Thermal Pyrolysis of Polystyrene Aided by a Nitroxide End-Functionality

processes

Article Thermal Pyrolysis of Polystyrene Aided by a Nitroxide End-Functionality. Experiments and Modeling

Almendra Ordaz-Quintero, Antonio Monroy-Alonso and Enrique Saldívar-Guerra *

Centro de Investigación en Química Aplicada, Blvd. Enrique Reyna 140, Saltillo Coahuila 25294, Mexico; [email protected] (A.O.-Q.); [email protected] (A.M.-A.) * Correspondence: [email protected]; Tel.: +52-844-439-4408

Received: 27 February 2020; Accepted: 31 March 2020; Published: 5 April 2020

Abstract: The thermal pyrolysis of polystyrene (PS) is gaining importance as the social pressure for achieving a circular economy is growing; moreover, the recovery of styrene monomer in such a process is especially relevant. In this study, a simple thermal pyrolysis process in the temperature range of 390–450 ◦C is developed. A working hypothesis is that by using a nitroxide-end functionalized PS (PS-T or dormant polymer), the initiation process for the production of monomer (unzipping) during the PS pyrolysis could be enhanced due to the tendency of the PS-T to activate at the nitroxide end. Two types of PS were used in this work, the ﬁrst one was synthesized by free-radical polymerization (FRP-dead polymer) and the second by nitroxide-mediated polymerization (NMP) using three levels of nitroxide to initiator ratio: 1.3, 1.1, and 0.9. Analysis of the recovered products of the pyrolysis by gas-mass spectroscopy shows that the yield of styrene increases from 33% in the case of dead ∼ polymer to 38.5% for PS-T. A kinetic and mathematical model for the pyrolysis of dead and dormant ∼ polymer is proposed and solved by the method of moments. After a parameter sensitivity study and data ﬁtting, the model is capable of explaining the main experimental trends observed.

Keywords: polystyrene; thermal pyrolysis; nitroxide mediated polymerization; mathematical modeling

1. Introduction Plastics are widely used materials due to their physical and chemical properties; [1] additionally, they are cheap, light, long-lasting [2], and relatively easy to produce [3]. These outstanding properties are the ones that have led to their excessive production and irresponsible consumption, originating severe ecological problems that could cause irreversible damage to the environment [4–6]. Essentially there are three traditional solutions for plastic waste management, each one presenting significant drawbacks: Landfill disposal, recycling, and combustion [6]. In landfill disposal, none of the material resources used to produce plastics are recovered. With respect to the second alternative, the recycling capacity of polymers is inherently limited since after a determined number of recycling processes the materials exhibit reduced mechanical properties and the plastic ends up being thrown away. Regarding the third option, recovering energy from combustion is feasible, but it causes negative effects on the environment and human health. Given the increasing social pressure to take care of the environment and the present trends towards circular economy, a better option to manage plastic waste would be to transform it into defined chemical compounds, especially the source monomer, with the required purity and quality to be used again as raw material in the same polymerization or in other processes. One such technique that permits the recovery of high-value compounds is pyrolysis, which can be finely tuned to favor

Processes 2020, 8, 432; doi:10.3390/pr8040432 www.mdpi.com/journal/processes Processes 2020, 8, 432 2 of 26 some product distribution over others. The practical implementation of a chemical recycling strategy requires a concerted effort among different sectors of the society (the general population, municipal governments, polymer producers, and recycling companies), but there are already some successful examples of this model. Pyrolysis of plastics can be carried out via thermal or catalytic routes, and the distribution of obtained products is influenced by a wide range of parameters, including the type of waste used, the type of reaction system, the residence time, the pressure range, the presence or absence of catalysts, the presence of hydrogen donor compounds, and several others that affect the composition and yield of the obtained products, which consist of complex mixtures that require further separation if individual products are desired. Purely thermal pyrolysis processes temperatures can reach up to 900 ◦C which enhance the rate of secondary decomposition reactions and reduce the liquid yield, producing low-quality products with no predictable composition of a broad range of compounds [7]. Catalytic pyrolysis, on the other hand, promotes the polymer decomposition at lower temperatures and shorter times, reducing the energy consumption and enhancing the selectivity of the obtained products, but implying extra costs of the catalyst and its regeneration. In the case of the present proposed process, despite the reactions taking place purely thermally (with temperatures above, but close to the ceiling point of polystyrene), the obtained distribution of products shows a selectivity similar to that of a catalytic process. Submitting polymers to heat deeply alters their main chain structure, their substituent atoms, and their side chains, provoking random scission at weak links in the main chain, at a chain-end, or in labile structures and, depending on the site of reaction in the polymer structure and the conditions chosen for the particular system, the resulting radicals follow different paths of reaction, from depolymerization of the main chain to inter- or intramolecular transfer reactions [8]. The pyrolysis of pyrolysis of polystyrene (PS) has been a topic of interest for many years, and it is not surprising that renewed interest is rising considering all of the above trends and given that PS is a thermoplastic used throughout the world in a variety of applications. Thermal degradation of PS is considered to be a chain radical process [9–11], but despite numerous studies and experiments performed to understand its mechanism, many aspects of the problem remain unresolved, making it still difficult to accurately predict the product distribution at a given set of conditions. Additionally, as the process system temperature increases, so does its complexity [8]. The recovery of styrene monomer in the highest possible yield is a desired output of the pyrolysis process of PS, since this would contribute to the consolidation of a circular economy. Therefore, any effort directed towards this goal should be welcome. It is our hypothesis that modifying PS with a nitroxide moiety at a chain-end could enhance the generation of monomer when this material is subjected to thermal pyrolysis at a temperature higher than the ceiling temperature of PS. In this paper, empirical evidence that supports this hypothesis is presented and a mechanistic and mathematical model that is consistent with the experimental observations is also presented and discussed. Before that, background information that supports and motivates this hypothesis is introduced. In the next paragraphs, a brief literature review is made, including a description of the reaction mechanism and previous experimental work. Some of the studies are of academic nature, while many others are associated with patents. Reaction Mechanism. It has been reported that the molecular weight of polystyrene decreases on heating in vacuum at 250 ◦C, and when heated above 300 ◦C volatile products are formed [8,10]. With regard to the initiation process, it is accepted that the radicals are formed by different mechanisms such as chain-end initiation, random mid-chain scission, and scission at irregular structures or “weak links” distributed along the polymer chain such as head-to-head linkages, chain branches, and unsaturated bonds [9,10,12,13]. Lehrle et al. [13] concluded that between 450 to 480 ◦C initiation occurs both by random mid-chain scission and at chain-ends, and Madorsky [14] pointed out that between 300 to 400 ◦C it degrades preferably at the ends and that simultaneously some random degradation takes place. Processes 2020, 8, 432 3 of 26

Propagation reactions include intramolecular transfer, which involves the transfer of a hydrogen atom within a single polymer chain and intermolecular hydrogen transfer, comprising the transfer of a hydrogen atom between polymer chains. Also included in this category is the depropagation reaction, where no hydrogen transfer occurs; this reaction is essentially the reverse of the polymerization propagation, also called unzipping or depolymerization, and is mainly responsible for the production of styrene monomer [10,15]. It has been claimed that the most frequent propagation reaction taking place in the 280–350 ◦C temperature range is β-scission [9], either at a reaction end or at a mid-chain position, a reversible reaction that becomes faster as the ceiling temperature is approached (310–390 ◦C depending on the source [16,17]); below 300 ◦C the production of styrene is hardly observed. The second most important reaction is the hydrogen abstraction from the main chain by any radical, producing a saturated chain-end from the attacking radical, and a new radical that undergoes β-scission [9]. At higher temperatures (600 and 700 ◦C), increased production of benzene, toluene, ethylbenzene, and α-methyl styrene and decreased production of styrene have been observed, probably due to the predominance of intramolecular hydrogen transfer reactions at this range of temperatures [18]. Concerning termination, some authors propose a mechanism involving the recombination [9,11], or the disproportionation reaction [19] between two radicals; while others suggest the occurrence of depropagation until the end of the polymer molecule [13]. Published studies related to the thermal degradation and pyrolysis of polystyrene [13,20–25] have been performed in a variety of reaction systems and sets of conditions, including reaction temperatures, reaction times, and molecular weights of the polystyrene sample, resulting in a broad collection of reaction yields and product distribution. The available reported experimental data fall into three types: chemical nature of the products (that help to elucidate the degradation mechanism), rate of evolution of products, and the change of molecular weight in the residue. Jellinek [22] reported experiments carried out in long open-to-air tubes containing 0.1 to 0.5 g of material, to show the influence of oxygen on the thermal degradation of polystyrene. The reactions were described as too violent above 220 ◦C, many side reactions set in and the material quickly became yellow and brown. They assumed that the first stage of the oxidation consisted of the formation of hydro peroxide groups, which led to chain scission, and that at later stages, the reaction slowed down due to the formation of antioxidants such as benzaldehyde. For practical application, to avoid the presence of oxygen in the resulting products, pyrolysis processes are designed to occur in an inert atmosphere. Ebert et al. [23] pyrolyzed samples of polydisperse polystyrene with molecular weights between 60,000 and 220,000 Da in ampoules free of oxygen at temperatures between 270 and 336 ◦C, assessing the sample weight loss as a function of degradation time. They also performed simulations of the thermal degradation assuming that scission and unzipping occur independently, and attributed scission as the main cause for the decrease in molecular weight, while unzipping as the sole cause of volatiles production. Additionally, in oxygen-free glass ampoules, Wegner and Patat [26] analyzed n-butyl lithium initiated polystyrene samples with molecular weights between 110,000 to 900,000 Da, submitting them to temperatures between 200 to 325 ◦C, concluding that the polystyrene molecule does not contain thermolabile bonds corresponding to the weak-links theory. Reactions conducted in a fluidized-bed reactor with operation temperatures in the range 450–700 ◦C, nitrogen atmosphere, heated by an external electric furnace, permitted liquid recovery above 90%, with the main components being styrene monomer, dimer and trimer, benzene, toluene, ethylbenzene, and α-methyl styrene [20]. Simard et al. [21] reported PS pyrolysis experiments using flat bottom vessels and temperatures between 370–420 ◦C, with a maximum reaction time of 45 min, observing that 85 to 95% of the obtained liquid fraction are styrene, dimer, and trimer. Anderson and Freeman [25] assessed the weight loss of samples heated from 246 to 430 ◦C, observing decomposition at 320 ◦C and 99% of weight loss at 430 ◦C. Processes 2020, 8, 432 4 of 26

Small scale PS pyrolysis experiments have also been performed. Richards and Salter [24] intentionallyProcesses 2020, synthesized8, 432 polystyrene containing weak links in its structure that differ from the usual4 of 27 head–tail links of the polymer backbone and should, therefore, have different thermal properties. head–tail links of the polymer backbone and should, therefore, have different thermal properties. The The pyrolysis experiments were performed at 276–329 ◦C, and the main product from both polymers waspyrolysis styrene, experiments although significant were performed quantities at of276–329 dimer °C and, and trimer the weremain alsoproduct formed, from the both main polymers difference was beingstyrene, that although the molecular significant weight quantities of the polymer of dimer with and weak trimer links were decreased also form moreed, rapidly. the main difference beingSeveral that the patents molecular regarding weight polystyrene of the polymer pyrolysis with haveweak also links been decreased filed [27 more–32] rapidly. reporting the use Several patents regarding polystyrene pyrolysis have also been filed [27–32] reporting the use of temperatures varying in a wide range, 330–870 ◦C, styrene monomer recovery from 33 to 80%, andof temperatures diverse processes, varying including in a wide the range, use of 330–870 a fluidized °C, styrene bed reactor monomer [31] andrecovery the use from of 33 solvents to 80%, like and toluenediverse [32 processes,]. including the use of a fluidized bed reactor [31] and the use of solvents like toluene [32].In general, pyrolysis temperature is a very important factor affecting the product distribution [20]; in all theIn general, studies bothpyrolysis higher temperature temperatures is a and very longer important reaction factor times affecting resulted the in product an increase distribution in the amount[20]; in ofall the the liquid studies yield both [21 higher] and atemperatures decrease in the and molecular longer reaction weight times of the resulted residue. in Many an increase of these in experimentsthe amount were of the done liquid with yield very [21] small and samples a decrease (micrograms in the molecular and milligrams) weight of and the none residue. of them Many had of anthese initial experiments charge greater were than done 25 with g. At very increasing small temperaturessamples (micrograms more monomer and milligrams) is formed and reaching none aof them had an initial charge greater than 25 g. At increasing temperatures more monomer is formed maximum at 600 ◦C where secondary reactions take place. reachingAs can a bemaximum seen, the at use 600 of °C solvents where secondary and catalytic reactions systems take allows place. high yield of styrene recovery, althoughAs can some be drawbacksseen, the use associated of solvents with and thesecatalytic processes systems are allows evident, high suchyield asof styrene high operating recovery, temperaturesalthough some and drawbacks a wide range associated of obtained with products these byprocesses purely thermalare evident, pyrolysis. such as high operating temperaturesOn the other and hand,a wide nitroxide-mediated range of obtained products polymerization by purely (NMP) thermal is pyrolysis. among the most popular techniquesOn the in theother category hand, nitroxide-mediated of RDRP (reversible polyme deactivationrization radical (NMP) polymerization) is among the thatmost provide popular livingtechniques character in the to freecategory radical of polymerizations RDRP (reversible and deactivation control the radical structure polymerization) of the resulting that polymers, provide enablingliving character the production to free of radical polymers polymerizations with narrow molecularand control weight the structure distributions of the and resulting the formation polymers, of well-definedenabling the polymer production architectures of polymers such with as block narrow copolymers molecular [33 weight]. Roughly distributions speaking, and these the techniques formation simultaneouslyof well-defined provide polymer some architectures of the advantages such as ofblock living copolymers polymerizations [33]. Roughly (structure speaking, control) andthese conventionaltechniques simultaneously radical polymerization provide (robustness some of the to ad impuritiesvantages and of living to protic polymerizations media). NMP, which(structure is especiallycontrol) suitedand conventional for styrene polymerization,radical polymerization is based on(robustness a reversible to terminationimpurities and mechanism to protic between media). theNMP, propagating which is growing especially species suited and for a stable styrene free polymerization, radical (nitroxide is radical),based on that a actsreversible as a control termination agent tomechanism generate a macrobetween alkoxyamine the propagating (dormant growing polymer) species as predominant and a stable species.free radical The dormant(nitroxide polymer radical), eventuallythat acts as regenerates a control aagent propagating to generate radical a macro and a alkoxyamine nitroxide radical (dormant through polymer) a homolytic as predominant breakage asspecies. the dormant The dormant and the polymer radical eventually species are regenerates in dynamic a equilibriumpropagating resultingradical and from a nitroxide the reversible radical deactivation-activationthrough a homolytic breakage reactions as just the described dormant (seeand Figurethe radical1). As species a result are of in the dynamic NMP process, equilibrium the finalresulting product from is mainlythe reversible constituted deactivation-activation by dormant polymer reactions end-functionalized just described with(see Figure a nitroxide 1). As moiety. a result Forof mostthe NMP practical process, purposes the final this product polymer is canmainly be prepared constituted to be by very dormant similar po tolymer the productend-functionalized produced viawith conventional a nitroxide free-radicalmoiety. For polymerizationmost practical purposes (except forthis its polymer dispersity). can be However, prepared its to nitroxide-endbe very similar functionalityto the product should produced make itvia more conventional favorable to free-rad depolymerizationical polymerization in a certain (except range for of temperaturesits dispersity). (viaHowever, unzipping) its nitroxide-end than the conventional functionality PS, due should to the make increased it more possibility favorable of chain-end to depolymerization initiation of thein a depolymerizationcertain range of reactiontemperatures via rupture (via ofunzipping) the oxygen–carbon than the bondconventional between PS, the nitroxidedue to the moiety increased and thepossibility rest of the of polymeric chain-end chain.initiation of the depolymerization reaction via rupture of the oxygen–carbon bond between the nitroxide moiety and the rest of the polymeric chain.

