Generalized kinetic master plots for the thermal degradation of polymers following a random scission mechanism

1  GENERALIZED KINETIC MASTER PLOTS FOR THE THERMAL DEGRADATION OF POLYMERS FOLLOWING A RANDOM SCISSION MECHANISM. Pedro E. Sánchez-Jiménez, Luis A. Pérez-Maqueda, Antonio Perejón and José M. Criado. Instituto de Ciencia de Materiales de Sevilla, C.S.I.C.-Universidad de Sevilla, C. Américo Vespucio nº49, 41092 Sevilla, Spain Abstract In this paper, the f(α) conversion functions for random scission mechanisms have been proposed in order to allow for the construction of generalized master plots suitable for these kind of mechanisms. The master plots have been validated by its application to simulated data and to the thermal degradation of polybutylenterephtalate, polyethylene and polytetrafluoroethylene. Keywords: Kinetics, Master plots, Random scission, Polymer degradation, Mechanisms   Corresponding author. Tel +34954489548 Fax +34954460665 e-mail address: [email protected] 2  1. Introduction The kinetic modelling of solid state reactions keeps raising a broad interest in materials science and engineering. A reliable evaluation of the kinetic parameters that govern a process provides a valuable insight into the mechanism followed by the reaction. A proper kinetic analysis calls for the determination of the kinetic triplet, namely, the activation energy, E; the pre-exponential factor, A; and the kinetic model, f(α). This latter parameter, also known as conversion function, is an algebraic expression that is associated with the physical model that describes the kinetics of a reaction.1-2 A number of different methods have been developed over the years for extracting kinetic information from experimental data, many requiring the experimental data to be obtained under certain experimental conditions.3-12 Thus, isothermal, non isothermal, or the recent combined analysis methods have been proposed. Recently, the so called “model-free” methods, that allow determining the activation energy of a process as a function of the degree of conversion without any previous assumption of the kinetic model, have attained great popularity.12-19 However, they do not directly yield the reaction kinetic model, although methods for its evaluation have been developed with the use of master plots. 20-23 Master plots are reference theoretical curves that depend on the kinetic model but are independent of the kinetic parameters, E and A. Experimental data can easily be transformed into experimental master plots and compared with the theoretical ones determined for the different kinetic models. Kinetic Analysis is equally important in the field of thermal stability of polymers. The development of workable models able to describe the decomposition processes and to determinate materials response to different thermal conditions has been the concern of many authors and extensive work has been produced during last decades.24 However, degradation of polymers is a complex phenomenon and despite the great deal of 3  research performed on this subject, a high controversy still remains. The difficulty of determining the proper kinetic model of polymer degradation reaction has prompted that most of the works found in the literature resort to model-free methods, 19,25-28 or just assume first or “n-order” kinetic models without reporting arguments that support this assumption.29-37 However, a recent work has proven that thermal degradation of polymers do not necessarily take place through first or “n-order” kinetics and other mechanisms such as diffusion or random scission can control the decomposition reaction.38 The use of master plots in thermal degradation of polymers would help to discriminate the kinetic model without the previous assumption of a particular conversion function that does not guarantee the proper description of the degradation mechanism. While a set of master plots applicable to experimental data recorded under any heating profile has already been proposed in this journal,23 they cannot be applied as proposed to random scission kinetic models due to the impossibility