Generalized master plots as a straightforward approach for determining the kinetic model: The case of cellulose pyrolysis
Abstract
The thermal degradation of cellulose is a complex reaction and, despite the large amount of work by many investigators during the last decades, the actual understanding of the thermal decomposition kinetics is still very limited. Thus, while several mechanisms have been proposed to describe the process, the real model has not yet been clearly identified. In this paper, a set of experimental curves recorded under different heating schedules, i.e., linear heating rate, isothermal and constant rate thermal analysis (CRTA), has been analyzed using isoconversional and master plots methodology to discriminate the kinetic model followed by the reaction.
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2 GENERALIZED MASTER PLOTS AS A STRAIGHTFORWARD APPROACH 15 FOR DETERMINING THE KINETIC MODEL: THE CASE OF CELLULOSE 16 PYROLYSIS. 17 Pedro E. Sánchez-Jiménez, Luis A. Pérez-Maqueda, Antonio Perejón, José M. 18 Criado 19 Instituto de Ciencia de Materiales de Sevilla, C.S.I.C.-Universidad de Sevilla, C. 20 Américo Vespucio nº49, 41092 Sevilla, Spain 21 Abstract 22 The thermal degradation of cellulose is a complex reaction and, despite the large 23 amount of work by many investigators during the last decades, the actual understanding 24 of the thermal decomposition kinetics is still very limited. Thus, while several 25 mechanisms have been proposed to describe the process, the real model has not yet been 26 clearly identified. In this paper, a set of experimental curves recorded under different 27 heating schedules, i.e, linear heating rate, isothermal and constant rate thermal analysis 28 (CRTA), has been analyzed using isoconversional and master plots methodology to 29 discriminate the kinetic model followed by the reaction. 30 Keywords: Kinetics, Cellulose, Thermal degradation, Random Scission, Master Plots, 31 Pyrolysis 32 33 34 35 Corresponding author. Tel +34954489548 Fax +34954460665 e-mail address: [email protected]
3 1. Introduction 36 A complete kinetic study of a solid state process requires the determination of the 37 kinetic triplet; namely, the activation energy E, the preexponential factor A and the 38 kinetic model f(α). The determination of the latter constitutes, perhaps, the most delicate 39 step, especially when dealing with highly complex reactions such as the degradation of 40 polymers or natural compounds. The knowledge of the kinetic model driving a process 41 provides a valuable insight regarding the reaction mechanism and it is useful for 42 controlling a process, determining optimum processing temperatures or for aging 43 studies. Procedures for determining the kinetic model most usually involve fitting the 44 experimental data, obtained from a set of curves recorded under different heating 45 profiles, into a predetermined set of kinetic models or equations [1]. Some of these 46 models have been proposed considering certain theoretical assumptions regarding the 47 geometry of the system or the reaction driving force while others are merely empirical 48 fitting equations without any underlying physical meaning. However, the simultaneous 49 fitting of experimental data proceeding from several curves to a single function is a 50 moderately complex affair and requires the use of fitting programs. The use of master 51 plots constitutes a simple and straightforward alternative for determining the kinetic 52 model. Master plots are reference theoretical curves independent of the kinetic 53 parameters, E and A, but dependent on the kinetic model [2, 3]. The method consists in 54 transforming the experimental data into an experimental master plot, which is 55 independent of the experimental conditions used for the experiment. Then, it is 56 compared with the theoretical master plots, which are drawn by assuming certain kinetic 57 models and serve as references. Thus, a simple graphical procedure is sufficient for 58 determining the kinetic model followed by a reaction or, at least, the general type of 59