CH2 -CH-CH2 -CH O-N CH2 -CH-CH2 -CH-O-N

FigureFigure 1. 1.Depolymerization Depolymerization initiation initiation reaction reaction via via oxygen–carbon oxygen–carbon bond bond rupture rupture (nitroxide (nitroxide radical radical activationactivation/deactivation)./deactivation).

WithWith respect respect to to the the thermal thermal pyrolysis pyrolysis process, process, it it is is known known that that chain-end chain-end initiation initiation in in mild mild temperaturetemperature conditions conditions or or moderatelymoderately aboveabove thethe polystyrenepolystyrene ceiling ceiling temperature temperature (310 (310 °C)◦C) leads leads to tounzipping unzipping and and increased increased production production of styrene of styrene monome monomerr over overother other products products of this of process. this process. Roland and Schmidt-Naake [34] studied the polymerization of styrene with TEMPO and benzoyl peroxide, claiming that the reversible capping with TEMPO can introduce a weak link at the end of the polymer, such as the bond between polymer and nitroxide. They conclude that polymer degradation for this material occurs in the same temperature range as for non-nitroxide polystyrene (400 °C). However the PS-T thermal degradation curve shows an additional step of mass loss at temperatures below 300

Processes 2020, 8, 432 5 of 26

Roland and Schmidt-Naake [34] studied the polymerization of styrene with TEMPO and benzoyl peroxide, claiming that the reversible capping with TEMPO can introduce a weak link at the end of the polymer, such as the bond between polymer and nitroxide. They conclude that polymer degradation for this material occurs in the same temperature range as for non-nitroxide polystyrene (400 ◦C). However the PS-T thermal degradation curve shows an additional step of mass loss at temperatures below 300 ◦C, suggesting two reactions, one apparently being the cleavage of the polystyrene–nitroxide bond followed by depolymerization, and the other being the breakage of the N-O bond in the TEMPO moiety, the importance of both reactions changing with increasing temperature. The lack of knowledge of the position and nature of the initial scission of the polystyrene thermal degradation has restricted its quantitative analysis. Taking into account this background, the goals of this work are twofold: (i) First, to define a process to achieve the thermal decomposition of conventional (free-radical) polystyrene in one step, and (ii) to compare the developed process for the thermal pyrolysis of conventional PS with another in which PS possessing a nitroxide end-functionality is used. This functionality is introduced in the polymer previously polymerizing styrene in the presence of the TEMPO nitroxide (PS-T) (TEMPO is (2,2,6,6-tetramethylpiperidin-1-yl) oxyl). For the comparison, precise pyrolysis conditions to induce depolymerization reactions that generate styrene monomer as main product and chemicals of high energetic value in the absence of solvents and catalysts at relatively mild conditions of temperature and pressure are first determined. TEMPO is used to provide living character to the free radical polymerization producing polystyrene with an extreme ended in a nitroxide moiety (PS-T) that, from the origin, has characteristics that favor depolymerization. As mentioned above, the hypothesis behind this work is that PS-T will depolymerize in a certain range of temperatures (via unzipping) by chain-end initiation of the reaction at the nitroxide functionality, promoting depropagation reactions in relatively mild temperature conditions or moderately above the polystyrene ceiling temperature. It is believed that above the Tc of polystyrene the uncapping (activation) reaction of the nitroxide moiety at the end of the polystyrene chain will leave a polystyryl radical that will undergo unzipping. The experiments seem to validate this hypothesis to some extent. On the other hand, a mathematical model of the pyrolysis reaction is developed to explain major effects observed and especially the effect of the end-nitroxide group in the PS-T chain.

2. Materials and Methods

2.1. Materials Styrene (99%) was puriﬁed using an inhibitor removal column (4-tert-butylcatechol). Benzoyl peroxide (98%), (2, 2, 6, 6-tetramethyl-1-piperidinyl-1-yl) oxyl (TEMPO), all reactants from Aldrich (St. Louis Missouri, USA) (Manufacturer name, city, state if US or Canada, country) and industrial grade methanol (90%) (Proquisa, Saltillo, Mexico) were used as received.

2.2. Reaction System The depolymerization system used in this work is shown in Figure2 and consists of a batch pressurized reactor and a condenser. The 50 mL reactor vase is made of stainless steel, and it comprises a heating source in the form of an electric mantle which provides homogeneous temperature increase inside the reactor and keeps the reaction temperature between 300 and 500 ◦C with the help of a thermocouple situated inside the reaction mixture and a digital controller that manipulates the current sent to the heating mantle. The reactor is also ﬁtted with a manometer, inlet and outlet valves (to feed inert gas to the system and generate a free oxygen atmosphere), and a third outlet valve that connects the reactor to a double tube condenser which uses a low temperature ﬂuid ( 5 C) that allows the rapid condensation of the − ◦ generated vapors. Processes 2020, 8, 432 5 of 27

°C, suggesting two reactions, one apparently being the cleavage of the polystyrene–nitroxide bond followed by depolymerization, and the other being the breakage of the N-O bond in the TEMPO moiety, the importance of both reactions changing with increasing temperature. The lack of knowledge of the position and nature of the initial scission of the polystyrene thermal degradation has restricted its quantitative analysis. Taking into account this background, the goals of this work are twofold: (i) First, to define a process to achieve the thermal decomposition of conventional (free-radical) polystyrene in one step, and (ii) to compare the developed process for the thermal pyrolysis of conventional PS with another in which PS possessing a nitroxide end-functionality is used. This functionality is introduced in the polymer previously polymerizing styrene in the presence of the TEMPO nitroxide (PS-T) (TEMPO is (2,2,6,6-tetramethylpiperidin-1-yl) oxyl). For the comparison, precise pyrolysis conditions to induce depolymerization reactions that generate styrene monomer as main product and chemicals of high energetic value in the absence of solvents and catalysts at relatively mild conditions of temperature and pressure are first determined. TEMPO is used to provide living character to the free radical polymerization producing polystyrene with an extreme ended in a nitroxide moiety (PS-T) that, from the origin, has characteristics that favor depolymerization. As mentioned above, the hypothesis behind this work is that PS-T will depolymerize in a certain range of temperatures (via unzipping) by chain-end initiation of the reaction at the nitroxide functionality, promoting depropagation reactions in relatively mild temperature conditions or moderately above the polystyrene ceiling temperature. It is believed that above the Tc of polystyrene the uncapping (activation) reaction of the nitroxide moiety at the end of the polystyrene chain will leave a polystyryl radical that will undergo unzipping. The experiments seem to validate this hypothesis to some extent. On the other hand, a mathematical model of the pyrolysis reaction is developed to explain major effects observed and especially the effect of the end-nitroxide group in the PS-T chain.

2. Materials and Methods

2.1. Materials Styrene (99%) was purified using an inhibitor removal column (4-tert-butylcatechol) . Benzoyl peroxide (98%), (2, 2, 6, 6-tetramethyl-1-piperidinyl-1-yl) oxyl (TEMPO), all reactants from Aldrich (St. Louis Missouri, USA) (Manufacturer name, city, state if US or Canada, country) and industrial grade methanol (90%) (Proquisa, Saltillo, Mexico) were used as received.

2.2. Reaction System ProcessesThe2020 depolymerization, 8, 432 system used in this work is shown in Figure 2 and consists of a batch6 of 26 pressurized reactor and a condenser.

Figure 2. Depolymerization reaction system.

2.3. Polystyrene Synthesis Methods Polystyrene that would be subjected to the pyrolysis experiments was synthesized by two different methods: by conventional free radical polymerization, and by nitroxide mediated radical polymerization (NMP). In the conventional free radical polymerization (FRP), the reaction was performed in a batch bulk process at constant temperature (90 ◦C) in a 150 mL glass reactor, inert atmosphere, and continuous agitation, using benzoyl peroxide as initiator. The agitation was implemented via a mechanical agitator that was used until the viscosity of the reaction media made it not possible to continue in this way. After this point, reached at conversions above 80%, the reaction was left without agitation until conversions of 99% were reached and then stopped by lowering the reaction temperature to ambient temperature. The polymer was subsequently recovered by adding acetone to it (around 10–15% wt. with respect to the total mass of polymer) to allow the material to flow, followed by precipitation of the reaction mixture in methanol. The precipitate was later filtered and dried, obtaining white polymer dust as product. Styrene polymerizations by the NMP method were performed using BPO as initiator and TEMPO as the stable free radical, varying the nitroxide (N) to initiator (I) ratio; three levels were established (N/I: 1.3, 1.1, and 0.9), they were all performed in a batch reactor at constant temperature (130 ◦C) with continuous agitation and under inert atmosphere. When conversions of 99% were reached, the reactions were stopped by submerging the reactor in a cooling ice bath, lowering the system temperature. The polymer from the reaction mixture was recovered following the same procedure described before for the PS produced by conventional FRP. The product was obtained as a white dust. The polymers obtained by FRP and NMP were analyzed by size exclusion chromatography (SEC) using an Agilent Technologies Mod G7810A chromatograph (Agilent Technologies, Santa Clara, 1 CA, USA) at 40 ◦C, with a sample concentration of 1 mg mL− , solvent flow of 1 mL/min and a polystyrene standard (Agilent Technologies, Santa Clara, CA, USA); the results are summarized in Table1. The experiments were designed to obtain approximately the same number average molecular weight in all the cases (40–50,000 Da).

Table 1. Properties of synthesized polystyrene.

Sample Mn Mw Mw/Mn NMP, N/I = 1.3 46,100 56,800 1.23 NMP, N/I = 1.1 44,800 56,900 1.27 NMP, N/I = 0.9 42,500 55,500 1.3 FRP 50,040 96,700 1.9 Processes 2020, 8, 432 7 of 26

2.4. Polystyrene Thermal Pyrolysis Experiments The process to depolymerize all the polystyrene samples was the same and started by introducing 10 g of the synthesized plastic material inside the reactor and displacing the oxygen atmosphere with nitrogen to guarantee that the experiment was developed under inert conditions. Once this was achieved, all the valves of the reactor were closed and the temperature was rapidly increased, provoking a pressure rise; the time needed to reach the defined temperature set point was between 5 to 7 min. When the pressure reached from 7 to 12 psig, the valve connecting the reactor to the condenser was turned open, allowing the vapor products to migrate from the reactor to the condenser, after which the products were recovered as a liquid mixture. After reaching the temperature set point, the reaction proceeded until the liquid product was no longer obtained (about 10 min). The total reaction time was considered from the point at which the heating started to the moment at which the liquid outflow stopped. Experiments were carried out varying the pyrolysis temperature: 390, 420, and 450 ◦C and the source polystyrene (see Table2), to investigate the e ffect of this variable on the prepared samples. After the reaction was finished and the reactor was cooled, two types of samples were obtained from each experiment: a liquid mixture from the condensed vapors and the residue left in the reactor in the form of a dark-colored oil with some solid degradation fragments (in the following discussion this will be referred as the solid fraction). Only the condensed liquid mixture was analyzed since it contains the most valuable products (including the monomer).

Table 2. Pyrolysis experiments performed.

Sample Pyrolysis Temperature

NMP, N/I = 1.3 390 ◦C * (Tri) 420 ◦C (Tri) 450 ◦C (Tri) NMP, N/I = 1.1 390 ◦C 420 ◦C 450 ◦C NMP, N/I = 0.9 390 ◦C 420 ◦C 450 ◦C FRP 390 ◦C (Tri) 420 ◦C (Tri) 450 ◦C (Tri) * (Tri) = Run and analyzed by triplicate.

3. Results

3.1. Product Analysis The obtained products from the polystyrene pyrolysis experiments were analyzed to determine the most abundant components present in the liquid mixture. A gas chromatograph (Thermo Finnigan by Thermo Fisher Scientiﬁc Waltham, MA, USA) equipped with a selective mass detector (DSQ Trace 2000) (Thermo Electron Corporation, Austin, TX, USA) and a Thermo TG-5ms column were used. All the samples were analyzed under the same conditions with helium as the carrier gas. The temperature program was as follows: the temperature was held at 60 ◦C for 3 min, then programmed to reach 300 ◦C 1 at a heating rate of 10 ◦C min− and held at that temperature for 10 min. The injector temperature was set at 250 ◦C. The transfer temperature line of the mass detector was set at 280 ◦C and the mass range was from 32 to 650 amu. The products were identiﬁed according to their fragmentation patterns using a library included in the software of the chromatograph.