to express f(α) as a function of the reacted fraction in a closed form. As random scission is one of the most usual mechanisms in degradation of polymeric materials, 36,39-45 the extension of the master plots in order to cover these situations is of the utmost interest. In this work, the original Simha-Wall equation for depolymerisation processes 46 has been reformulated in such a way that the reaction rate can be directly expressed as a function of f(α), time or temperature. Then, making use of the new equations, the generalized master plots 23 have been extended to random scission mechanisms, The evaluation of the proposed random scission kinetic model and its corresponding master plot has been carried out by simulated and experimental curves alike. The latter were obtained from the thermal degradation of three commercial polymers: polybutylenterephtalate (PBT), polyethylene (PE) and polytetrafluoroethylene (PTFE). 4  2. Proposal of a new kinetic model for random scission mechanisms The reaction rate, dα/dt, can be described by the following equation:     fRTEAkf d t d exp (1), where A is the Arrhenius pre-exponential factor, R is the gas constant, E the activation energy, α the reacted fraction, T is the process temperature and f(α) the kinetic model, which accounts for the reaction rate dependence on α. Table 1 shows the functions corresponding to the most common models in the literature. Decomposition of a polymer by random scission implies a random cleavage of bonds along the polymer chains, producing fragments of progressively shorter length that will eventually evaporate when the size is small enough.24 According to Simha-Wall 46, the cleavage of bonds follows first order kinetics and the following expressions hold true: )1()1( xAexk dt dx RT E   (2)              N LLN xx L1 111 1  (3), where x, N and L are the fraction of bonds broken, the initial degree of polymerization and the minimum length of the polymer that is not volatile, respectively. As L is usually negligible in comparison to N, Eq. (3) can be simplified to:   )1(111 1 Lxx L  (4) 5  Most thermal degradation studies are carried out by thermogravimetry because the mass lost can be easily related to the conversion. However, in the case of random scission mechanisms only the broken bonds that produce fragments small enough would actually evaporate and therefore be detected as mass loss. That problem is solved by Eq. (4), which establishes a relationship between the detected mass loss and the actual reacted fraction in terms of fraction of bonds broken. This relationship is shown graphically in Figure 1, assuming L values ranging from 2 to 8. However, as x cannot be measured by conventional techniques and L is very difficult to obtain experimentally, the application of Eq. (4) has been severely limited. Nevertheless, by differentiating Eq. (4), and incorporating Eq. (2) we get: )1()1()1( 2xkxxLL dt dL   (5) This way, taking into account Eq. (1), we can get from Eq. (5) the conversion function f(α) which is characteristic of a random scission model: 1 )1()1()(   L xxLLf  (6) Many kinetic analysis methods involve the fitting of experimental data to a certain kinetic model. This requires the f(α) functions for the different models to be previously known. Thus, if random scission mechanisms are to be used in this way, f(α) must be determined. However, a symbolic solution can only be reached for L=2. In this latter case, from Eq. (1) and Eq. (6) we obtain: )(2 2/1    k dt d (7) Therefore, f(α) must be: 6  )(2)( 2/1  f (8) Taking into account the relationship between x and α as established in Eq. (4), for any given L and assigning values to α, from Eq. (4) and (6) it is possible to calculate numerically the corresponding f(α) conversion functions, which are plotted against α in Figure 2a. For the sake of comparison, Figure 2 also includes the f(α) conversion functions corresponding to the most common kinetic models in literature: “n order” (2b), diffusion controlled (2c) and nucleation and growth kinetic models (2d). Random scission functions have a characteristic shape which is quite different from the other models. Since the results of a kinetic analysis are heavily dependent on the kinetic model considered, random scission driven reactions could never be adequately described by other models, and in particular by “n-order” models, as it is often done in literature, and doing so will only result in obtaining incorrect kinetic parameters. However, the random scission kinetic model could be described by the modified SestakBerggren expression that was proposed as a fitting equation for the combined kinetic analysis procedure. 