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Evaluation of Simha's approach to thermal decomposition of a linear polymer system

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Evaluation of Simha's approach to thermal decomposition of a linear polymer system

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Content EVALUATION OF SIMHA'S APPROACH TQ THERMAL DECOMPOSITION OF A LINEAR POLYMER SYSTEM by Jae-Min Liao A Thesis Presented to the FACULTY OF THE SCHOOL' OF ' ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN CHEMICAL ENGINEERING August 1982 UMI Number: EP41811 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI EP41811 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 This thesis, written by Jae-Min Liao under the guidance o/h i s Faculty Committee and approved by a ll its members, has been presented to and accepted by the School of Engineering in partial fu lfillm e n t of the re quirements fo r the degree of ring Dean Date. Faculty Committee ACKNOWLEDGEMENTS The author is deeply grateful to Professor R, Salovey for his continual guidance and encouragement in this work. He would like to thank Professors W. V. Chang and Y. C, Yortsos for serving on his dissertation committee and providing many helpful suggestions. "Also, he owes a great deal to his friends, Mr. Shie- Ning Wang for his helpful discussions and comments on the manuscript; Mr. and Mrs. Huang for their spiritual encour agement., He sincerely appreciates the Hydril Company for the financial assistance which made this work possible. ii TABLE OF CONTENTS Acknowledgements ii Table of Contents . iii List of F i g u r e s ....................... v List of Tables vi Nomenclature . . . . . . . . . . . ................ viii Abstract 1 1, Introduction . . ...................... ....... 3 2, Theoretical Treatment: Review and Analysis . . . 9 2.1 Random Chain Scission ............. . . 9 2.2 MacCallum’s Work: A Rigid Model in the Depropagation Step ..... ................ 13 2.3 Thermal Decomposition with Four-Step Mechanism 16 3, Preparation of Computer W o r k ...........................24 3.1 Basic Assumptions. Made in this W o r k ............24 3.2 Estimation of Rate Constants...................24 3.3 Procedure for Numerical Calculation ......... 33 4, Results and Discussion . . . . . . . . ............ 36 5, Conclusion ................. 56 iii Appendices References 59 iv LIST QF FIGURES Page Figure la: Figure lb; Figure 2a.: Figure 2b; Molecule with n links is obtained by breaking 2 links in a molecule with nr t links ..................... 0 A molecule with n links is obtained Only by breaking 1 link in a molecule with nr t links ......... .. 0 11 11 The smooth curve, Rgg versus t, drawn through thirteen discrete data points which are listed on Table VII . . 46 The same curve as in Fig. (2a) consisting of many very steep small segments, which are shown as very large dR^/dt in each cycle of the numerical calculation of program A ...... 47 v Table Table Table Table Table Table LIST OF TABLES I The bond dissociation energies of C”H and O C (Kcal/mole), 1 Kcal = 4,184 Joules . . . » ........... . . II Activation energies of depolymeri zation of some vinyl monomers (Kcal mol-1) ..... ..................... III The chain transfer constants for transfer agent, cyclohexane at temperatures 130 PC and 200°C ......... IV Frequency factor, activation energy and rate constant at temperature 400QC for each reaction step ...... V For K * 1012, K. = 107 I mole"1, 7 - 1 K = 10 £'mole , the reaction times 4 and overall radical concentrations at seyeral stages of conversions are calculated by program A with four different TF1s which are 0.25, 0.1, 0.01, and 0.001 ..... .............. VI The dimensionless reaction times and overall radical concentrations are calculated by program A and program B With K = 10+6, Ki - 10+3 1 mole-1 + 13 -1 and K^ = 1 0 i mole respectively, at several- fractions of pn (°)• Results are also calculated by using a random process with = 10“^ g— 1.. ............ Table VIZ The concentrations for B-10/ R50' R^q and R at different conversions are calculated by program A with 14 12 -i K - 10 , K- = 10 £ mole and K = 17 -r 10 £ mole .... .............. 45 Table VIII The molecular weight fractions for eleven selected species and reaction times calculated by program C with K2 = 10+14, K3 = 10+12 £ mole-1 and K.^ = 10+17 £ mole 1 at three differ ent ratios of m^ to m^ when conver sion is 4 % ............................ 49 Table XX The molecular weight fractions for eleven species and reaction times are calculated by program C with K2 = 1014, K3 = 1012 £ mole-1 and = 10+17 £ mole-1 at three different ratios of m2 to m^ when conversion reaches 20% ...... 