3.2. Kinetic Model As described before, the radical process of conventional polystyrene pyrolysis comprises the typical steps of initiation, propagation, transfer to polymer, and termination: Although initiation can occur due to several causes, there seems to be general agreement that consists of the formation of free radicals after bond breakage by the action of heat mainly either at a chain-end (which results in the production of monomer) or at a mid-chain position along the polymer backbone. Propagation covers the competition of three diﬀerent reaction mechanisms: unzipping, intramolecular, and intermolecular hydrogen transfer, the ﬁrst one is sometimes also called depolymerization and is taken to be the reverse Processes 2020, 8, 432 8 of 26

of chain growth, while transfer to polymer consists in the abstraction of a hydrogen from the same or another molecule. The termination step can occur via recombination and disproportionation reactions. Processes 2020, 8, 432 8 of 27 Our proposed model, which aims to describe both conventional and nitroxide-modiﬁed Processes 2020, 8, 432 8 of 27 polystyreneProcesses 2020, pyrolysis8, 432 behavior, contains the following reactions: mid-chain random scission, end chain8 of 27 scission), (transfer to polymer + β-scission, simplified mechanism), (depropagation), scissionProcesses 2020 or activation,, 8, 432 transfer to polymer, β-scission, depropagation, and termination by combination8 of 27 (terminationscission), by (transfer combination), to polymer and + β-scission, (termination simplified by mechanism),disproportionation). (depropagation), To simplify the Processesandscission), by 2020 disproportionation, , 8, 432 (transfer toas polymer can beseen + β-scission, in Table3 .simplifiedP denotes mechanism), living polymer with (depropagation), length n, while8 of R 27 , Processes(termination 2020, 8, 432by combination), and (terminationn by disproportionation). To simplify8 ofthe 27n description,scission), some (transfer assumptions to polymer and approximations + β-scission, simplified are made: mechanism), (depropagation), S(terminationProcessesn, Dn, 2020M,, and 8 , 432byM combination),denote live polymerand of (termination length n with by nitroxide disproportionation). in one end, dormant To simplify polymer8 ofthe 27 scission),description,Dormant some polymer (transfer· assumptions with to polymer one andor two+ approximations β-scission, nitroxide-function simplified are made:alized mechanism), ends will be lumped (depropagation), together into a of(terminationdescription, length n, deadsome by combination), polymerassumptions of length and and approximations n, monomer (termination and are monomermade: by disproportionation). radical, respectively. To simplify The kinetic the singlescission),Dormant quantity. polymer (transferUpon mid-chain with to polymer one orra ndomtwo+ β-scission, nitroxide-function scission simplified of thesealized speciesmechanism), ends to willform be two lumped (depropagation), radicals, together it will into be a (terminationscission), by (transfercombination), to polymer and + β-scission, (termination simplified by mechanism),disproportionation). (depropagation), To simplify the constantsdescription,Dormant are some denotedpolymer assumptions as:withkb one(mid-chain and or twoapproximations nitroxide-function random scission), are made:alizedkbe (end-chain ends will be scission), lumpedk trtogetherβ (transfer into to a description,assumed(terminationsingle quantity. that some onlyby Uponassumptionscombination), one radicalmid-chain containsand and raapproximationsndom a nitroxide-fu scission(termination areofnctionalized thesemade: by species disproportionation). end. to Noticeform two that radicals, asTo the simplify number it will the ofbe (terminationDormantβ bypolymer combination), with one orand two nitroxide-function (terminationk byalized disproportionation). endsk will be lumped To together simplify into the a polymerdescription,singleassumed quantity.+ that -scission,some only Upon assumptions one simpliﬁed mid-chainradical containsand mechanism), ra approximationsndom a nitroxide-fu scissionrev (depropagation), of are nctionalizedthese made: species end. tcto(termination formNotice two that radicals, byas combination),the number it will be of chaindescription,Dormant ends increases some polymer assumptions in with the system one andor twodue approximations nitroxide-functionto the pyroly sisare process, made:alized endsthe probability will be lumped of a chain together ending into ina andsingleassumedktd quantity.(termination that only Upon one by mid-chainradical disproportionation). contains random a nitroxide-fu scission To simplify ofnctionalized these the species description, end. to formNotice sometwo that radicals, assumptions as the number it will and beof singleachain nitroxideDormant quantity.ends moiety increases polymer Upon decreases. inmid-chain with the system oneThis orra introduces ndomtwodue nitroxide-functionto scissionthe some pyroly oferror, sisthese process, butalized species it will endsthe beto probability willformneglected be two lumped offorradicals, a simplicity chain together it ending will intosince be ina approximationsassumedchainDormant ends that increases only polymer are one made: in radicalwiththe system one contains or duetwo toanitroxide-function nitroxide-fu the pyrolysisnctionalized process,alized theends end. probability will Notice be lumped that of aas chain togetherthe number ending into ofin a itsinglea is nitroxide assumed quantity. moiety that Upon on decreases. the mid-chain averag Thise the ra introduces ndomerror introduced scission some oferror, isthese small. but species it will beto formneglected two forradicals, simplicity it will since be assumedchainsingle ends quantity. that increases only Upon one in radical mid-chainthe system contains ra duendom ato nitroxide-fu the scission pyroly ofsisnctionalized these process, species the end. probabilityto Notice form two that of radicals, asa chainthe number endingit will of bein assumedait nitroxide is assumed that moiety thatonly ondecreases.one the radical averag This containse the introduces error a nitroxide-fu introduced some error,nctionalized is small. but it will end. be neglectedNotice that for as simplicity the number since of chainaassumed nitroxideTable ends that 3.increases moietyPolystyrene only decreases.one in the radical depolymerization system This contains dueintroduces to a kinetic thenitroxide-fu pyroly some mechanism error,sisnctionalized process, but including it willthe end. probabilitybe the neglected presenceNotice thatof of for a nitroxide-end aschain simplicity the endingnumber since in of chainit is assumedTable ends 3. increases Polystyrene that on inthe thedepolymerization averag systeme the due error to kinetic theintroduced pyroly mechanissis is process, msmall. including the theprobability presence of nitroxide-enda chain ending in aitchain nitroxide is assumedpolystyrene. ends moiety increases that ondecreases. thein the averag system Thise theintroduces due error to theintroduced some pyroly error,sis is process, small.but it will the be probability neglected offor a simplicity chain ending since in a nitroxidepolystyrene.Table 3.moiety Polystyrene decreases. depolymerization This introduces kinetic some mechanis error,m but including it will bethe neglected presence offor nitroxide-end simplicity since ita is nitroxide assumedTable 3. moiety thatPolystyrene on decreases. the averagdepolymerization Thise the introduces error kineticintroduced some mechanis error, is small.m but including it will bethe neglected presence of for nitroxide-end simplicity since it is assumedpolystyrene. that onMechanism the average the error introduced is small. Reaction it is assumedTablepolystyrene. 3. Polystyrene that on the Mechanismdepolymerization average the error kinetic introduced mechanis is msmall. including theReaction presence of nitroxide-end Tablepolystyrene. 3. Polystyrene depolymerizationMechanism kinetic mechanism including the Reactionpresence of nitroxide-end Table 3. Polystyrene Mechanismdepolymerization kinetic mechanism including theReaction presence of nitroxide-end polystyrene.MidTable chainMid 3. chainPolystyrene random random scission, depolymerization scission, dormant dormant polymer kinetic polymer mechanis m including the presence of nitroxide-end polystyrene. Mechanism Reaction polystyrene.Mid chain random scission, dormant polymer Mid chain randomMechanism scission, dormant polymer Reaction Mechanism Reaction MidMid chain chain random randomMechanism scission, scission, dormant dead polymer polymer Reaction MidMid chainMid chain chain random random random scission, scission, scission, dead dormant polymerdead polymerpolymer MidMid chain chain random random scission, scission, dormant dead polymer polymer Mid chain random scission, dormant polymer MidEnd chain chain random scission, scission, dead dead polymer polymer Mid chainEnd chainrandom scission, scission, dead dead polymer polymer EndMid chain Endchain chain scission, random scission, dead scission, polymerdead dead polymer polymer Mid chain random scission, dead polymer * End chainEnd scission chain scission,or activation, dead dormant polymer polymer * End chainEnd scissionchain scission, or activation, dead polymer dormant polymer * End chainEnd scission chain scission, or activation, dead dormantpolymer polymer End chain scission, dead polymer * End* chain End chain scission scission or activation, or activation, dormant dormant polymer polymer * End chain scission or activation, dormant polymer * End chain scission or activation, dormant polymer * End chainTransfer scission to polymer or activation, + β-scission dormant (1) polymer

Transfer to polymer + β-scission (1) Transfer to polymer + β-scission (1) Transfer to polymer + β-scission (1)

TransferTransfer to polymer to polymer+ β-scission + β-scission (1) (1) Transfer to polymer + β-scission (1) Transfer to polymer + β-scission (1)

Transfer to polymer + β-scission (2)

Transfer to polymer + β-scission (2) Transfer to polymer + β-scission (2) Transfer to polymer + β-scission (2) Transfer to polymer + β-scission (2) TransferTransfer to polymer to polymer+ β-scission + β-scission (2) (2) Transfer to polymer + β-scission (2)

Transfer to polymer + β-scission (3)

Transfer to polymer + β-scission (3) Transfer to polymer + β-scission (3)

Transfer to polymer + β-scission (3) Transfer to polymer + β-scission (3) Transfer to polymer + β-scission (3) TransferTransferTransfer to polymer to to polymer polymer+ β-scission + + β β-scission-scission (3) (4) (3) Transfer to polymer + β-scission (4) Transfer to polymer + β-scission (4)

Transfer to polymer + β-scission (4) Transfer to polymer + β-scission (4) Transfer toDe-propagation polymer + β-scission (4) Transfer to polymer + β-scission (4) De-propagation De-propagation

De-propagation

De-propagation De-propagation De-propagation

Processes 2020, 8, 432 8 of 27 Processes 2020, 8, 432 8 of 27 scission), (transfer to polymer + β-scission, simplified mechanism), (depropagation), (terminationscission), by (transfer combination), to polymer and + β-scission, (termination simplified by mechanism),disproportionation). (depropagation), To simplify the description,(termination some by assumptionscombination), and and approximations (termination are made: by disproportionation). To simplify the description,Dormant some polymer assumptions with one andor two approximations nitroxide-function are made:alized ends will be lumped together into a singleDormant quantity. polymer Upon mid-chain with one orra twondom nitroxide-function scission of thesealized species ends to willform be two lumped radicals, together it will into be a assumedsingle quantity. that only Upon one radicalmid-chain contains random a nitroxide-fu scission ofnctionalized these species end. to Noticeform two that radicals, as the number it will ofbe chainassumed ends that increases only one in theradical system contains due to a thenitroxide-fu pyrolysisnctionalized process, the end. probability Notice that of a aschain the numberending in of achain nitroxide ends moiety increases decreases. in the system This introduces due to the some pyroly error,sis process, but it will the be probability neglected offor a simplicitychain ending since in ita isnitroxide assumed moiety that on decreases. the averag Thise the introduces error introduced some error, is small. but it will be neglected for simplicity since it is assumed that on the average the error introduced is small. Table 3. Polystyrene depolymerization kinetic mechanism including the presence of nitroxide-end polystyrene.Table 3. Polystyrene depolymerization kinetic mechanism including the presence of nitroxide-end polystyrene. Mechanism Reaction Mechanism Reaction Mid chain random scission, dormant polymer Mid chain random scission, dormant polymer

Mid chain random scission, dead polymer Mid chain random scission, dead polymer

End chain scission, dead polymer End chain scission, dead polymer

* End chain scission or activation, dormant polymer * End chain scission or activation, dormant polymer

Transfer to polymer + β-scission (1) Transfer to polymer + β-scission (1)

Transfer to polymer + β-scission (2) Transfer to polymer + β-scission (2)

Processes 2020, 8, 432 9 of 26

Transfer to polymer + β-scission (3)Table 3. Cont. Transfer to polymer + β-scission (3) Mechanism Reaction

TransferTransfer to polymer to polymer+ β-scission + β-scission (4) (4) Transfer to polymer + β-scission (4)

De-propagation De-propagationDe-propagation

Processes 2020, 8, 432 9 of 27

ProcessesProcesses 20202020,, 88,, 432432 9 9of of 2727

TerminationTermination by combination by combination TerminationTermination byby combinationcombination

Termination by disproportionation TerminationTerminationTermination by disproportionation byby disproportionationdisproportionation

Termination with monomeric radicals

TerminationTerminationTermination with with monomericwith monomericmonomeric radicals radicalsradicals

* Notice that this is the same reaction illustrated in Figure 2. ** NoticeNotice* Notice thatthat that thisthis this isis is thethe the samesame same reactionreaction illustrated illustratedillustrated in Figureinin FigureFigure2. 2.2. Initiation includes the possible formation of free radicals either in the terminal or in a mid-chain Dormant polymer with one or two nitroxide-functionalized ends will be lumped together into randomInitiationInitiation position. includesincludes The corresponding thethe possiblepossible formationformation kinetic coefficien ofof freefree raratsdicalsdicals could eithereither be different inin thethe terminalterminal depending oror ininon aa the mid-chainmid-chain position a single quantity. Upon mid-chain random scission of these species to form two radicals, it will be randomrandom(end-chain position.position. or mid-chain) TheThe correspondingcorresponding since the radicals kinetickinetic coefficienproducedcoefficientsts in couldcould each bebecase differentdifferent are of different dependingdepending nature, onon thethe mid-chain positionposition assumed that only one radical contains a nitroxide-functionalized end. Notice that as the number of (end-chain(end-chainscission forms oror mid-chain)mid-chain) one primary sincesince radical thethe radicals radicalsand one produced producedsecondar y inin benzyl eacheach case caseradical, areare ofwhileof differentdifferent end-chain nature,nature, scission mid-chainmid-chain forms chainscission ends forms increases one primary in the systemradical dueand toone the secondar pyrolysisy benzyl process, radical, the probability while end-chain of a chain scission ending forms in a scissiona secondary forms benzylone primary radical; radical both and reactions one secondar can occury benzyl on dormant radical, whileor dead end-chain polymer scission ( , forms ) as nitroxide moiety decreases. This introduces some error, but it will be neglected for simplicity since it is aadescribed secondarysecondary in benzylTablebenzyl 3. radical; radical;Transfer bothboth to polymer reactionsreactions followed cancan occuroccur by β on-scissionon dormantdormant can occuroror deaddead between polymerpolymer any ((of𝑆 two,, 𝐷 types)) asas assumeddescribed that in Table on the 3. averageTransfer the to polymer error introduced followed is by small. β-scission can occur between any of two types describedof living inchains Table (normal, 3. Transfer ,to or polymer with a followednitroxide-end by β-scission functionality, can occur ) between and one any of oftwo two types types of of livingInitiation chains includes (normal, the possible, or with formation a nitroxide-end of free radicals functionality, either in the ) terminaland one or of in two a mid-chain types of ofinactive living chains (dormant,(normal, 𝑃, ,or or withdead, a nitroxide-end). In the case functionality, of dormant polymer,𝑅) and theone initiation of two types can also of random position. The corresponding kinetic coefficients could be different depending on the position inactiveinactiveoccur at chainschains the nitroxide (dormant,(dormant, end 𝑆 via,, oror the dead,dead, activation 𝐷).). InIn reaction. thethe casecase Theofof dormantdormant rest of the polymer,polymer, reactions, thethe depropagationinitiationinitiation cancan also alsoand (end-chain or mid-chain) since the radicals produced in each case are of different nature, mid-chain occuroccurtermination, atat thethe nitroxidenitroxide are the end endsame viavia thetheas activationthoseactivation appearing reaction.reaction. in TheThe the restrest standard ofof thethe reactions,reactions, mechanism depropagationdepropagation of free-radical andand termination,termination,polymerization, areare withthethe samesubtlesame asasvariations thosethose appearingappearingdepending in inon thethe the standardstandardpossible presencemechanismmechanism of aofof nitroxide-end free-radicalfree-radical polymerization,polymerization,functionality in onewithwith or subtle subtleboth reacting variationsvariations chains. dependingdepending onon thethe possiblepossible presencepresence ofof aa nitroxide-endnitroxide-end functionalityfunctionality inin oneone oror bothboth reactingreacting chains.chains. 3.3. Mathematical Model 3.3.3.3. MathematicalMathematical ModelModel Based on the described reaction scheme, the corresponding mathematical model was formulated writingBasedBased a mass onon thethe balance describeddescribed equation reactionreaction for scheme,scheme, each species. thethe cocorresponding rrespondingThen, the method mathematicalmathematical of moments modelmodel was waswas used formulatedformulated to keep writingwritingtrack of aa changes massmass balancebalance of the averageequationequation molecular forfor eacheach species.species.weights Then, Then,as a function thethe methodmethod of time. ofof momentsmoments waswas usedused toto keepkeep tracktrack ofof changeschanges ofof thethe averageaverage molecularmolecular weightsweights asas aa functionfunction ofof time.time.