42,47 3. Generalized Master Plots In a previous paper the generalized kinetic equations introduced by Ozawa 48 was used for the proposal of universal master plots that were valid for experimental data recorded under any heating profile.23 Thus, if the generalized time is defined as 49: dt RT E t      0exp  (9), 7  where, considering the integral of Eq. (1), it is clear that θ represents the time needed to reach a certain α value at infinite temperature. By differentiating Eq. (9) the following equation can be obtained:       RT E dt dexp  (10) The combination of Eq. (1) and Eq. (10) leads to: )(    Af d d (11), which can also be expressed in the following way:       RT E dt d d dexp    (12), dα/dθ being the generalized reaction rate that, according to Eqs. (1), (11) and (12), represents the reaction rate extrapolated at infinite temperature as previously shown by Ozawa.49 Since the previous knowledge of the activation energy allows for the extrapolation to infinite temperature of experimental data recorded under any heating profile, Eq. (12) should be valid for the analysis of any data, independently of the temperature profile under which they were obtained. From Eq. (11) and taking α = 0.5 as a reference we get:  )5.0( )( / / 5.0 f f dd dd      (13) As f(0.5) is constant for a certain kinetic model, Eq (13) indicates that for a given α, the reduced-generalized reaction rate, (dα/dθ)/(dα/dθ)α=0.5, would be equivalent to f(α)/f(0.5) when the proper f(α) is selected to describe the process. From Eq. (12) and 8  Eq. (13), the relationship between the generalized reaction rate and the experimental data can be established:    )/exp( /exp / / / / 5.0 5.05.0 RTE RTE dtd dtd dd dd       (14), where T0.5 represents the temperature corresponding to α = 0.5. In the case of experimental data obtained under isothermal conditions, the exponential term of the second half in Eq. (14) cancels and the equations becomes:  5.05.0 / / / / dtd dtd dd dd       (15) On the other hand, for experimental data recorded under non isothermal conditions, the previous knowledge of the activation energy is required in order to construct the experimental master plots. By plotting together versus α the generalized reaction rate, as calculated from Eq. (14) (or Eq. (15) for isothermal conditions), and the fraction f(α)/f(0.5), corresponding to different theoretical kinetic models, it is possible to deduce by comparison the kinetic model followed by the process. It must be noted that, according to Eq. (14), for non isothermal experiments a single activation energy value is assumed. Therefore, for this analysis procedure to be valid, the studied process must obey single step kinetics. Here resides the importance of the previous isoconversional analysis, checking that the activation energy does not vary with alpha in a significant way. 9  4. Random Scission Master Plots The reduced-generalized reaction rate for random scission models is derived from Eq. (5) by taking α = 0.5 as a reference:   )1( )1( )1( )1( 5.0 2 5.05.0 2 5.0 xk xk xx xx dtd dtd L L         (16) According to Eq (15), in the case of isothermal conditions, (dα/dθ)/(dα/dθ)α=0.5 and (dα/dt)/ (dα/dt)α=0.5 are equivalent and Eq (16) becomes:   1 5.05.0 1 5.0 )1( )1(     L L xx xx dtd dtd   (17) The generalized master plots for random scission mechanisms can now be constructed numerically by giving values to x and plotting Eq (17) against α. The resulting curves are plotted in Fig 3 for different values of L and compared with the master plots corresponding to the rest of the kinetic models in Table 1. As it can be clearly noticed, random scission master plots could be distinguished easily due to the maximum they show at α values of around 0.275. Diffusion and “n-order” master plots present no maximum while the master plots corresponding to nucleation and growth laws have it at α = 0.4 or higher. 