50 Table X Overall radical concentrations and reaction times calculated by program .8 K. A with k = 10■, a,nd - 1011 £ mole-1 conversions .... 10^ £ mole 1 at'several 53 Table XI The molecular weight fractions for eleven selected species and reaction times calculated by program A and' program c separately with K~ = l0+a, Ko - 10+6 £ mole-1, and 4-11 —1 = 10 £ mole at conversion of 4 % and 25,8% vii NOMENCLATURE Degree of degradation Crs(A,B) Conversion calculated by using random scission mechanism (by program A, program B) c, tr Transfer constant DP Number average of degree of polymerization: kl Rate constant for initiation k2 Rate constant for depropagation k3 Rate constant for chain transfer k4 Rate constant for termination K2 The ratio of to k^ K3 The ratio of k^ to k^ K4 The ratio of k^ to k^ k s Random chain scission rate constant k, , td Rate constant for termination by at ion disproportion- m Mass of a (mole of) monomer no Number (moles) of chain links in time 0 sample at nt Number (moles) of chain links in time t sample at nwf t Weight average chain length viii N . . . Number (moles) of the molecules with n degree n 'f ■ of polymerization (at time t) Ntot Total number of molecules in sample P Polymeric concentration of the species with n degree of polymerization n R Overall radical concentration Rn Radical concentration of the species with degree of polymerization n t The reaction time T Dimensionless reaction time, i.e. T = k^t m ,dRn. 1 v TF Max (-~ • ~ ) • AT n W Weight of the sample W_ . Weight fractions for the molecules with n n, t degree of polymerization at time t Z Kinetic chain length ABSTRACT « In this work, three different approaches to analyze the kinetics of thermal degradation have been reviewed. In ]YlacCallum, s work‘ d, the assumption that the "most probable" molecular size distribution holds during the random scis sion process for a polymer with initial "most probable" molecular size distribution has been verified, as shown in 2 Appendix A. Simha's work assumed a four-step mechanism, and will be discussed in detail due to its generality. His Suggestion that the radical concentration for each species is constant if we neglect the induction period to-be reexam ined here. Three computer programs have been developed to calculate the MWD and the conversion during the reaction with and without the steady state concentration assumption for each radical species. A comparison of results from the different programs has been made. We find that the varia tion of each radical concentration is very limited within a reasonable time interval after the induction period as long as the rate constants are suitably selected. Also, we ffnd that the differences in computer calculated results between above two cases with and without the steady state 1 concentration assumption are very small. 1. Introduction The degradation of polymers is a chemical reaction in which a macromolecule will transform into smaller molecules by breaking covalent bonds under specific circumstances. The factors causing the polymer to degrade can be classi fied as thermal effect, thermal oxidative effect, photo effect, radiative effect, mechanical effect and chemical effect. Generally, the degradation of polymers results from the combined effects of the above types. Due to the fact that a polymer will generally lose its utility when it degrades, it is very important to know how to prevent and avoid degradation in order to prolong the life of the poly mer. In this article, the field of thermal degradation will be considered. Since the late 1940's, the importance of the thermal degradation of polymers was recognized. Around 1950, re searchers attempted to understand the thermal degradation of polymers by observing the weight loss, the production of volatile products and the change in viscosity of the degraded sample. The relationship between the average molecular weight and the conversion became a criterion to 3 judge the extent of random chain scission. Two extreme 3 cases of thermal degradation were proposed by Grassie. 4 The thermal degradation of polymethyl methacrylate (PMMA) ' 5,6,7,8 . , 1 ,__.4,9 , . . 