Processes 2020, 8, 432 10 of 26 scission forms one primary radical and one secondary benzyl radical, while end-chain scission forms a secondary benzyl radical; both reactions can occur on dormant or dead polymer (Sn, Dn) as described in Table3. Transfer to polymer followed by β-scission can occur between any of two types of living chains (normal, Pn, or with a nitroxide-end functionality, Rn) and one of two types of inactive chains (dormant, Sn, or dead, Dn). In the case of dormant polymer, the initiation can also occur at the nitroxide end via the activation reaction. The rest of the reactions, depropagation and termination, are the same as those appearing in the standard mechanism of free-radical polymerization, with subtle variations depending on the possible presence of a nitroxide-end functionality in one or both reacting chains.

3.3. Mathematical Model Based on the described reaction scheme, the corresponding mathematical model was formulated writing a mass balance equation for each species. Then, the method of moments was used to keep track of changes of the average molecular weights as a function of time. Dead polymer, n = 1, ... , ∞

dDn P∞ P∞ = k nDn k Dn + ktr Pn m Dm ktr n Dn Pm dt − b − be β − β ! m=1 m=1 P∞ P∞ 1 P∞ +ktrβ Ps m Dm m + ktrβPn m Sm s=1 !m=n+1 m=1 ! ktrβ P∞ P∞ 1 ktrβ P∞ P∞ 1 + 2 Pr m Sm m + 2 Rr m Sm m = = + = = + (1) r 1 !m n 1 r 1 m n 1 P∞ P∞ 1 P∞ +ktrβ Rr m Dm m ktrβ nDn Rm r=1 m=n+1 − m=1 n 1 ktc P− P∞ P∞ + 2 PmPn m + ktdPn Pm + ktdPn Rm + ktcPn 1 M m=1 − m=1 m=1 − +ktdPn M

Dormant polymer, n = 1, ... , ∞ ! dSn P∞ ktrβ P∞ P∞ 1 dt = kbnSn kaSn + kdMPn ktrβ n Sn Pm + 2 Pr m Sm m − − − m=1 r=1 m=n+1 P∞ P∞ ktr n Sn Rm + ktr Rn m Sm − β β m=!1 m=1 ktrβ P∞ P∞ 1 P∞ + 2 Rr m Sm m + ktrβRn m Dm (2) r=1 m=n+1 m=1 n 1 n 1 1 P− 1 P− P∞ + 2 ktc PmRn m + 2 ktc RmRn m + ktdRn Pm m=1 − m=1 − m=1 P∞ +ktdRn Rm + ktcRn 1M + ktdRnM m=1 − ·

Live polymer, n = 1, ... , ∞ dPn P∞ 1 P∞ 1 dt = kaSn kdNPn +2kb m Dm m + kbeDn+1 + kb m Sm m − m=n+1 m=n+1 P∞ krevPn + krevP + ktr Pn m Sm − n 1 − β ! m=1 ! ktrβ P∞ P∞ 1 ktrβ P∞ P∞ 1 + 2 Pr m Sm m + 2 Rr m Sm m (3) r=1 m=n+1 r=1 m=n+1 P∞ P∞ P∞ 1 ktrβPn m Dm + ktrβ Pr m Dm m − m=n+1 r=1 m=1 P∞ P∞ 1 P∞ P∞ +ktrβ Rr m Dm m ktPn Pm ktPn Rm ktPnM r=1 m=n+1 − m=1 − m=1 − Processes 2020, 8, 432 11 of 26

Live polymer with a nitroxide at the chain-end, n = 1, ... , ∞ ! dRn P∞ 1 ktrβ P∞ P∞ 1 P∞ = k m Sm + Pm r Sr ktr Rn m Sm dt b m 2 r − β m=n+1 ! m=1 r=n+1 m=1 ktrβ P∞ P∞ 1 P∞ + 2 Rr m Sm m ktrβRn m Dm + krevRn+1 (4) r=1 m=n+1 − m=1 P∞ P∞ krevRn ktRn Pm ktRn Rm ktRnM − − m=1 − m=1 −

Monomer

dM X∞ 1 X∞ X∞ 1 X∞ = krev Pm + k M Pm + krev Rm + k M Rm (5) dt 2 td · 2 td · m=1 m=1 m=1 m=1

Monomeric radicals

dM X∞ 1 X∞ X∞ 1 X∞ = krev Pm + k M Pm + krev Rm + k M Rm (6) dt 2 td · 2 td · m=1 m=1 m=1 m=1

Nitroxide radicals dN = kaSn k N Pn (7) dt − d · Ethyl benzene dEt B 1 X∞ 1 X∞ − = k M Pm + k M Rm (8) dt 2 td · 2 td · m=1 m=1 Four polymer populations exist in this system: Dead polymer, dormant polymer, growing polymeric radical and polymer radical with a nitroxide end functionality; moments for these species are deﬁned in Equations (9)–(12) respectively: X i λi = r Dr (9) r X i ξi = r Sr (10) r X i µi = r Pr (11) r X i ψi = r Rr (12) r The method of moments applied to the previously described balance equations generates Equations (13)–(15) for the zero-th, ﬁrst, and second moments of dead polymer, respectively, Equations (16)–(18) for the moments of dormant polymer, Equations (19)–(21) for polymeric radical moments, and Equations (22)–(24) for the moments of polymeric radicals with a nitroxide end functionality.

dλ0 2 ktc 2 = k λ k λ + k µ + µ + k µ ψ + ktcM µ + k M µ dt − b 1 − be 0 td 0 2 0 td 0 0 · 0 td · 0 +ktrβµ0λ1 ktrβµ0λ1 + ktrβ(λ1 λ0)(µ0 + ψ0) + ktrβµ0ξ1 ktrβ − − (13) + (µ + ψ )B (ξ ξ ) ktr λ ψ 2 0 0 0 1 − 0 − β 1 0 B0 = 1

dλ1 = k λ k λ + k µ µ + ktcµ µ + k µ ψ + ktcM (µ + µ ) + k M µ dt − b 2− be 1 td 1 0 1 0 td 1 0 1 0 td · 1 (µ0+ψ0) +ktr [µ λ λ µ + (λ λ ) + µ ξ (14) β 1 1 − 2 0 2 2 − 1 1 1 + 1 (µ + ψ )(ξ ξ ) λ ψ ] 4 0 0 2 − 1 − 2 0 Processes 2020, 8, 432 12 of 26

dλ2 2 = k λ k λ + k µ µ + ktc µ µ + µ + k µ ψ + ktcM (µ + 2µ dt − b 3− be 2 td 2 0 0 2 1 td 2 0 · 2 1 +µ0) + ktdM µ2 + ktrβ[µ2λ1 λ3µ0 1· 1 1 − (15) +(µ0 + ψ0) λ3 λ2 + λ1 + µ2ξ1 3 − 2 6 + 1 (µ + ψ ) 1 ξ 1 ξ + 1 ξ λ ψ ] 2 0 0 3 3 − 2 2 6 1 − 3 0 dξ0 ktc ( ) ( ) 2 dt = kbξ1 kaξ0 + kdNµ0 + 2 µ0ψ0 + ψ0ψ0 + ktd ψ0µ0 + ktdψ0 + ktcM − − ktrβ ψ + k ψ M ktr ξ µ ktr ξ ψ + (µ + ψ )(ξ ξ ) (16) · 0 td 0 · − β 1 0 − β 1 0 2 0 0 1 − 0 +ktrβψ0(ξ1 + λ1)

dξ1 ktc = k ξ kaξ + k Nµ + (µ ψ + µ ψ ) + ktcψ ψ + k (ψ µ ) dt − b 2− 1 d 1 2 1 0 0 1 1 0 td 1 0 +ktdψ1ψ0 + ktcM (ψ0 + ψ1) + ktdψ1M ktrβξ2µ0 ktrβξ2ψ0 (17) ktrβ · − − + (µ + ψ )(ξ ξ )+ktr ψ (ξ + λ ) 4 0 0 2 − 1 β 1 1 1 dξ2 ktc 2 = k ξ kaξ + k Nµ + (µ ψ + 2µ ψ + µ ψ ) + ktc ψ ψ + ψ dt − b 3− 2 d 2 2 2 0 1 1 0 2 2 0 1 +ktd(ψ2µ0) + ktdψ2ψ0 + ktcM (ψ2 + 2ψ1 + ψ0) + ktdψ2M (18) ktrβ 1 1 1 ktr ξ µ ktr ξ ψ + (µ + ψ ) ξ ξ + ξ − β 3 0 − β 3 0 2 0 0 3 3 − 2 2 6 1 +ktrβψ2(ξ1 + λ1)

dµ0 2 dt = kaξ0 kdNµ0 +kbeλ0 ktµ0 ktµ0ψ0 ktµ0M ktrβµ0ξ1 − − − − − 1 +ktr (µ + ψ )(λ λ ) ktr µ λ + ktr (µ + ψ )( ξ ξ ) (19) β 0 0 1 − 0 − β 0 1 2 β 0 0 1 − 0 +2k [(λ λ )+(ξ ξ )] b 1 − 0 1 − 0

dµ1 = kaξ k N µ + k (λ λ ) krevµ + krev(µ µ ) ktµ µ ktµ ψ ktµ M dt 1 − d 1 be 1 − 0 − 1 1 − 0 − 1 0 − 1 0 − 1 h 1 i 1 h 1 i h 1 i (20) ktr µ λ + ktr (µ + ψ ) (λ λ ) ktr µ ξ + ktr (µ + ψ ) (ξ ξ ) + k (λ λ ) + (ξ ξ ) − β 1 1 β 0 0 2 2 − 1 − β 1 1 2 β 0 0 2 2 − 1 b 2 − 1 2 2 − 1

dµ2 ( ) ( ) dt = kaξ2 kdNµ2 +kbe λ2 2λ1 + λ0 krevµ2 + krev µ2 2µ1 + µ0 ktµ2µ0 − − − − h 1 −1 1 i ktµ2ψ0 ktµ2M ktrβµ2λ1 + ktrβ(µ0 + ψ0) 3 λ3 2 λ2 + 6 λ1 − − 1 − h 1 1 1 i − (21) ktrβµ2ξ1 + ktrβ(µ0 + ψ0) ξ3 ξ2 + ξ1 − h 2 3 − 2 6 i +2k 1 λ 1 λ + 1 λ + 1 ξ 1 ξ + 1 ξ b 3 3 − 2 2 6 1 3 3 − 2 2 6 1 dψ0 2 1 = ktψ µ ktψ ktψ M + k (ξ ξ ) + ktr (µ + ψ )(ξ ξ ) dt − 0 0 − 0 − 0 b 1 − 0 2 β 0 0 1 − 0 (22) ktr ψ (ξ + λ ) − β 0 1 1 dψ1 kb ( ) dt = krevψ0 ktψ1µ0 ktψ1ψ0 ktψ1M + 2 ξ2 ξ1 − − 1 − h 1− i − (23) + ktr (µ + ψ ) (ξ ξ ) ktr ψ (ξ + λ ) 2 β 0 0 2 2 − 1 − β 1 1 1 dψ2 ( ) dt = krev ψ2 2ψ1 + ψ0 krevψ2 ktψ2µ0 ktψ2ψ0 ktψ2M − h 1 −1 1 −i 1 − −h 1 1 1 i +k ξ ξ + ξ + ktr (µ + ψ ) ξ ξ + ξ (24) b 3 3 − 2 2 6 1 2 β 0 0 3 3 − 2 2 6 1 ktr ψ (ξ + λ ) − β 2 1 1 From the previous equations, the number average molecular weight can be calculated using Equation (25) λ1 + ξ1 + µ1 + ψ1 Mn = (25) λ0 + ξ0 + µ0 + ψ0 Notice that the moment equations are not closed since the second moment depends on the third moment. To solve this problem, the well-known expression of Saidel and Katz [35] is used to estimate the third moment in terms of the lower moments.