5. Experimental Commercial Polybutyleneterephtalate (Aldrich, product number 435147), polytetrafluoroethylene (Aldrich, product number 182478) and polyethylene (Aldrich, product number 332119, medium density d=0.940) were used in this work. 16  (16)Mamleev,V.;Bourbigot,S.;LeBras,M.;Lefebvre,J.Threemodel‐freemethods forcalculationofactivationenergyinTG.JournalofThermalAnalysisandCalorimetry2004, 78,1009. (17)Opfermann,J.R.;Kaisersberger,E.;Flammersheim,H.J.Model‐freeanalysisof thermoanalyticaldata‐advantagesandlimitations.ThermochimicaActa2002,391,119. (18)Vyazovkin,S.Modificationoftheintegralisoconversionalmethodtoaccount forvariationintheactivationenergy.JournalofComputationalChemistry2001,22,178. (19)Vyazovkin,S.;Sbirrazzuoli,N.Isoconversionalkineticanalysisofthermally stimulatedprocessesinpolymers.MacromolecularRapidCommunications2006,27,1515. (20)Criado,J.M.;Malek,J.;Ortega,A.Applicabilityofthemasterplotsinthe kineticanalysisofnon‐isothermaldata.ThermochimicaActa1989,147,377. (21)Criado,J.M.;Perez‐Maqueda,L.A.;Gotor,F.J.;Malek,J.;Koga,N.Aunified theoryforthekineticanalysisofsolidstatereactionsunderanythermalpathway.Journalof ThermalAnalysisandCalorimetry2003,72,901. (22)PerezMaqueda,L.A.;Ortega,A.;Criado,J.M.Theuseofmasterplotsfor discriminatingthekineticmodelofsolidstatereactionsfromasingleconstant‐ratethermal analysis(CRTA)experiment.ThermochimicaActa1996,277,165. (23)Gotor,F.J.;Criado,J.M.;Malek,J.;Koga,N.Kineticanalysisofsolid‐state reactions:Theuniversalityofmasterplotsforanalyzingisothermalandnonisothermal experiments.JournalofPhysicalChemistryA2000,104,10777. (24)Pielichowski,J.;Njuguna,J.ThermalDegradationofPolimericMaterials:Rapra TechnologyLtd:Shawbury,2005. (25)Habibi,A.;DeWilde,J.Kineticmodelingofthethermaldegradationof methacrylatecopolymersbythermogravimetricmethods.InternationalJournalofChemical ReactorEngineering2007,5,18. (26)Mamleev,V.;Bourbigot,S.;LeBras,M.;Yvon,J.;Lefebvre,J.Model‐free methodforevaluationofactivationenergiesinmodulatedthermogravimetryandanalysisof cellulosedecomposition.ChemicalEngineeringScience2006,61,1276. (27)Vyazovkin,S.;Sbirrazzuoli,N.Isoconversionalanalysisofcalorimetricdataon nonisothermalcrystallizationofapolymermelt.JournalofPhysicalChemistryB2003,107, 882. (28)Vyazovkin,S.;Dranca,I.;Fan,X.W.;Advincula,R.Degradationandrelaxation kineticsofpolystyrene‐Claynanocompositepreparedbysurfaceinitiatedpolymerization. JournalofPhysicalChemistryB2004,108,11672. (29)Gao,Z.M.;Kaneko,T.;Hou,D.Y.;Nakada,M.Kineticsofthermaldegradation ofpoly(methylmethacrylate)studiedwiththeassistanceofthefractionalconversionatthe maximumreactionrate.PolymerDegradationandStability2004,84,399. (30)Paik,P.;Kar,K.K.Thermaldegradationkineticsandestimationoflifetimeof polyethyleneparticles:Effectsofparticlesize.MaterialsChemistryandPhysics2009,113,953. (31)Ceamanos,J.;Mastral,J.F.;Millera,A.;Aldea,M.E.Kineticsofpyrolysisofhigh densitypolyethylene.Comparisonofisothermalanddynamicexperiments.Journalof AnalyticalandAppliedPyrolysis2002,65,93. (32)Inoue,E.;Tsuchiya,M.;Ishimaru,K.;Kojima,T.Thermogravimetricstudieson poly(methylmethacrylate),poly(tetrahydrofuran)andtheirblends.JournalofThermalAnalysis andCalorimetry2002,70,747. (33)Conesa,J.A.;Font,R.Polytetrafluoroethylenedecompositioninairand nitrogen.PolymerEngineeringandScience2001,41,2137. (34)Liu,N.;Chen,H.;Shu,L.;Statheropoulos,M.Errorevaluationofintegral methodsbyconsiderationontheapproximationoftemperatureintegral.JournalofThermal AnalysisandCalorimetry2005,81,99. (35)Gao,Z.M.;Amasaki,I.;Nakada,M.Athermogravimetricstudyonthermal degradationofpolyethylene.JournalofAnalyticalandAppliedPyrolysis2003,67,1. 