight sample), where the initial polymer mass was partitioned into two polymer groups. If, instead, the full MWD had just been determined from the total polymer moments, the model would not have been able to reproduce any bimodal distributions. To determine the conditions needed to evolve a bimodal distribution from a unimodal distribution, pyrolysis of a theoretical polystyrene sample that conformed to a unimodal Schultz distribution initially was modeled. The initial PDI and initial molecular weight of this theoretical sample were varied within the model. It was determined that polystyrene with an initial molecular weight between 10 000 and 500 000 g/mol and with an initial PDI below 1.1 evolves to a distribution with significant bimodal characteristics at low conversions before evolving to a unimodal distribution at high conversions. With an initial PDI below 1.1, a distinct low molecular weight shoulder forms within the model distribution, as is demonstrated in Figure 12b [with a very low initial PDI (

Modeling the Evolution of the Full Polystyrene Molecular Weight

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