λ2 2 λ3 = 2λ2λ0 λ1 (26) λ0λ1 − Processes 2020, 8, 432 13 of 27

= ( −2 + ) − − − − 1 1 1 1 1 1 1 + − + + ( + ) − + (24) 3 2 6 2 3 2 6 −( +) From the previous equations, the number average molecular weight can be calculated using Equation (25)

+ + + = (25) + + + Notice that the moment equations are not closed since the second moment depends on the third moment. To solve this problem, the well-known expression of Saidel and Katz [35] is used to estimate the third moment in terms of the lower moments. Processes 2020, 8, 432 13 of 26 = (2 −) (26) The mathematical model formulation yielded a set of ordinary differential equations (ODE’s) that The mathematical model formulation yielded a set of ordinary differential equations (ODE’s) were numerically solved using the FORTRAN programming language with the DDASSL routine for that were numerically solved using the FORTRAN programming language with the DDASSL routine the integration of the equations. Kinetic rate constant values were estimated to match the recorded for the integration of the equations. Kinetic rate constant values were estimated to match the recorded experimental data as explained below. experimental data as explained below. 3.4. Experimental Results 3.4. Experimental Results Since the experiments at the extreme levels of N/I (0 and 1.3) were run by triplicate, it was possible Since the experiments at the extreme levels of N/I (0 and 1.3) were run by triplicate, it was to estimate a pooled variance (σ2) for the relevant responses (one for each temperature), and therefore possible to estimate a pooled variance (σ2) for the relevant responses (one for each temperature), and error bars of 1σ are included in most of the plots. therefore error bars of 1σ are included in most of the plots. Figure3 shows the relative yields of the solid, liquid, and gaseous fractions obtained in the Figure 3 shows the relative yields of the solid, liquid, and gaseous fractions obtained in the pyrolysis experiments using a given N/I ratio or by FRP carried at different temperatures. It can be pyrolysis experiments using a given N/I ratio or by FRP carried at different temperatures. It can be noted that at higher pyrolysis temperatures the liquid fraction has a tendency to increase while the gas noted that at higher pyrolysis temperatures the liquid fraction has a tendency to increase while the and solid fractions diminish. Comparing all the samples, the ones corresponding to the N/I ratio of gas and solid fractions diminish. Comparing all the samples, the ones corresponding to the N/I ratio 1.3 exhibit the higher values of recovered liquid fraction, the maximum value reached being 88% at of 1.3 exhibit the higher values of recovered liquid fraction, the maximum value reached being 88% 420 ◦C, while the lowest values correspond to the FRP samples. These results point out to a favorable at 420 °C, while the lowest values correspond to the FRP samples. These results point out to a effect of the presence of the nitroxide moiety at the end of the polymer chains on the pyrolysis process; favorable effect of the presence of the nitroxide moiety at the end of the polymer chains on the more will be discussed below. All the percentages are referred to the total amount of polymer charged pyrolysis process; more will be discussed below. All the percentages are referred to the total amount at the beginning of the experiments. of polymer charged at the beginning of the experiments.

100 NMP (N/I =1.3) 100 NMP (N/I = 1.1) 90 90 80 80 70 70 60 60 50 50 % wt.

40 % wt. 40 30 30 20 20 10 10 0 0 Solid (%wt.) Liquid (%wt.) Gas (%wt.) Solid (%wt.) Liquid (%wt.) Gas (%wt.)

Processes 2020, 8, 432 (a) (b) 14 of 27

100 NMP (N/I 0.9) 100 FRP 90 90 80 80 70 70

60 60 50 50 % wt. % wt. 40 40 30 30 20 20 10 10 0 0 Solid (%wt.) Liquid (%wt.) Gas (%wt.) Solid (%wt.) Liquid (%wt.) Gas (%wt.)

FigureFigure 3.3. YieldsYields of of recovered recovered fractions fractions of of the the pyrolyzed pyrolyzed samples: samples: (a)N (a/)I N/I= 1.3, = 1.3, (b)N (b/I) =N/I1.1, = (1.1,c)N (/cI) =N/I0.9 =, and0.9, (andd) FRP. (d) FRP.

Figure 4 shows the identified products of the PS pyrolysis in the liquid fraction; their relative amounts are plotted for each PS pyrolysis experiment at three reaction temperatures (390, 420, and 450 °C). The main product in all cases is styrene followed by styrene dimer in second place; an extra category is included denoted as “mixture”, which corresponds to the sum of all the components that could not be identified, but individually amount to less than 1 wt.%. In the case of the FRP samples, stilbene and benzoic acid are also generated and the compounds in the non-identified mixture reach the highest values of all the samples.

NMP (N/I = 1.3) NMP (N/I = 1.1)

45 45 40 40 35 35 30 30 25 25

% wt. 20

% wt. 20 15 15 10 10 5 5 0 0 Mixture Toluene Styrene Benzene Ethyl benzene Ethyl Styrene Dimer Methyl Styrene Methyl Diphenylmethane 390°C 390°C Diphenylpropane

420°C 420°C cyclopropane Diphenyl

450°C Compound 450°C Compound DihydrophenylNaphtalene

(a) (b)

Processes 2020, 8, 432 14 of 27

100 NMP (N/I 0.9) 100 FRP 90 90 80 80 70 70 60 60 50 50 % wt. % wt. 40 40 30 30 20 20 10 10 0 0 Solid (%wt.) Liquid (%wt.) Gas (%wt.) Solid (%wt.) Liquid (%wt.) Gas (%wt.)

ProcessesFigure2020, 3.8, 432Yields of recovered fractions of the pyrolyzed samples: (a) N/I = 1.3, (b) N/I = 1.1, (c) N/I 14= of 26 0.9, and (d) FRP.

Figure4 4 shows shows thethe identifiedidentified productsproducts ofof thethe PSPS pyrolysis pyrolysis inin the the liquid liquid fraction; fraction; theirtheir relativerelative amounts are are plotted plotted for for each each PS PS pyrolysis pyrolysis experime experimentnt at three at three reaction reaction temperatures temperatures (390, (390, 420, 420,and and450 °C). 450 ◦TheC). main The main product product in all in cases all cases is styrene is styrene followed followed by styrene by styrene dimer dimer in second in second place; place; an extra an extracategory category is included is included denoted denoted as “mixture”, as “mixture”, which which corresponds corresponds to the to sum the sumof all of the all components the components that thatcould could not notbe identified, be identified, but butindividu individuallyally amount amount to toless less than than 1 1wt.%. wt.%. In In the the case case of of the the FRP FRP samples, samples, stilbene and benzoic acid are also generated and thethe compounds in the non-identifiednon-identified mixture reach thethe highesthighest valuesvalues ofof allall thethe samples.samples.

NMP (N/I = 1.3) NMP (N/I = 1.1)

45 45 40 40 35 35 30 30 25 25

% wt. 20

% wt. 20 15 15 10 10 5 5 0 0 Mixture Toluene Styrene Benzene Ethyl benzene Ethyl Styrene Dimer Methyl Styrene Methyl Diphenylmethane 390°C 390°C Diphenylpropane

420°C 420°C cyclopropane Diphenyl

450°C Compound 450°C Compound DihydrophenylNaphtalene

Processes 2020, 8, 432 15 of 27 (a) (b)

NMP (N/I=0.9) FRP 45 45 40 40 35 35 30 30 25 25

%wt. 20

%wt. 20 15 15 10 10 5 5 0 0

Mixture Stilbene Styrene Toluene Benzene Benzoic acid Benzoic Ethyl benzene Ethyl Styrene dimer Methyl styrene Methyl Diphenyl propane Diphenyl Diphenyl Methane Diphenyl 390°C

390°C cyclopropane Diphenyl 420°C 420°C Compound Compound naphtalene Dihydrophenyl 450°C 450°C (c) (d)

Figure 4. % wt. yield of the polystyrene depolymerization products in the liquid fraction: (a) N/I = 1.3, Figure 4. % wt. yield of the polystyrene depolymerization products in the liquid fraction: (a)N/I = 1.3, (b) N/I = 1.1, (c) N/I = 0.9, and (d) FRP. (b)N/I = 1.1, (c)N/I = 0.9, and (d) FRP.

FigureFigure 5 showsshows thethe absolute absolute yields yields of styreneof styrene monomer monomer and styreneand styrene dimer dimer of the diof ﬀtheerent different samples samples(with respect (with torespect the total to the load total of PS,load not of only PS, not with on respectly with to respect the liquid to the fraction liquid recovered), fraction recovered), indicating indicatingthat at higher that at reaction higher temperaturesreaction temperatures the styrene the dimer styrene concentration dimer concentration tends to tends lower to for lower the reactionfor the reaction with N/I = 1.3, but it is the opposite for the reaction with N/I = 0.9. The monomer recovery yield exhibits a more complex behavior with temperature. For samples N/I = 1.1 and N/I = 0.9, the monomer recovery yield increases with higher temperatures, while for the N/I = 1.3 and the FRP samples maxima for monomer yield are exhibited. Clearly, not all the effects observed are significant, but in some cases the differences are evident, although generally moderate.

41 22 39 21 37 20 35 19 33 18 31 17 %wt. styrene 29 16 27 %wt. styrene dimer 25 15 23 14 380 400 420 440 460 380 400 420 440 460 Temperature (°C) Temperature (°C) (a) (b)

Figure 5. Absolute yields of (a) styrene and (b) styrene dimer of the analyzed samples.

Comparing all the samples at 450 °C, the highest styrene yield (38.5%) is obtained using N/I = 1.1, and the lowest using N/I = 1.3 (around 33%) or, almost identical, FRP. Interestingly, at 420 °C all the samples with nitroxide show higher styrene yield than the FRP sample.

Processes 2020, 8, 432 15 of 27

NMP (N/I=0.9) FRP 45 45 40 40 35 35 30 30 25 25

%wt. 20

%wt. 20 15 15 10 10 5 5 0 0 Mixture Stilbene Styrene Toluene Benzene Benzoic acid Benzoic Ethyl benzene Ethyl Styrene dimer Methyl styrene Methyl Diphenyl propane Diphenyl Diphenyl Methane Diphenyl 390°C

390°C cyclopropane Diphenyl 420°C 420°C Compound Compound naphtalene Dihydrophenyl 450°C 450°C (c) (d)

Figure 4. % wt. yield of the polystyrene depolymerization products in the liquid fraction: (a) N/I = 1.3, (b) N/I = 1.1, (c) N/I = 0.9, and (d) FRP.

ProcessesFigure2020, 85, 432shows the absolute yields of styrene monomer and styrene dimer of the different15 of 26 samples (with respect to the total load of PS, not only with respect to the liquid fraction recovered), indicatingwith N/I = that1.3, at but higher it is reaction the opposite temperatures for the reaction the styrene with dimer N/I = concentration0.9. The monomer tends to recovery lower for yield the reactionexhibits awith more N/I complex = 1.3, but behavior it is the with opposite temperature. for the Forreaction samples with N /N/II = 1.1= 0.9. and The N/I monomer= 0.9, the monomerrecovery yieldrecovery exhibits yield a increases more complex with higher behavior temperatures, with temperature. while for theFor N samples/I = 1.3 and N/I the = 1.1 FRP and samples N/I = maxima0.9, the monomerfor monomer recovery yield areyield exhibited. increases Clearly, with higher not all thetemperatures, effects observed while are for significant, the N/I = but1.3 inand some the casesFRP samplesthe differences maxima are for evident, monomer although yield are generally exhibited. moderate. Clearly, not all the effects observed are significant, but in some cases the differences are evident, although generally moderate.

41 22 39 21 37 20 35 19 33 18 31 17 %wt. styrene 29 16 27 %wt. styrene dimer 25 15 23 14 380 400 420 440 460 380 400 420 440 460 Temperature (°C) Temperature (°C) (a) (b)

FigureFigure 5. 5. AbsoluteAbsolute yields yields of of ( (aa)) styrene styrene and and ( (bb)) styrene styrene dimer dimer of of the the analyzed analyzed samples. samples.

ComparingComparing all thethe samplessamples atat 450 450◦ C,°C, the the highest highest styrene styrene yield yield (38.5%) (38.5%) is obtainedis obtained using using N/I N/I= 1.1, = 1.1,and and the the lowest lowest using using N/I N/I= 1.3 = 1.3 (around (around 33%) 33%) or, almostor, almost identical, identical, FRP. FRP. Interestingly, Interestingly, at 420 at 420◦C all°C theall theProcessessamples samples 2020 with, 8with, 432 nitroxide nitroxide show show higher higher styrene styrene yield yield than than the FRPthe FRP sample. sample. 16 of 27 Pyrolysis reaction time is also aﬀected by the use of PS samples containing nitroxide. Compared withPyrolysis FRP PS, the reaction pyrolysis time of is these also samples affected results by the in us shortere of PS reactionsamples times containing at comparable nitroxide. temperatures, Compared with FRP PS, the pyrolysis of these samples results in shorter reaction times at comparable except in the case of FRP PS at 450 ◦C (Figure6). temperatures, except in the case of FRP PS at 450 °C (Figure 6).

Time (minutes) 12

8 380 390 400 410 420 430 440 450 460 Temperature (°C)

Figure 6. Reaction time (min) vs. temperature temperature (C).

During the experimentation, special emphasis was put in obtaining reproducible results, and therefore the experimental design included replicate runs. Additionally, by performing replicate experiments, we were able to observe clear qualitative differences in the presence and absence of nitroxide in the samples. For example, for the blank reactions, the pressure increase was more abrupt, and the liquid recovery began in all cases at around 380 °C (during the temperature increase ramp). On the other hand, for the dormant polymer samples, especially those of N/I = 1.3, the pressure increase was smoother and more gradual, and the liquid recovery began before, at temperatures between 370–380 °C. These and other observations, as well as some of the quantitative results, reinforce the conclusion that the pyrolysis of PS samples with and without nitroxide ends behave differently.

3.5. Mathematical Modeling Results The mathematical model developed here was intended as a first approach to qualitatively and quantitatively understand the experimental results observed. At this stage, only main experimental trends, particularly reaction time and the effect of nitroxide chain-ends on reaction time, were taken into account to tune the model parameters, since not much detailed information, such as time evolution of the molecular weight distribution or of the product distribution (in terms of individual chain-lengths), was available. Ongoing work in our group is directed towards getting more detailed experimental information as well as more detailed mathematical models. With respect to the kinetic parameters, values are reported in the literature for some of them. Table 4 summarizes some rate constants for PS depolymerization available from the literature that were estimated in different reaction systems.

Processes 2020, 8, 432 16 of 26

During the experimentation, special emphasis was put in obtaining reproducible results, and therefore the experimental design included replicate runs. Additionally, by performing replicate experiments, we were able to observe clear qualitative diﬀerences in the presence and absence of nitroxide in the samples. For example, for the blank reactions, the pressure increase was more abrupt, and the liquid recovery began in all cases at around 380 ◦C (during the temperature increase ramp). On the other hand, for the dormant polymer samples, especially those of N/I = 1.3, the pressure increase was smoother and more gradual, and the liquid recovery began before, at temperatures between 370–380 ◦C. These and other observations, as well as some of the quantitative results, reinforce the conclusion that the pyrolysis of PS samples with and without nitroxide ends behave diﬀerently.