17  (36)Lu,L.;Yu,H.Y.;Wang,S.F.;Zhang,Y.ThermalDegradationBehaviorof Styrene‐Butadiene‐StyreneTri‐BlockCopolymer/MultiwalledCarbonNanotubesComposites. JournalofAppliedPolymerScience2009,112,524. (37)Garcia,N.;Hoyos,M.;Guzman,J.;Tiemblo,P.Comparingtheeffectof nanofillersasthermalstabilizersinlowdensitypolyethylene.PolymerDegradationand Stability2009,94,39. (38)Sanchez‐Jimenez,P.E.;Perez‐Maqueda,L.A.;Perejon,A.;Criado,J.M. Combinedkineticanalysisofthermaldegradationofpolymericmaterialsunderanythermal pathway.PolymerDegradationandStability2009,94,2079. (39)Choukourov,A.;Grinevich,A.;Polonskyi,O.;Hanus,J.;Kousal,J.;Slavinska,D.; Biederman,H.VacuumThermalDegradationofPoly(ethyleneoxide).JournalofPhysical ChemistryB2009,113,2984. (40)Caceres,C.A.;Canevarolo,S.V.ChainScissionintheThermoMechanical DegradationofPolystyreneunderMultipleExtrusions.Polimeros‐CienciaETecnologia2008, 18,348. (41)Zhou,X.M.ThermalBehaviorofPES/PEESCopolymerswithDifferent SequenceDistribution:ComparativeStudyoftheKineticsofDegradation.JournalofApplied PolymerScience2009,111,833. (42)Sanchez‐Jimenez,P.E.;Perez‐Maqueda,L.A.;Perejon,A.;Criado,J.M.Anew modelforthekineticanalysisofthermaldegradationofpolymersdrivenbyrandomscission. PolymerDegradationandStability2010,95,733. (43)Daraboina,N.;Madras,G.Thermalandphotocatalyticdegradationof poly(methylmethacrylate),poly(butylmethacrylate),andtheircopolymers.Industrial& EngineeringChemistryResearch2008,47,6828. (44)Liu,Y.;Wei,R.Q.;Wei,J.;Liu,X.N.ThermalDegradationofPoly(L‐Lactide) withHighMolecularWeightintoL‐Lactide.ProgressinChemistry2008,20,1588. (45)Wallis,M.D.;Bhatia,S.K.Thermaldegradationofhighdensitypolyethylenein areactiveextruder.PolymerDegradationandStability2007,92,1721. (46)Simha,R.;Wall,L.A.KineticsofchaindepolymerizationJournalofPhysical Chemistry1952,56,707. (47)Perez‐Maqueda,L.A.;Criado,J.M.;Sanchez‐Jimenez,P.E.Combinedkinetic analysisofsolid‐statereactions:Apowerfultoolforthesimultaneousdeterminationofkinetic parametersandthekineticmodelwithoutpreviousassumptionsonthereactionmechanism. JournalofPhysicalChemistryA2006,110,12456. (48)Ozawa,T.AnewmethodforanalyzingthermogravimetricdataBulletinofthe ChemicalSocietyofJapan1965,38,1881. (49)Ozawa,T.NonisothermalkineticsandthegeneralizedtimeThermochimica Acta1986,100,109. (50)Criado,J.M.;Perez‐Maqueda,L.A.;Dianez,M.J.;Sanchez‐Jimenez,P.E. Developmentofauniversalconstantratethermalanalysissystemforbeingusedwithany thermoanalyticalinstrument.JournalofThermalAnalysisandCalorimetry2007,87,297. (51)Alcala,M.D.;Criado,J.M.;Gotor,F.J.;Ortega,A.;PerezMaqueda,L.A.;Real, C.Developmentofanewthermogravimetricsystemforperformingconstantratethermal analysis(CRTA)undercontrolledatmosphereatpressuresrangingfromvacuumto1bar. Thermochim.Acta1994,240,167. (52)Dianez,M.J.;Perez‐Maqueda,L.A.;Criado,J.M.Directuseofthemassoutput ofathermobalanceforcontrollingthereactionrateofsolid‐statereactions.Reviewof ScientificInstruments2004,75,2620. (53)Perez‐Maqueda,L.A.;Criado,J.M.;Gotor,F.J.;Malek,J.Advantagesof combinedkineticanalysisofexperimentaldataobtainedunderanyheatingprofile.Journalof PhysicalChemistryA2002,106,2862.  18  TABLE1.f()kineticfunctionsforthemostwidelyusedkineticmodels,includingthenewly proposedrandomscissionmodel.  Mechanism Symbol f() Phase boundary controlled reaction (contracting area) R2 21 )1(   Phase boundary controlled reaction (contracting volume) R3   32 )1(   Random nucleation followed by an instantaneous growth of nuclei. (Avrami-Erofeev eqn. n =1) F1)1(   Random nucleation and growth of nuclei through different nucleation and nucleus growth models. (Avrami-Erofeev eqn ≠1.) An   n n11 )1ln()1(     Two-dimensional diffusion D2       1ln1  Three-dimensional diffusion (Jander equation) D3         3/1 3/2 112 )1(3    Three-dimensional diffusion (Ginstling-Brounshtein equation) D4    112 3 31     Random Scission L=2 L2       21 2 Random Scission L>2 Ln  Nosymbolicsolution  19   TABLE 2. Activation energy values at different values of conversion and their correlation coefficients, obtained by the Friedman isoconversional analysis of the thermal decomposition of polytetrafluoroethylene (PTFE), polyethylene (PE) and poly(1,4-butylen) terephtalate (PBT).   