4 model. The main advantage of this approach is the lack of a previous assumption 60 regarding the kinetic model followed by the reaction, thereby preventing errors arising 61 from the fit to inappropriate kinetic models[4, 5]. The only requirements are the 62 previous knowledge of the activation energy and that such parameter is maintained 63 constant along the entire conversion range. Thus, the master plots method is usually 64 employed together with isoconversional kinetic analysis procedures, which serve to 65 evaluate whether the requirements are met [6-8]. 66 The master plots approach might prove especially useful for studying the pyrolysis of 67 natural materials, which degrade through complex reactions and for which finding the 68 correct model has proven difficult. The degradation of cellulose is one of such cases, 69 having attracted a great deal of attention in the last decades [9-17]. The decomposition 70 of cellulose is a complex process involving complicated chemical pathways, mass and 71 heat transfer phenomena and possible intermediates. Several models have been 72 proposed for describing the reaction, including multistep kinetic models comprising 73 several consecutive or competitive processes [18-20], simple first and nth order kinetic 74 laws [12, 21-23], autoaccelerated models such as Avrami-Erofeev or Prout-Tompkins, 75 commonly related to nucleation and growth mechanisms [10, 11, 22, 24, 25] and, 76 recently, chain scission driven mechanisms [26]. The importance of determining the 77 correct kinetic model is illustrated by the wide disparity in activation energies published 78 in the literature [11, 15, 22, 26] which are most probably due to the nature of the 79 sample, experimental errors and an incorrect selection of models. 80 Thus, in this work, the kinetics of cellulose pyrolysis has been studied by a combined 81 isoconversional analysis and master plots procedure, showing that it is a fairly easy to 82
5 implement methodology, which does not require the use of complicated mathematical 83 procedures and yet produce pretty accurate results despite the complexity of the process. 84 85 2. Experimental 86 Commercial microcrystalline cellulose from Aldrich, (product number 435236) was 87 used for performing the study. Thermogravimetric measurements were carried out with 88 a TA instruments Q5000 IR electrobalance (TA Instruments, Crawley, UK) connected 89 to a gas flow system to work in inert atmosphere (150 mL min-1 N 2). Small mass 90 samples (8-10 mg) were employed in order to minimize heat and mass transfer 91 phenomena so that the kinetic parameters obtained are really representative of the 92 forward reaction. Three different heating schedules were used to obtain the 93 experimental data: linear heating rate at 1, 2, 5 and 10 K min-1, isothermal runs at 533 94 and 548 K and CRTA at constant rates of 0.006 and 0.009 min-1. Constant Rate Thermal 95 Analysis (CRTA) consists in controlling the sample temperature in such a way that the 96 reaction rate is maintained at a constant value, previously selected by the user, along the 97 entire process. This way, by selecting a low enough decomposition rate, the mass and 98 heat transfer phenomena occurring during the reaction are minimized. This advantage 99 can prove especially valuable for studying the pyrolysis of cellulose, which seems to be 100 quite susceptible to mass and heat inhomogeneities [9, 15]. As a consequence, the 101 results obtained by CRTA are more representative of the forward reaction than those 102 obtained from conventional methods such as linear heating programs or isotherms [27-103 29]. 104 105
6 3. Theory 106 3.1 Theoretical background 107 The reaction rate, dα/dt, of a solid state reaction can be described by the following 108 equation: 109 fRTEA d t d exp (1), 110 where A is the Arrhenius pre-exponential factor, R is the gas constant, E the activation 111 energy, α the reacted fraction, T is the process temperature and f(α) accounts for the 112 reaction rate dependence on α. The kinetic model f(α) is an algebraic expression which 113 is usually associated with a physical model that describes the kinetics of the solid state 114 reaction. Table 1 lists the functions corresponding to the most commonly used 115 mechanisms found in literature. In this work, the reacted fraction, α, has been expressed 116 with respect to the degradable part of the cellulose, as defined below: 117 f ww ww 0 0 (2), 118 where wo is the initial mass of cellulose, wf the mass of residual char and w the sample 119 mass at an instant t . 120 Eq. (1) is a general expression that describes the relationship among the reaction rate, 121 reacted fraction and temperature independently of the thermal pathway used for 122 recording the experimental data. Thus, experimental data extracted from all runs, 123 whatever the heating profile, can be simultaneously fitted by Eq. (1). 124 125