3.5. Mathematical Modeling Results The mathematical model developed here was intended as a first approach to qualitatively and quantitatively understand the experimental results observed. At this stage, only main experimental trends, particularly reaction time and the effect of nitroxide chain-ends on reaction time, were taken into account to tune the model parameters, since not much detailed information, such as time evolution of the molecular weight distribution or of the product distribution (in terms of individual chain-lengths), was available. Ongoing work in our group is directed towards getting more detailed experimental information as well as more detailed mathematical models. With respect to the kinetic parameters, values are reported in the literature for some of them. Table4 summarizes some rate constants for PS depolymerization available from the literature that were estimated in different reaction systems.

Table 4. Reaction kinetic constants of polystyrene depolymerization from literature sources.

Reference [36][37][38] E b E b E b Processes A a,b A a,b A a,b (kcal/mol) (kcal/mol) (kcal/mol) Chain fission s 1 5.0 1013 63.7 7.98 1015 2.3 1.0 1016 2.3 − × × × Chain fission allyl s 1 5.0 1012 58.6 - - 5.5 1013 2.3 − × × L 7 H- abstraction of end chain mol seg 5.0 10 13.5 - - - - × H-abstraction of mid chain L 5 107 16.5 - - - - mol seg × H-abstraction - - 1.5 108 13.3 2.1 106 12 × × Mid chain β-scission s 1 1.2 1013 27.0 2.62 1012 12.9 4.1 1012 11.4 − × × × End chain β-scission s 1 1.6 1013 25.8 2.62 1012 12.9 4.1 1012 11.4 − × × × L 6 9 11 Radical recombination mol seg 5.0 10 14.0 2.2 10 2.3 1.1 10 2.3 × × × Disproportionation L -- 1.14 108 2.3 5.5 109 2.3 mol seg × × a Units compatible with those shown in first column. b A and E are pre-exponential and activation energy in the corresponding Arrhenius expression for the kinetic constant.

Literature values for the kinetic constants were tested using the mathematical model in terms of moments described by the Equations (5)–(8) and (13)–(24); however, the sets of parameter values available in the literature were either incomplete or corresponded to kinetic models with differences with respect to the one proposed here. When they were tried in the present kinetic model, they yielded inconsistent results in the sense of high stiffness and/or non-convergence of the numerical integration of the differential equations, as well as some results without proper physical meaning.

4. Discussion

4.1. Sensitivity Analysis Given the lack of useful values from the literature, it was decided to perform a sensitivity analysis to ﬁt a consistent set of kinetic parameter values of the model. The results of the sensitivity analysis are summarized in Figures7–11, which show essentially the sensitivity of the pyrolysis model in terms of Processes 2020, 8, 432 17 of 26

the evolution of the polymer Mn and the monomer generation with the time of reaction upon varying different reaction coefficients. The base set of used kinetic constants at 350 ◦C in all cases is listed in Table5. In the value of M n plotted the definition given by Equation (25) was used, which represents a lumped value of average molecular weight that includes all the present polymer populations (naturally, the terms corresponding to dormant polymer and live polymer with a nitroxide end are zero for the dead polymer case). The criterion used to choose the set of parameters of Table5, among other viable sets, was that they produced qualitative results that were consistent with the experimental observations and also that their values fell in the vicinity of similar parameters reported in the literature when available (same order of magnitude). Clearly, the proper selection of kinetic parameters for this system is a difficult and still unsolved problem which requires further investigation. Ideally, the kinetic parameters should be evaluated by independent experiments as model-free as possible; however, this goal is quite challenging for systems exhibiting the complexity of the present one. The proposed kinetic parameters are assumed as a good starting point for further research, and are valuable to show the feasibility of the mathematical model presented and for parameter sensitivity studies, but it is important to mention that this parameter set is not unique and there may be other sets of values similarly viable and consistent with the experimental observations. The discussion of Figures6–11 is done by groups of plots. In each case analyzed, four plots are presented in each figure: two of them correspond to pyrolysis experiments of dead polymer having no functionality; the other two correspond to the pyrolysis of dormant polymer with a nitroxide functionality at its end. In the first two plots, the evolution (degradation) of Mn of the dead polymer and of the dormant polymer respectively is shown; in the other two plots, the evolution of monomer for the dead and dormant polymer is respectively exhibited.

Table 5. Depolymerization kinetic constants at 350 ◦C.

1 1 Mechanism (L mol− s− Unless Indicated) Variable Value 1 2 End Chain scission (s− ) kbe 8.0 10 × 2 Transfer + β-scission ktrb 1.1 10 1 × 1 Depropagation (s− ) krev 1.1 10− × 1 Termination by combination ktc 5.8 10 × 1 Termination by disproportionation ktd 2 10 1 × 1 Mid chain scission (s− ) kb 1.5 10− × 7 Deactivation of dormant species (at 130 ◦C) kd 4.78 10 * 1 × 3 Activation of dormant species (s ) (at 130 C) ka 3.16 10 * − ◦ × − 9 16,000 13 124,000 1 1 * Arrhenius expressions: k = 5 10 exp ; ka = 4 10 exp ; R in J mol K [39]. d × − RT × − RT − −

Figure7 shows the e ffect of the variation of the value of kbe (end-chain scission) on the pyrolysis of dead and dormant polymer. Unexpectedly, in both cases, a smaller value of this kinetic coefficient results in a faster decomposition rate and in lower Mn at the end of the reaction time. In the following discussion the case of dead polymer will be taken as an illustration for the explanation, but the case of dormant polymer follows a similar pattern. To understand why this behavior is observed, the evolution of moments 0 and 1 of live and dead polymer were investigated and it was found that almost instantaneously most of the dead chains are activated; that is, they go from a dead state to live polymer (produced by the different available initiation mechanisms). There are however differences in the dynamics of this activation depending on the values of kbe. When lower values of this constant are used, the live polymer concentration rises more slowly (and the dead polymer concentration also decreases more slowly), but it reaches a higher maximum and then a higher stationary value, at the same time that the dead polymer concentration is less depleted in this case (higher stationary concentration). In fact, higher remaining concentration of dead polymer in these conditions favors a relatively faster steady formation of live polymer (via chain-end activation) and leads to a higher quasi-stationary h i1/2 kbeλ0 concentration of live polymer (roughly proportional to µ0 , the lower value of kbe is more ≈ kt than compensated by a higher λ0). Finally, an ultimate higher concentration of live polymer leads to Processes 2020, 8, 432 18 of 27

no functionality; the other two correspond to the pyrolysis of dormant polymer with a nitroxide functionality at its end. In the first two plots, the evolution (degradation) of Mn of the dead polymer and of the dormant polymer respectively is shown; in the other two plots, the evolution of monomer for the dead and dormant polymer is respectively exhibited.

Table 5. Depolymerization kinetic constants at 350 °C.

Mechanism (L mol−1s−1 Unless Indicated) Variable Value −1 End Chain scission (s ) 8.0 × 10 Processes 2020, 8, 432 Transfer + β-scission 1.1 × 10 18 of 26 −1 Depropagation (s ) 1.1 × 10 Termination by combination 5.8 × 10 2×10 faster depropagationTermination and higher monomerby disproportionation production. The eﬀect of kbe is more pronounced for the −1 pyrolysis of dead polymerMid than chain for dormantscission (s polymer.) This is expected since1.5 × 10 a large portion of the 4.78 × 10 initial decompositionDeactivation step of the of dormantdormant species polymer (at should 130 °C) occur at the nitroxide end, * making the other −1 Activation of dormant species (s ) (at 130 °C) 3.16 × 10 * contribution less important. , , * Arrhenius expressions: =5×10 exp − ; =4×10 exp − ; R in J mol−1K−1 [39].

80000 kbe - D = 1000 40000 kbe - D = 800 kbe - S = 8000 70000 35000 kbe - D = 600 kbe - S = 3000 kbe - D = 400 60000 30000 kbe - S = 1000 kbe - S = 800 50000 25000

40000 20000 30000 15000

20000 10000

10000 5000

0 0 0 150 300 450 600 750 900 0 150 300 450 600 750 900 Time (sec) Time (sec)

(a) (b) 0.045 0.045 kbe - D = 1000 0.04 kbe - D = 800 0.04 0.035 kbe - D = 600 0.035 kbe - D = 400 0.03 0.03

0.025 0.025

0.02 0.02

Monomer (mol) 0.015 (mol) Monomer 0.015 kbe - S = 8000 kbe - S = 3000 0.01 0.01 kbe - S = 1000 0.005 0.005 kbe - S = 800

0 0 0 150 300 450 600 750 900 0 150 300 450 600 750 900 Time (sec) Time (sec)

Figure 7. Effect of the variation of the chain-end scission coefficient, kbe, value on the pyrolysis Mn Figure 7. Eﬀect of the variation of the chain-end scission coeﬃcient, kbe, value on the pyrolysis Mn evolution of (a) dead polymer and (b) dormant polymer, and on the monomer generation of (c) dead evolution of (a) dead polymer and (b) dormant polymer, and on the monomer generation of (c) dead polymer and (d) dormant polymer. polymer and (d) dormant polymer.

Figure8 shows the sensitivity of the M n evolution when the value of the transfer to polymer + β-scission coefficient, ktrb, is changed. As this value increases, the polymer decomposes faster, but the behavior shows significant quantitative differences for the dead and the dormant polymer. In both cases the value of Mn goes through a minimum, then increases to reach a maximum and finally steadily decreases; however, for the dormant polymer cases the initial minima are much more pronounced and the subsequent maxima are less marked, indicating a more effective decomposition in the case of dormant polymer. This is confirmed by the plots of monomer evolution that grows faster and reaches higher values (roughly double) for the case of dormant polymer pyrolysis compared to that of dead polymer. Processes 2020, 8, 432 19 of 27

Figure 7 shows the effect of the variation of the value of kbe (end-chain scission) on the pyrolysis of dead and dormant polymer. Unexpectedly, in both cases, a smaller value of this kinetic coefficient results in a faster decomposition rate and in lower Mn at the end of the reaction time. In the following discussion the case of dead polymer will be taken as an illustration for the explanation, but the case of dormant polymer follows a similar pattern. To understand why this behavior is observed, the evolution of moments 0 and 1 of live and dead polymer were investigated and it was found that almost instantaneously most of the dead chains are activated; that is, they go from a dead state to live polymer (produced by the different available initiation mechanisms). There are however differences in the dynamics of this activation depending on the values of kbe. When lower values of this constant are used, the live polymer concentration rises more slowly (and the dead polymer concentration also decreases more slowly), but it reaches a higher maximum and then a higher stationary value, at the same time that the dead polymer concentration is less depleted in this case (higher stationary concentration). In fact, higher remaining concentration of dead polymer in these conditions favors a relatively faster steady formation of live polymer (via chain-end activation) and leads to a higher / quasi-stationary concentration of live polymer (roughly proportional to ≈ , the lower value of is more than compensated by a higher ). Finally, an ultimate higher concentration of live polymer leads to faster depropagation and higher monomer production. The effect of kbe is more pronouncedProcesses 2020, 8 ,for 432 the pyrolysis of dead polymer than for dormant polymer. This is expected since19 of 26a large portion of the initial decomposition step of the dormant polymer should occur at the nitroxide end, making the other contribution less important. Figure9 shows the e ffect of the termination by combination value (ktc) over the Mn evolution. Figure 8 shows the sensitivity of the Mn evolution when the value of the transfer to polymer + β- For dead polymer the effect is quite complex. At early stages of the reaction (less than 100 s) Mn values scission coefficient, ktrb, is changed. As this value increases, the polymer decomposes faster, but the reach a maximum, which is higher and is reached faster for higher values of k (an expected output); behavior shows significant quantitative differences for the dead and the dormanttc polymer. In both however, as the activation reactions proceed, the faster dynamics apparently promoted by higher values cases the value of Mn goes through a minimum, then increases to reach a maximum and finally of k accelerate the polymer degradation leading faster to lower values of M . These shorter chains steadilytc decreases; however, for the dormant polymer cases the initial miniman are much more imply larger number of dead chain-ends that can undergo end-scission, accelerating the degradation of pronounced and the subsequent maxima are less marked, indicating a more effective decomposition the polymer and the monomer production. Similar effects are present in the case of dormant polymer, in the case of dormant polymer. This is confirmed by the plots of monomer evolution that grows but they are significantly attenuated by the competing mechanism of deactivation/degradation at the faster and reaches higher values (roughly double) for the case of dormant polymer pyrolysis nitroxide end, which tends to dominate, especially at lower values of k . compared to that of dead polymer. tc

30000 70000 ktrb - D = 150 ktrb - D = 110 ktrb - S = 150 ktrb - D = 50 25000 ktrb - S = 110 ktrb - D = 20 60000 ktrb - S = 50 20000 ktrb - S =20 50000 15000

10000 40000 5000

30000 0 0 150 300 450 600 750 900 0 150 300 450 600 750 900 Time (sec) Time (sec) Processes 2020, 8, 432 (a) (b) 20 of 27

0.025 0.05 ktrb - S = 150 ktrb - D = 150 0.045 ktrb - S = 110 ktrb - D = 110 ktrb - S = 50 0.02 0.04 ktrb - D = 20 ktrb - S =20 ktrb - D = 50 0.035 0.015 0.03 0.025 0.01 0.02 Monomer (mol) Monomer (mol) Monomer 0.015 0.005 0.01 0.005 0 0 0 150 300 450 600 750 900 0 150 300 450 600 750 900 Time (sec) Time (sec) (c) (d)

Figure 8. Effect of the variation of transfer to polymer + β-scission (ktrb) values on the pyrolysis Mn Figure 8. Eﬀect of the variation of transfer to polymer + β-scission (ktrb) values on the pyrolysis Mn evolution of ( (a)) dead dead polymer polymer and and ( (bb)) dormant dormant polymer, polymer, and on the monomer monomer generation generation of of ( (c)) dead dead polymer and ( d) dormant polymer.