9,10,11,12 ,polyethylene (PE) and polystyrene have been widely investigated during this period. From the middle of 1960, methods used to analyze ther mal degradation have improved a great deal. First, non- thermal thermogravimetry at programmed rates of heating gradually replaced traditional isothermal weight loss mea surements. Second, differential thermal analysis, which is the measurement of the difference in temperature between a substance and a reference material as the temperature of their surrounding - is varied in a controlled manner, is used to detect the onset of thermal degradation. Third, thermal and mass chromatography, which can separate and ’identify the components of a gas phase simultaneously are used in the detection and identification of volatile components evolved from the thermal degradation of polymers. Because of these improved methods, the phenomenon of thermal degra dation has been studied extensively for many polymers. For instance, the thermal degradation of copolymers, poly- 13 14 styrene-acrylonitrile and butadiene-acrylonitrile , have been investigated by Grassie using techniques which include thermal volatilization analysis (TVA), thermogravimetry analysis (TGA), differential scanning calorimetery (DSC), 4 gas chromatography (GC) and infrared (IR) spectrometry. The currently accepted theories of thermal degradation gf polymers are all based on the work of Simha et al. in 1950. Prior to 1950, thermal degradation was discussed in terms of a simple model of random scission. Random chain scission is analogous to a condensation type polymerization which is a stepwise process. Generally it is assumed that the probability of breaking any bond in the main chain is equal. Therefore, it can be predicted that the average mo lecular weight of the sample will decrease rapidly during the early stage of this process. The year, 1950, was the turning point for theoretical research on thermal degradation. At that time, Simha et 2 a,l, proposed a four-step mechanism, involving initiation, depropagation, chain transfer and termination, for thermal decomposition. This provides a more complete and flexible approach than before to account for the phenomena of thermal degradation. Based on the mechanism suggested, two sets of differ- 2 ential equations were derived in Simha's work , one for polymer species and the other for radical species. The complexity of these equations arises from serious coupling effects on the variations of each polymer and radical species with time. Due to this, the. assumption that each 5 radical species attains a steady state was introduced to linearize the differential equations for polymer species. In the set of linearized equations, rate constants can be summarized by two parameters, one is the transfer coeffi cient which is the ratio of the rate of chain transfer re action to that of initiation reaction and the other is related to the kinetic chain length. Then, the change in each polymer species concentration is not dependent on rate constants explicitly. Therefore, corresponding states for the thermal degradation of polymers were established. Further, Simha et al. have solved analytical results nu- 15 merxcally and obtained the important kinetic parameters by fitting the experimental data with a regression computer program. T7 1 R IQ Boyd ' ’ ' extended Simha1s work during the perie od from 1959 to 1967. He used the same mechanism as Simha. However, he removed some limitations from Simha's approach. First, in addition to the disproportionation termination reaction, he also considered the effects of recombination and first order termination reactions. Second, he assumed that the density of the sample is invariant, and then the volume of degraded sample will decrease as the amount of volatile components increases. Third, the initial molecu lar weight distribution (MWD) of the sample can not be monodispersed. Instead, a "most probable" molecular size 6 distribution was applied. The defficiency of Boyd's work is that he assumed that the molecular weight distribution function remains the same as the initial one regardless of the degree of reaction, which is not always true in thermal degradation. Only some special cases were discussed in his work. 20 1 21 McCallum ' ' studied the kinetics of thermal degra dation from another approach different from Simha's. He assumed that once any radical forms, it will unzip Z mono mers before it becomes deactive, where Z is the average un zipped length. Based on the above assumption, together with a simple mass balance equation (eq. 10 in sec. 2), he obtained an expression for the rate of change of number average degree of polymerization and solved it numerically. Details of his approach will be discussed in Section 2 ' ; . - After a comparison of previous theories, Simha's theo ry was chosen for further study. Three computer programs were developed to calculate the MWD and conversion through out the reaction; Program A, which is used to calculate the variation of polymeric and radical concentrations for each species with time simultaneously, is determined without a steady state concentration assumption for each radical species; Program B is written to calculate the change of MWD of the sample with the steady state radical concentra tion for each species during the reaction; Program C is 7 modified from Program A with the assumption of a local steady state radical concentration for each species. Sev eral sets of rate constants have been chosen to test the results of cases with and without the steady state assump tion. This will also be discussed in detail in section 4 ‘ . . 