7 3.2. Isoconversional Analysis 126 Isoconversional methods, also known as “model-free”, are used for determining the 127 activation energy as a function of the reacted fraction without any previous assumption 128 on the kinetic model fitted by the reaction. The Friedman isoconversional method is a 129 widely used differential method that, unlike conventional integral isoconversional 130 methods, provides accurate values of the activation energies even if they were a 131 function of the reacted fraction [6]. Eq. (1) can be written in logarithmic form: 132 RT E Af dt d )(lnln (3) 133 At a constant value of , f( ) would also be constant and Eq. (3) could be written in the 134 form: 135 RT E Const dt d ln (4) 136 The activation energy at a constant value can be determined from the slope of the plot 137 of the left hand side of Eq. (3) against the inverse of the temperature, at constant values 138 of α. 139 140 3.3. Determination of the kinetic model by means of generalized master plots 141 It has been shown that universal master plots, valid for experimental data recorded 142 under any heating profile, can be obtained by the introduction of the generalized time, θ, 143 defined as [2, 3, 30]: 144
8 dt RT E t 0exp (5), 145 that can also be written: 146 RT E e dt d (6), 147 148 that after substitution into Eq. (1) leads to: 149 )( Af d d (7) 150 From Eq. (7) and taking α = 0.5 as a reference it follows: 151 )5.0( )( / / 5.0 f f dd dd (8) 152 Eq. (8) shows that for a given α, the reduced-generalized reaction rate, (dα/dθ)/ 153 (dα/dθ)α=0.5, is equivalent to f(α)/f(0.5) if the proper f(α) is considered. 154 Additionally, the generalized reaction rate can be written as: 155 RT E dt d d dexp (9) 156 Thus, the relationship between the generalized reaction rate and the experimental data 157 can be established: 158
9 )/exp( /exp / / / / 5.0 5.05.0 RTE RTE dtd dtd dd dd (10), 159 being T0.5 the temperature corresponding to α= 0.5. The equation above implies that, for 160 experimental data recorded under non isothermal conditions, the knowledge of the 161 activation energy is required to construct the experimental master plots. Alternatively, 162 in the case of isothermal conditions, the previous knowledge of the activation energy is 163 no longer required and Eq. (10) is simplified to: 164 5.05.0 / / / / dtd dtd dd dd (11) 165 It is possible to deduce the kinetic model the reaction obeys by simply comparing the 166 plot of the generalized reaction rate vs. the reacted fraction, as calculated from the 167 experimental data and Eq. (10), with the plots corresponding to the different theoretical 168 solid state reaction models, as deducted from the fraction f(α)/f(0.5) (Eq. (8)). The plots 169 mastering the kinetic models represented in Table 1 are shown in Fig. 1, including the 170 very recently proposed random scission models [3, 31]. 171 3.4. The random scission kinetic model 172 This model for depolymerisation processes was initially developed by Simha-Wall for 173 the case of polyethylene. The model implies the random cleavage of the bonds, 174 following first order kinetics, along the polymer chains, producing fragments of 175 progressively shorter lengths that are eventually released when their size is small 176 enough to evaporate [32]. However, given that most degradation reactions are studied 177 by thermogravimetry, a relationship between the fraction of bonds broken and the mass 178 lost by volatilization must be established before the model can actually be applied. 179