FigureRemarkably, 9 shows no the reasonable effect of resultsthe termination can be obtained by combination by using typicalvalue ( valuesktc) over of the this M coen evolution.ﬃcient at 8 1 1 Fortemperatures dead polymer in the the range effect of is 100–200 quite complex.◦C, which At areearly in thestages order of the of 10reactionLmol −(lesss− than[40] 100 and s) much Mn values lower reachvalues a hadmaximum, to be used which in is the higher simulations; and is reached otherwise faster numerical for higher convergence values of ktc for (an the expected solution output); of the however,equations as was the not activation possible. reactions Some researchers proceed, that the havefaster modeled dynamics the apparently thermal pyrolysis promoted of PS by omit higher this valuesreaction of [ 41ktc ]accelerate at all; others, the polymer for example degradation Kruse et al.leading [37,38 faster], use to a standardlower values Arrhenius of Mn. expressionThese shorter for chainsthis constant imply extrapolatinglarger number its valueof dead at thechain-ends high pyrolysis that can temperatures; undergo end-scission, however, they accelerating report to have the degradationused a gel eﬀ ofect the expression polymer modifyingand the monomer the Arrhenius producti expressionon. Similar which effects would are present lower the in valuethe case of koftc dormantby several polymer, orders of magnitude.but they are They significantly do not report attenuated the resulting by rangethe competing of values formechanism this constant, of deactivation/degradation at the nitroxide end, which tends to dominate, especially at lower values of ktc.

(a) (b)

Processes 2020, 8, 432 20 of 27

Figure 8. Effect of the variation of transfer to polymer + β-scission (ktrb) values on the pyrolysis Mn Processes 2020, 8, 432 20 of 26 evolution of (a) dead polymer and (b) dormant polymer, and on the monomer generation of (c) dead polymer and (d) dormant polymer. but they may lie close to the values used in this work. Since it is not clear that diffusion limitations are presentFigure at these 9 shows conditions, the effect it was of preferredthe termination here not by to combination speculate and value use instead(ktc) over a constantthe Mn evolution. value that Forwas dead adequate polymer for thethe numericaleffect is quite solution complex. of the At model early stages and resulted of the reaction in reasonable (less than outputs. 100 s) In M ourn values next reachstudy a this maximum, issue will which be explored is higher further. and is reached faster for higher values of ktc (an expected output); however, as the activation reactions proceed, the faster dynamics apparently promoted by higher In Figure 10 the effect of the reverse propagation value (krev) over the Mn evolution is shown. valuesAs for otherof ktc accelerate parameters, the for polymer example degradation those shown leading in Figures faster7 to and lower8, the values pyrolysis of M processn. These forshorter the chainsdormant imply polymer larger is fasternumber than of fordead the chain-ends dead polymer. that Thecan dormantundergo polymerend-scission, swiftly accelerating reaches avery the degradation of the polymer and the monomer production. Similar effects are present in the case of low value of Mn, which is lower the higher the value of krev, while for dead polymer the value of Mn dormantexhibits a graduallypolymer, decreasingbut they valueare significantly even at the final attenuated time of the by process. the competing The generation mechanism of monomer of deactivation/degradationin both cases is consistent at with the thenitroxide decrease end, of which the molecular tends to weight.dominate, especially at lower values of ktc.

Processes 2020, 8, 432 (a) (b) 21 of 27

Figure 9. Effect of the variation of the termination by combination coefficient, ktc, value on the Figure 9. Eﬀect of the variation of the termination by combination coeﬃcient, ktc, value on the pyrolysis pyrolysis Mn evolution of (a) dead polymer and (b) dormant polymer, and on the monomer Mn evolution of (a) dead polymer and (b) dormant polymer, and on the monomer generation of (c) generationdead polymer of (c and) dead (d) polymer dormant and polymer. (d) dormant polymer.

Remarkably, no reasonable results can be obtained by using typical values of this coefficient at temperatures in the range of 100–200 °C, which are in the order of 108 Lmol−1s−1 [40] and much lower values had to be used in the simulations; otherwise numerical convergence for the solution of the equations was not possible. Some researchers that have modeled the thermal pyrolysis of PS omit this reaction [41] at all; others, for example Kruse et al. [37,38], use a standard Arrhenius expression for this constant extrapolating its value at the high pyrolysis temperatures; however, they report to have used a gel effect expression modifying the Arrhenius expression which would lower the value of ktc by several orders of magnitude. They do not report the resulting range of values for this constant, but they may lie close to the values used in this work. Since it is not clear that diffusion limitations are present at these conditions, it was preferred here not to speculate and use instead a constant value that was adequate for the numerical solution of the model and resulted in reasonable outputs. In our next study this issue will be explored further. In Figure 10 the effect of the reverse propagation value (krev) over the Mn evolution is shown. As for other parameters, for example those shown in Figures 7 and 8, the pyrolysis process for the dormant polymer is faster than for the dead polymer. The dormant polymer swiftly reaches a very low value of Mn, which is lower the higher the value of krev, while for dead polymer the value of Mn exhibits a gradually decreasing value even at the final time of the process. The generation of monomer in both cases is consistent with the decrease of the molecular weight. The sensitivity of the outputs to ktd is not included here since it results in effects closely related to those observed with ktc.

Processes 2020, 8, 432 21 of 26

Processes 2020, 8, 432 22 of 27

60000 40000 krev - S = 0.11 35000 krev - S = 0.8 55000 krev - S = 0.05 30000 krev - S = 0.01 50000 25000

45000 20000 15000 40000 krev - D = 0.11 krev - D = 0.08 10000 35000 krev - D = 0.05 5000 krev - D = 0.01 30000 0 0 150 300 450 600 750 900 0 150 300 450 600 750 900 Time (sec) Time (sec)

(a) (b) 0.025 0.045 krev - S = 0.11 krev - D = 0.11 krev - S = 0.8 krev - D = 0.08 0.04 krev - S = 0.05 0.02 krev - D = 0.05 0.035 krev - S = 0.01 krev - D = 0.01 0.03 0.015 0.025

0.02 0.01

Monomer (mol) Monomer Monomer (mol) 0.015

0.005 0.01 0.005

0 0 0 150 300 450 600 750 900 0 150 300 450 600 750 900 Time (sec) Time (sec)

Figure 10. EffectEﬀect of the variation of the reverse propagationpropagation coecoefficientﬃcient value,value,k krevrev,, onon thethe pyrolysispyrolysis MMnn evolution of ( a) dead polymer and ( b) dormant polymer, and on the monomer generation of (c) dead polymer and (d) dormant polymer.polymer.

TheFigure sensitivity 11 shows of the the effect outputs of tothektd mid-chainis not included random here scission since it value results (k inb) on eff ectsthe closelynumber related average to thosemolecular observed weight with andktc monomer. generation evolution. Mid-chain β-scission has been identified as a key stepFigure in the 11 showsreduction the of eff molecularect of the mid-chainweight of polymers, random scission resulting value in the (k bformation) on the number of an end-chain average molecularpolymer radical weight and and a stable monomer polymer generation with a double evolution. bond Mid-chain at the chainβ-scission end and hasit is beenconsidered identified a major as a keycontribution step in the to reductionthe formation of molecular of styrene weight monomer of polymers, from PS. resulting [42] In the in sensitivity the formation calculations of an end-chain for the polymerdead polymer radical case and athe stable behavior polymer is somewhat with a double complex bond atand the non-linear, chain end andbut itthe is consideredmagnitude a of major the contributioneffects are relatively to the formation small. These of styrenesmall effects, monomer combined from PS.with [42 the] In fact the that sensitivity the Mn value calculations combines for the deadpopulations polymer of caselive theand behavior dead polymer, is somewhat partly complexexplains andthe complexity non-linear, observed. but the magnitude Notice that of the euseffects of arekb values, relatively far small.from those These shown small e ffinects, Figure combined 11, resulted with the in factno convergence that the Mn value of the combines integration the populationsalgorithm; therefore, of live and the dead values polymer, studied partly were explains restricted the to complexity a small range. observed. On the Notice other that hand, the for use the of kdeadb values, polymer far from case, those the monomer shown in generation Figure 11, resultedgrows with in no an convergence increase in the of thevalues integration of kb, as algorithm;expected. therefore,With respect the valuesto the dormant studied were polymer restricted case, to the a smallbehavior range. of OnMn theis simpler, other hand, as this for theoutput dead decreases polymer case,faster the the monomer larger the generation value of k growsb, as it withcould an be increase expected; in theit seems values that of k bthe, as presence expected. of With the activation respect to thereaction dormant at the polymer nitroxide case, end thehelps behavior in removing of M nsmallis simpler, non-linear as this effects. output Consistent decreases with faster this thebehavior, larger the generation of monomer increases with larger values of kb.

Processes 2020, 8, 432 22 of 26

the value of kb, as it could be expected; it seems that the presence of the activation reaction at the nitroxide end helps in removing small non-linear eﬀects. Consistent with this behavior, the generation of monomer increases with larger values of kb. Processes 2020, 8, 432 23 of 27

90000 20000 kb - S = 0.2 80000 18000 kb - S = 0.15 70000 16000 kb - S = 0.1 kb - S = 0.05 14000 60000 12000 50000 10000 40000 8000 30000 6000 kb - D = 0.2 20000 kb - D = 0.15 4000 kb - D = 0.1 10000 2000 kb - D = 0.05 0 0 0 150 300 450 600 750 900 0 150 300 450 600 750 900 Time (sec) Time (sec)

(a) (b) 0.025 kb - D = 0.15 0.045 kb - D = 0.05 0.04 kb - D = 0.1 0.02 kb - D = 0.2 0.035

0.03 0.015 0.025

0.02 0.01 kb - S = 0.2 Monomer (mol)

Monomer (mol) Monomer 0.015 kb - S = 0.15 0.01 0.005 kb - S = 0.1 0.005 kb - S = 0.05 0 0 0 150 300 450 600 750 900 0 150 300 450 600 750 900 Time (sec) Time (sec)

Figure 11. Effect of the variation of the mid-chain random scission coefficient, kb, value on the Figure 11. Eﬀect of the variation of the mid-chain random scission coeﬃcient, kb, value on the pyrolysis pyrolysis Mn evolution of (a) dead polymer and (b) dormant polymer, and on the monomer Mn evolution of (a) dead polymer and (b) dormant polymer, and on the monomer generation of (c) dead polymergeneration and of ((dc) dormantdead polymer polymer. and (d) dormant polymer.

Notice that in all of the previous analyzed figure ﬁgures,s, the values of the constants tested can vary by several orders of magnitude, with no clear indication as to why the curves for the shown values were selected. TheThe generalgeneral criterion criterion used used to showto show curves curves for speciﬁc for specific ranges ranges of values of ofvalues the kinetic of the constants kinetic wasconstants to maximize was to maximize the sensitivity the sensitivity of the model of the to the mode parametersl to the parameters analyzed;that analyzed; is, the curvesthat is, werethe curves more sensitivewere more for sensitive the ranges for the of values ranges shown, of values and shown, for others and for not others shown not the shown sensitivity the sensitivity was very was small very or almostsmall or null. almost null.

4.2. Base Case Simulation As discussed before,before, givengiven thethe didifficultiesﬃculties to to ﬁt fit kinetic kinetic parameters parameters for for this this system system and and the the lack lack of detailedof detailed quantitative quantitative information information from from experiments, experiments, no no attempt attempt was was made made toto comparecompare inin detaildetail the simulations with the experimental data. The goal at this stage was just to obtain agreement of the simulations with gross measurements of reaction progress, specifically the monomer generation and the time of reaction. In this section simulation results for the base case are presented and discussed. In a more detailed study, it would be convenient to gather data of Mn evolution with time, but this is experimentally difficult as it implies sampling in rather short reaction times and separate the polymeric species from the reaction mixture. This will be attempted in a future study. Figures 12 and

Processes 2020, 8, 432 23 of 26 simulations with the experimental data. The goal at this stage was just to obtain agreement of the simulations with gross measurements of reaction progress, specifically the monomer generation and the time of reaction. In this section simulation results for the base case are presented and discussed. In a more detailed study, it would be convenient to gather data of Mn evolution with time, but this is experimentally difficult as it implies sampling in rather short reaction times and separate the polymeric Processesspecies 2020 from, 8, the432 reaction mixture. This will be attempted in a future study. Figures 12 and 1324 show of 27 the outputs of the simulations with the base case. The base case was assumed to have a Mn of 50,000 13 show the outputs of the simulations with the base case. The base case was assumed to have a Mn with dispersity of 1.9 (same as the FRP case in Table1). For the dormant polymer a livingness of 100% of 50,000 with dispersity of 1.9 (same as the FRP case in Table 1). For the dormant polymer a livingness was assumed. As commented above, the goal was to find a set of kinetic parameters that at least of 100% was assumed. As commented above, the goal was to find a set of kinetic parameters that at semi-quantitatively could reproduce the observed experimental trends; in particular, the decrease of M least semi-quantitatively could reproduce the observed experimental trends; in particular, then and the generation of around 30–40 wt.% of monomer. Figure 12 shows the decrease of Mn (a) for both decrease of Mn and the generation of around 30–40 wt.% of monomer. Figure 12 shows the decrease the dead and the dormant polymer cases, as well as the generation of monomer (b). The Mn produced of Mn (a) for both the dead and the dormant polymer cases, as well as the generation of monomer (b). from the degradation of dormant polymer goes down to very low values, while that produced from The Mn produced from the degradation of dormant polymer goes down to very low values, while dead polymer remains relatively high (around 35,000), although, as discussed below, this is mainly live that produced from dead polymer remains relatively high (around 35,000), although, as discussed polymer in relatively low concentration. More monomer is generated in the case of dormant polymer, below, this is mainly live polymer in relatively low concentration. More monomer is generated in the as experimentally observed; out of the initial polymer present, the model predicts the conversion to case of dormant polymer, as experimentally observed; out of the initial polymer present, the model monomer of about 43% (vs. 38.5% experimental) of the total polymer for the pyrolysis of dormant predicts the conversion to monomer of about 43% (vs. 38.5% experimental) of the total polymer for polymer (0.092 initial moles expressed as moles of monomer in the polymeric chain), and of 23% for the pyrolysis of dormant polymer (0.092 initial moles expressed as moles of monomer in the the pyrolysis of dead polymer (vs. 33% experimental). The difference only qualitatively agrees with polymeric chain), and of 23% for the pyrolysis of dead polymer (vs. 33% experimental). The difference the experimental observation; the quantitative deviation can be explained by the oversimplification of only qualitatively agrees with the experimental observation; the quantitative deviation can be the model with respect to the final small molecules formed, such as styrene dimers and trimers, which explained by the oversimplification of the model with respect to the final small molecules formed, are not explicitly taken into account in the model, in particular when they are formed by recombination such as styrene dimers and trimers, which are not explicitly taken into account in the model, in of monomer molecules. particular when they are formed by recombination of monomer molecules.