2• Theoretical Treatment: Review and Analysis 2.1 Random Chain Scission The mechanism of random chain scission is shown as follows: ' ks . ktd P , V-* R + R r 1* P + P (1) m+n 1 m n 2 m n i Generally, the reaction rate of the second step is assumed to be much faster than that of the first so that eg. (1) can be rewritten as k P . S-> P + P (2) m+n • m n The other assumptions made to facilitate the analytical treatment of this process are: ii) The polymer sample is initially monodisperse. ii) The probability of breaking any main chain link between two adjacent repeating units is equal. 22 The analytical treatment with a statistical method, is shown as follows: For a molecule with n^ + 1 repeating units initially, the degree of degradation in this process is defined as a, which is the ratio of the average number of main chain Tinks which have been scissioned at time t, n^ - n^, to the 9 original number of main chain links n^, that is n0 ‘ nt a - The rate of chain scission is proportional to the num ber of existing links n^_, i.e., dn_ k n^ (4) dt s t If we rearrange eq. (3) , we get nfc = n^(l - a); Inserting it into eq. (4) and integrating, we obtain -k t a = 1 - e st‘ (5) A polymer molecule of n + 1 chain length can be ob tained from an original molecule with n^ + 1 chain length by the following two ways: i) Two links are broken and n main chain links be tween them remain intact, shown in Fig. la. 2 n The probability of this event existing is a (1 - a) and the number of ways is nQ - n - 1. ii) One link is broken and a smaller molecule with n links intact is formed starting from one end of an original chain, shown in Fig. lb. The probability of this event occurring is a(l - a)n and the number of ways is 2. Therefore, from the above two cases, the total number of polymer species with n links in main chain is 10 tig links r O--o— o— -o-fc-o--------o-)(-o— o— o— o V ' J V n links o : Carbon atoms• — : Link between two adjoint Carbon atoms. X : Location where a link breaks. Fig. 1(a). Molecule with n links is obtained by breaking 2 links in a molecule with n^ links. n^ links c o—-o— o— Q-K-O--------G— o— o— o— o v J n links Fig. 1(b). A molecule with n links is obtained only by breaking 1 link in a molecule with- n^ links. 11 Nn t = N0 a(1 " a)n{(n0 ~ n ~ 1)a + 2} (6-1} % , t - V 1 - “,n° <6-2> The number average chain length is equal to (n^ +?D/(rig -^+3) By multiplying eqns (6.1) and (6.2) with (n+1)/(n^+1)N and 1/Nq respectively, the weight fraction can be obtained as Wn t = (n + l)/(nQ +1) • a (1 - a)n[ (nQ - n)a + 2] (7.1) W = (1 - a) n° (7.2) 0' and the weight average chain length is given by n0 n , = . S w. (i + 1) (8) w, t i=0 i The results shown above were also obtained by solving 23 a set of differential equations in Simha's work. Although only an initially monodisperse system was considered in above work, the results can be applied to initially polydisperse ones by treating it as a specific combination of different monodisperse systems. The general form of molecular size distribution for initial polydisperse system is expressed as follow: 12 N n N n, 0 (1 - a)n + p=n+l p,0 00 y % Z N a(l - a) [(p - n)a + 2] (9) Once the initial molecular size distribution and the reac tion extent of degradation are known, the molecular size distribution at time t can be determined by eq. (9). 2.2. MacCallum's Work: A Rigid Model in the Degradation In MacCallum's work, the basic assumption made is that once any radical forms, it will unzip Z times before it becomes deactive, where Z is the average unzipped length and defined as the ratio of the rate of depropagation to the sum of the rates of chain transfer and termination re actions. The starting point to explore the kinetics of thermal degradation in this work is the following relationship: After differentiating eq. (10) with respect to time t, we obtain Step W = n. 4 tot Df> * m (10) 13 1 ' I V , - . dc? , ^ dNtot ,,,, m 3t Ntot -dF + D? “ at- (11) , * v rJ _i dN ’ . 