10 Thus, in a recent work, the model was reformulated to facilitate their use with 180 conventional kinetic methods [31] and the following conversion function was proposed: 181 182 1 )1()1()( L xxLLf (12), 183 184 where x and L are the fraction of bonds broken and the minimum length of the polymer 185 that is not volatile, respectively. Unfortunately, Eq (12) only has a symbolic solution for 186 L=2. The problem can be sorted out by calculating numerically the f() functions for 187 L≠2, just by giving values to both L and x. 188 189 4. Results and Discussion 190 Fig. 2 shows a set of α-T curves corresponding to the thermal degradation of 191 microcrystalline cellulose. Experiments were carried out using three different heating 192 schedules: linear rise in temperature (Fig. 2a), isothermal (Fig. 2b) and constant reaction 193 rate (Fig. 2c). The curves have been constructed from the thermogravimetric 194 experiments using Eq (2) to determine the α values. It should be noted that after the 195 main degradation step, involving the loss of more that 90% of the initial mass, the mass 196 keeps decreasing but at a much reduced rate so it is difficult to set a clear value for wf. 197 Such behaviour is clearly illustrated in Figure 2a where a stable plateau at α=1 is not 198 reached. Thus, a final residue of 8,5% of the initial mass was assumed as wf. The curves 199 recorded under linear rise in temperature show the typical sigmoidal shape of any 200 kinetic curve obtained under this kind of temperature schedule. Isothermal runs were 201
17 Figure Captions 385 Fig. 1 : Generalized master plots corresponding to the different kinetic models in Table 1 386 as constructed from Eq. (8): (a) Random scission models; (b) diffusion controlled 387 models; (c) “n order” models and (d) nucleation and growth models. 388 Fig. 2: Experimental curves (dotted lines) obtained for the thermal decomposition of 389 microcrystalline cellulose under N2 gas flow and the following experimental conditions: 390 (a) linear heating rate of 1, 2, 5 and 10 K min-1; (b) isotherm at 533 and 548 K; and (c) 391 CRTA degradation rate of 0.006 and 0.009 min-1. 392 Fig. 3: Friedman plots resulting from the simultaneous model-free analysis of all the 393 experimental curves included in Fig 2 corresponding to cellulose pyrolysis under different 394 experimental conditions; linear heating rate, isothermal and controlled reaction rate. 395 Fig. 4: Evolution of the activation energy with the conversion obtained using the 396 Friedman isoconversional method according to Eq. (4). 397 Fig. 5: Comparison between theoretical master plots constructed from the ideal kinetic 398 models in Table 1 and the generalized master plots built from experimental curves 399 recorded under (a) linear heating rate, (b) isothermal conditions and (c) controlled reaction 400 rate. 401 402 403 404 405 406 407 408
18 TABLE 1. f( ) kinetic functions for the most widely used kinetic models, including the 409 newly proposed random scission model. 410 Mechanism Symbol f( ) Phase boundary controlled reaction (contracting area) R2 21 )1(2 Phase boundary controlled reaction (contracting volume) R3 32 )1(3 First order law or 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. (AvramiErofeev 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 [31] L2 21 2 Random Scission L>2 [31] Ln Nosymbolicsolution
19 FIGURE 1 411 412 413 Fig. 1 : Generalized master plots corresponding to the different kinetic models in Table 1 414 as constructed from Eq. (8): (a) Random scission models; (b) diffusion controlled 415 models; (c) “n order” models and (d) nucleation and growth models. 416 417 418 419 420 421 422 423
20 FIGURE 2 424 425 426 427
21 Fig. 2: Experimental curves (dotted lines) obtained for the thermal decomposition of 428 microcrystalline cellulose under N2 gas flow and the following experimental conditions: 429 (a) linear heating rate of 1, 2, 5 and 10 K min-1; (b) isotherm at 533 and 548 K; and (c) 430 CRTA degradation rate of 0.006 and 0.009 min-1. 431 432 433 FIGURE 3 434 435 Fig. 3: Friedman plots resulting from the simultaneous model-free analysis of all the 436 experimental curves included in Fig. 2 corresponding to cellulose pyrolysis under different 437 experimental conditions; linear heating rate, isothermal and controlled reaction rate. 438 439 440
22 FIGURE 4 441 442 443 444 445 Fig. 4: Evolution of the activation energy with the conversion obtained using the 446 Friedman isoconversional method according to Eq. (4). 447 448 449 450 451 452
23 FIGURE 5 453 454 455 456
24 Fig. 5: Comparison between theoretical master plots constructed from the ideal kinetic 457 models in Table 1 and the generalized master plots built from experimental curves 458 recorded under (a) linear heating rate, (b) isothermal conditions and (c) controlled reaction 459 rate. 460 461