(a) (b)

Figure 12. (a) Mn evolution, and (b) monomer generation for the pyrolysis of dead and dormant Figure 12. (a)Mn evolution, and (b) monomer generation for the pyrolysis of dead and dormant polymer predicted by the model with the base set of kinetickinetic constantsconstants ofof TableTable5 5..

Figure 13a,b13a,b show the predicted evolution of live and dead polymer concentration (zero-th moment) and of the first first moments for both popula populationstions during during the the pyrolysis pyrolysis of of dead dead polymer, polymer, indicating that that the the model model predicts predicts a a very very rapid rapid activa activationtion of of the the dead dead polymer polymer in inthe the first first instants instants of theof the reaction reaction followed followed by by reverse reverse propagation propagation and and mid-chain mid-chain breaking breaking reactions. reactions. Initially, Initially, the the first first moment of dead polymer isis aroundaround 0.0920.092 mol,mol, butbut thisthis is is not not visible visible in in plot plot (b) (b) because because in in less less than than 1 1 s sthis this polymer polymer is is activated activated and and converted converted intointo livelive (activated)(activated) polymerpolymer which is gradually degraded. degraded. Similarly, plotsplots (c) (c) and and (d) show(d) show the predicted the predicted evolution evolution of the zero-th of the and zero-th first moments, and first respectively, moments, respectively,of the four polymer of the populationsfour polymer existing populations during exis theting pyrolysis during of the dormant pyrolysis polymer. of dormant Although polymer. more Although more complex than in the former case, the overall dynamics of the process are equivalent. Looking at plot (d) all the dormant polymer is rapidly converted into live polymer or live polymer with a nitroxide-end, and these two populations undergo reverse propagation decaying gradually, but faster than in the case of dead polymer pyrolysis. In the process some dead polymer is formed, but in concentrations several orders of magnitude lower than the live polymer.

Processes 2020, 8, 432 24 of 26 complex than in the former case, the overall dynamics of the process are equivalent. Looking at plot (d) all the dormant polymer is rapidly converted into live polymer or live polymer with a nitroxide-end, and these two populations undergo reverse propagation decaying gradually, but faster than in the case of dead polymer pyrolysis. In the process some dead polymer is formed, but in concentrations several orders of magnitude lower than the live polymer. Processes 2020, 8, 432 25 of 27

(a) (b)

Figure 13. Model predictions for (a) zero-th moment evolution and (b) first moment evolution of live Figure 13. Model predictions for (a) zero-th moment evolution and (b) ﬁrst moment evolution of andlive anddead dead polymer polymer for forthe thepyrolysis pyrolysis of ofdead dead PS. PS. Also, Also, model model predictions predictions for for (c (c) )zero-th zero-th moment moment evolutionevolution andand (d(d)) ﬁrst first moment moment evolution evolution of live,of live, dead, dead, dormant, dormant, and live and with live a nitroxide-endwith a nitroxide-end polymer polymerstarting fromstarting dormant from dormant PS. PS.

5.5. Conclusions Conclusions AA simplesimple processprocess for for the the thermal thermal pyrolysis pyrolysis of PS of has PS been has presented. been presented. PS-T (dormant PS-T (dormant PS) undergoes PS) undergoesthermal pyrolysis thermal and pyrolysis monomer and formation monomer more formation effectively more thaneffectively conventional than conventional (dead) PS, presumably (dead) PS, presumablydue to the additional due to the activationadditional producedactivation atproduc the nitroxideed at the endnitroxide at temperatures end at temperatures above the above Tc of the PS, Tsupportingc of PS, supporting the hypothesis the hypothesis of this work. of this With work. the nitroxide-functionalizedWith the nitroxide-functionalized PS the monomer PS the production monomer productionincreases from increases33 to from38.5 ∼ wt.%33 to with ∼38.5 respect wt.% to with the initialrespect polymer, to the initial an effi ciencypolymer, enhancement an efficiency of ∼ ∼ enhancementroughly 15–16%. of roughly A kinetic 15–16%. and mathematical A kinetic and model ma wasthematical developed model to explain was developed the observed to explain differences, the observedand, after differences, some parameter and, fitting,after some the model parameter is capable fitting, of produce the model semi-quantitative is capable of agreementproduce semi- with quantitative agreement with the experimental observations regarding time of reaction and monomer production, starting either from dead or dormant polymer. It is a difficult task to develop a predictive model for polymer pyrolysis due the complex polymer structure, the multiple reactions that take place and the large amount of intermediate compounds with different size and compositions that are formed in this intricate process, but with the help of population balance equations and the method of moments it is in principle possible to track the formation of the pyrolysis products and, in particular, the evolution of monomer.

Processes 2020, 8, 432 25 of 26 the experimental observations regarding time of reaction and monomer production, starting either from dead or dormant polymer. It is a diﬃcult task to develop a predictive model for polymer pyrolysis due the complex polymer structure, the multiple reactions that take place and the large amount of intermediate compounds with diﬀerent size and compositions that are formed in this intricate process, but with the help of population balance equations and the method of moments it is in principle possible to track the formation of the pyrolysis products and, in particular, the evolution of monomer.

Author Contributions: Conceptualization and methodology, E.S.-G., A.O.-Q. and A.M.-A.; software, E.S.-G. and A.O.-Q.; validation, A.M.-A. and A.O.-Q.; formal analysis, E.S.-G.; resources, E.S.-G.; writing—original draft preparation, A.O.-Q. and E.S.-G.; writing—review and editing, E.S.-G. All authors have read and agreed to the published version of the manuscript. Funding: This research was partially funded by Consejo Nacional de Ciencia y Tecnología México (CONACYT), grant numbers 256358 and 278829. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Saldívar-Guerra, E.; Vivaldo-Lima, E. Polymers and polymer types. In Handbook of Polymer Synthesis, Characterization and Processing; Chapter 1; Saldívar-Guerra, E., Vivaldo-Lima, E., Eds.; John Wiley and Sons: Hoboken, NJ, USA, 2013. 2. McCrum, N.G.; Buckley, C.P.; Bucknall, C.B. Principles of Polymer Engineering, 2nd ed.; Chapter 1; Oxford Science Publications: Oxford, UK, 1997. 3. Villalobos, M.A.; Debling, J. Bulk and solution processes. In Handbook of Polymer Synthesis, Characterization and Processing; Chapter 13; Saldívar-Guerra, E., Vivaldo-Lima, E., Eds.; John Wiley and Sons: Hoboken, NJ, USA, 2013. 4. Tullo, A.H. Fighting ocean plastics at the source. Chem. Eng. News 2018, 96, 28–34. 5. Scott, A. Not-so-fantastic plastic. Chem. Eng. News 2018, 96, 16–18. 6. Achilias, D.S.; Andriotis, L.; Koutsidis, I.A.; Louka, D.A.; Nianias, N.P.; Siafaka, P.; Tsagkalias, I.; Tsintzou, G. Recent advances in the chemical recycling of polymers (PP, PS, LDPE, HDPE, PVC, PC, Nylon, PMMA). In Material Recycling Trends and Perspectives; Achilias, D., Ed.; InTechOpen: London, UK, 2012. 7. Al-Salem, S.M.; Antelava, A.; Constantinou, A.; Manos, G.; Dutta, A. A review on thermal and catalytic pyrolysis of Plastic Solid Waste (PSW). J. Environ. Manag. 2017, 197, 177–198. [CrossRef][PubMed] 8. David, C. Thermal degradation of polymers. In Comprehensive Chemical Kinetics; Bamford, C.H., Tipper, C.F.H., Eds.; Elsevier Scientiﬁc Publishing Company: New York, NY, USA, 1957; Volume 14, pp. 1–173. 9. Guyot, A. Recent developments in the thermal degradation of polystyrene—A review. Polym. Degrad. Stab. 1986, 15, 219–235. [CrossRef] 10. Grassie, N.; Kerr, W.W. The thermal depolymerization of polystyrene. Part 1—The reaction mechanism. Trans. Faraday Soc. 1957, 53, 234–239. [CrossRef] 11. Faravelli, T.; Bozzano, G.; Colombo, M.; Ranzi, E.; Dente, M. Kinetic modeling of the thermal degradation of polyethylene and polystyrene mixtures. J. Anal. Appl. Pyrolysis 2003, 70, 761–777. [CrossRef] 12. Malhotra, S.L.; Hesse, J.; Blanchard, L.P. Thermal decomposition of polystyrene. Polymer 1975, 16, 81–93. [CrossRef] 13. Lehrle, R.S.; Peakman, R.E.; Robb, J.C. Pyrolysis-gas-liquid-chromatography utilised for a kinetic study of the mechanisms of initiation and termination in the thermal degradation of polystyrene. Eur. Polym. J. 1982, 18, 517–529. [CrossRef] 14. Madorsky, S.L. Rates of thermal degradation of polystyrene and polyethylene in a vacuum. J. Polym. Sci. 1952, 9, 133–156. [CrossRef] 15. Guaita, M. Thermal degradation of polystyrene. Br. Polym. J. 1986, 18, 226–230. [CrossRef] 16. Odian, G. Principles of Polymerization; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2004. 17. Stevens, M.P. Polymer Chemistry: An Introduction, 3rd ed.; Oxford University Press: New York, NY, USA, 1999. 18. Audisio, G.; Bertini, F. Molecular weight and pyrolysis products distribution of polymers. I. Polystyrene. J. Anal. Appl. Pyrolysis 1992, 24, 61–74. [CrossRef] Processes 2020, 8, 432 26 of 26

19. Beyler, C.L.; Hirschler, M.M. Beyler_Hirschler_SFPE_Handbook_3. In SFPE Handbook of Fire Protection Engineering; National Fire Protection Association: Quincy, MA, USA, 2002; pp. 111–131. 20. Liu, Y.; Qian, J.; Wang, J. Pyrolysis of polystyrene waste in a ﬂuidized-bed reactor to obtain styrene monomer and gasoline fraction. Fuel Process. Technol. 2000, 63, 45–55. [CrossRef] 21. Simard, Y.D.M.; Kamal, M.R.; Cooper, D.G. Thermolysis of Polystyrene. J. Appl. Polym. Sci. 1995, 58, 843–851. [CrossRef] 22. Jellinek, H.H.G. Thermal degradation of polystyrene and polyethylene. Part III. J. Polym. Sci. 2003, 4, 13–36. [CrossRef] 23. Ebert, K.H.; Ederer, H.J.; Schroder, U.K.O.; Hamielec, A.W. On the kinetics and mechanism of thermal degradation of polystyrene. Makromol. Chem. 1982, 183, 1207–1218. [CrossRef] 24. Richards, D.H.; Salter, D.A. Thermal degradation of vinyl polymers II—The synthesis and degradation of polystyrene containing thermally weak bonds. Polymer 1967, 8, 139–152. [CrossRef] 25. Anderson, D.A.; Freeman, E.S. The kinetics of the thermal degradation of polystyrene and polyethylene. J. Polym. Sci. 1961, 54, 253–260. [CrossRef] 26. Wegner, J.; Patat, F. Thermal degradation of polystyrene. J. Polym. Sci. Part C 1970, 31, 121–135. [CrossRef] 27. Paisley, M.A.; Litt, R.D. Monomer Recovery from Polymeric Materials. U.S. Patent 5,326,919, 5 July 1994. 28. Luo, X.; Yie, F.; Tan, W. Method for Retrieving Styrene Monomer from Discarded Polystyrene Scrap through Pyrolytic Reduction. U.S. Patent 5,072,068, 10 December 1991. 29. Lee, S.; Gencer, M.A.; Fullerton, K.L.; Azzam, F.O. Depolymerization Process. U.S. Patent 5,386,055, 31 January 1995. 30. Tippet, J.; Butler, J.; Assef, J.; Ashbaugh, J.; Clark, J.; Duc, M.; Cary, J.-B. Depolymerization of Plastic Materials. U.S. Patent 9,650,313, 15 May 2017. 31. Northemann, A. Recovery of Styrene from Waste Polystyrene. U.S. Patent 5,672,794, 30 September 1997. 32. Yoon, B.T.; Choi, M.J.; Kim, S.B.; Lee, S.B. Method for Preparing Fermentable Sugar from Wood-Based Biomass. U.S. Patent 8,449,725, 24 March 2013. 33. Nicolas, J.; Guillaneuf, Y.; Lefay, C.; Bertin, D.; Gigmes, D.; Charleux, B. Nitroxide-mediated polymerization. Prog. Polym. Sci. 2013, 38, 63–235. [CrossRef] 34. Roland, A.I.; Schmidt-Naake, G. Thermal degradation of polystyrene produced by nitroxide-controlled radical polymerization. J. Anal. Appl. Pyrolysis 2001, 58–59, 143–154. [CrossRef] 35. Saidel, G.M.; Katz, S. Dynamic analysis of branching in radical polymerization. J. Polym. Sci. Part A-2 1968, 6, 1149–1160. [CrossRef] 36. Marongiu, A.; Faravelli, T.; Ranzi, E. Detailed kinetic modeling of the thermal degradation of vinyl polymers. J. Anal. Appl. Pyrolysis 2007, 78, 343–362. [CrossRef] 37. Kruse, T.M.; Woo, O.S.; Broadbelt, L.J. Detailed mechanistic modeling of polymer degradation: Application to polystyrene. Chem. Eng. Sci. 2001, 56, 971–979. [CrossRef] 38. Kruse, T.M.; Woo, O.S.; Wong, H.W.; Khan, S.S.; Broadbelt, L.J. Mechanistic modeling of polymer degradation: A comprehensive study of polystyrene. Macromolecules 2002, 35, 7830–7844. [CrossRef] 39. Lemoine-Nava, R.; Flores-Tlacuahuac, A.; Saldívar-Guerra, E. Optimal operating policies for the nitroxide-mediated radical polymerization of styrene in a semibatch reactor. Ind. Eng. Chem. Res. 2006, 45, 4637–4652. [CrossRef] 40. Beuermann, S.; Buback, M. Rate coeﬃcients of free-radical polymerization deduced from pulsed laser experiments. Prog. Polym. Sci. 2002, 27, 191–254. [CrossRef] 41. Sterling, W.J.; Walline, K.S.; McCoy, B.J. Experimental study of polystyrene thermolysis to moderate conversión. Polym. Degrad. Stabil. 2001, 73, 75–82. [CrossRef] 42. Simha, R.; Wall, L.A.; Blatz, P.J. Depolymerization as a chain reaction. J. Polym. Sci. 2003, 5, 615–632. [CrossRef]

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