PFTE  PE PBT α REa (kJmol‐1) α REa (kJmol‐1) α REa (kJmol‐1) 0.10.998299±60.10.997246±80.10.999182±4 0.21.000286±30.20.996246±160.21.000182±4 0.31.000282±40.30.999259±100.31.000180±4 0.40.999286±60.40.998256±130.41.000180±4 0.50.999285±50.50.998255±130.51.000181±3 0.61.000282±40.60.999259±90.61.000182±3 0.70.999280±40.70.998254±110.71.000183±2 0.80.998291±80.80.998255±100.81.000185±2 0.90.998296±90.90.997260±110.90.999195±5 20  Figure Captions Figure 1: Relationship between the actual fraction of bonds broken (x) and the conversion, α, for different random scission kinetic functions, according to Eq. (4). The rightmost curve was plotted assuming L=2 while the leftmost for L=8. Curves assuming L=3 to L=7 lie in-between them. Figure 2: The f(α) conversion functions for: (a) the newly proposed random scission model, plus the different kinetic models most commonly used in literature, (b) diffusion controlled, (c) “n order” and (d) nucleation and growth kinetic models. Figure 3: Generalized master plots corresponding to the different kinetic models in Table 1 as constructed from Eq. (13) and (17). (a) Random scission models; (b) diffusion controlled models; (c) “n order” models and (d) nucleation and growth models. Figure 4. Curves simulated assuming a random scission L2 model, E = 150kJmol-1, A=1011 s-1 and the following heating profiles: (a) linear heating rate of 1, 2 and 5 K min1 and (b) controlled rate of 0.06 min-1. Figure 5. Comparison between the generalized master plots constructed for the different simulated curves included in Figure 4 (symbols) and the master plots corresponding to some of the ideal kinetic models included in table 1 (solid lines). Figure 6. Experimental curves obtained for the thermal decomposition of polytetrafluoroethylene under the following experimental conditions: (a) linear heating rate of 1, 2 and 5 K min-1 and (b) sample controlled degradation rate of 5 10-4 min-1. 21  Figure 7. Comparison between the generalized master plots corresponding to the experimental curves in Fig 6 with the theoretical master plots constructed from the ideal kinetic models in Table 1. Figure 8. Experimental curves obtained for the thermal decomposition of polyethylene under the following experimental conditions: (a) linear heating rate of 1, 2 and 10 K min-1 and (b) sample controlled degradation rate of 1.6 10-4 min-1. Figure 9. Comparison between the generalized master plots corresponding to the experimental curves in Fig 9 with the theoretical master plots constructed from the ideal kinetic models in Table 1. Figure 10. Experimental curves obtained for the thermal decomposition of poly(1,4butylene)terephtalate under the following experimental conditions: (a) linear heating rate of 1, 2 and 5 K min-1 and (b) sample controlled degradation rate of 1.4 10-5 min-1. Figure 11. Comparison between the generalized master plots corresponding to the experimental curves in Fig 10 with the theoretical master plots constructed from the ideal kinetic models in Table 1. 22  Figure 1: Relationship between the actual fraction of bonds broken (x) and the conversion, α, for different random scission kinetic functions, according to Eq. (4). The rightmost curve was plotted assuming L=2 while the leftmost for L=8. Curves assuming L=3 to L=7 lie in-between them. 23  Figure 2: The f(α) conversion functions for: (a) the newly proposed random scission model, plus the different kinetic models most commonly used in literature, (b) diffusion controlled, (c) “n order” and (d) nucleation and growth kinetic models. 24  Figure 3: Generalized master plots corresponding to the different kinetic models in Table 1 as constructed from Eq. (13) and (17). (a) Random scission models; (b) diffusion controlled models; (c) “n order” models and (d) nucleation and growth models. 25  Figure 4. Curves simulated assuming a random scission L2 model, E = 150kJmol-1, A=1011 s -1 and the following heating profiles: (a) linear heating rate of 1, 2 and 5 K min-1 and (b) controlled rate of 0.06 min-1. 32  Figure 11. Comparison between the generalized master plots corresponding to the experimental curves in Fig 10 with the theoretical master plots constructed from the ideal kinetic models in Table 1. 33  TABLE OF CONTENTS IMAGE