1 dW _ 1 dDP 1 tot 1* ' w at — at »tot at From eq. (10) or eq. (11), obviously, it can be ob served that only the variations of global properties during the thermal degradation have been considered and that the l . dW ^Ntot uncomplicated expressions for and ■ — must be found dDP first before "d^ ; * ' s deduced. The cases having been dis cussed by J. R. MacCallum are listed as follow: i) The type of initiation reaction is random scis sion and no molecules are lost in unzipping pro- , 20 cess is assumed. ii) The type of initiation reaction is also random scission and that all molecules being scissioned 20 are lost. In addition to above two assumptions, the subcase with the assumption that the initial molecular weight distribution is exponential one 20 has also been discussed. iii) The initiation reaction is chain-end scission and that no molecules will lose in unzipping , 20 process is assumed. iv) The initiation reaction is chain-end scission and 20 that all molecules being scissioned will lose. 14 v) The mechanism of degradation is random scission, which has been discussed in section 2.1. The initial molecular size distribution is the "most probable" one. Two assumptions have been made. One is the molecules whose chain length are equal to or less than L will volatile from de graded system, the other is the "most probable" molecular size distribution is kept during de gradation process.^ iv) Modifying case (v) by adding unzipping-process after random scission occurs.^ With regard to applications of the above results, we can briefly catagorize them as follows: 1) Case (i) and case (iii) are suitably applied to the system in*which all the molecules with a very short average unzipped length have long enough chain lengths. Generally, both of these will occur at the early stage of thermal degradation. 2) Case (ii) and case (iv),in contrary to case (i) and case (iii), are applied to the system in which all the molecules with a very long average unzipped length have short enough molecular chain lengths. 3) Case (v) can be used to simulate the process of random chain scission of the sample with "most probable" molecular size distribution throughout the reaction. 15 4) Case (vi) is more general case than case (v). The depropagation reaction, in which any radical will unzip Z times during its existence, has been considered in this case. 2.3 Thermal Decomposition with Four-Step Mechanism The general mechanism of thermal degradation proposed 2 by Simha et al. in 1950 is: Initiation: kl p;. , . * R. + R. 2 < i rj < N - 2 (1 + 3) 1 3 — J - Depropagation: k2 R -> R . + P, (= M) f n n - 2 2 Chain transfer k3 R + P + P + P + R 4 < x + y <N -2 n x + y n x y — — 2 < x, y Termination k4 Rs + Rt * Ps + Pt t: volatile product In this mechanism, several assumptions are made: i) The probability of chain scission is equal for any link in main chain. 16 ii) Monomer is the only volatile product, this as sumption had been modified after 1950. iii) Intramolecular chain tranfer, i.e., rearrange ment reaction, is neglected, iv) Recombination termination is ignored. According to above mechanism, two sets of differential equations are derived for the concentration of each radical 2 and polymeric species respectively and simplified as follows: dRN - 2//dt kfPN kdRM - 2 N dRN - 3^dt = kf N-lPi “ kdRN. - 3 (12) dV dt = kf nl2 Pi + k2Rn + 2 " kdPn « < n < N - 4 N dV dt = kf | pi + V s " (kd - k2)R3 N dR2/dt = kf | Pi + k2R4~ ^kd ~ V R2 dPN/dt = -(N - 3) (kf - k1)PN dpN _ V t = ~(N - 4) <kf - k] _)PN dPn/dt = -(n - 3) (kf - k1)Pn + (kd - k2)Rn N + k_R P. 3 n+2 l N-2 N dP2/dt = k2 | R± + (kd - k2) R2 + k3R I R± (13) 17 where kf = 2kx + k3E N kd = k2 + k4 + k3 4 (i -3)Pi -Eqs, (12) and (13) divided by k^ can be simplified further aRN-2/aT = KfPH “ KdRN-2 N dRH-3/aT “ Kf „El Pi - V n -3 aRn/dT - Kf J2 p. + KA + 2 - KA 4 < n < N - 4 (14) N - - dR3/dT = Kf |:R + K2E5 - (Kd - K2)R3 aV dT = Kf 4 Pi + K2R4 - (Kd ' K2)R2 where dPM/dT ■= -(N - 3) (Ki - 1)P„ N f N dPN-l/dT = ~(N - 4)(Kf - 1)PN_1 dP /dT = -(n - 3) (K_ - 1)P^ + (K, - K )R (15) N f n d i n N + K-R ^ „ P. 3 < n < N - .2 3 n+2 x — —"v. N-2 N dP2/dT.= K2 I R. + (Kfl - K2)R2 + K3R|P. T = kxt K2 = k2/k1 K3 = k3/k1 K4 = k4/kl 18 3) P . To transform eqs. (14) and (15) into difference equations we obtain AT AT l N-3,T AT 2 n+ 2 ’ n,T n, T AT '2,T AT (16) [ (N - 3) (K AT N,T [ (N - 4) (K 1)P. AT [ (n - 2) (K n,T AT n+2 AT (17) where T + AT 19 Summation of all radical differential equations in (12), / shown m Appendix A, leads ” to“' N dR/dt = 2k1 4 (i - 3)Pi - k4R .(18) At steady state, i.e. dR/dt = 0, eq. (18) becomes N „ 2k -2 (i - 3) p . - k R = 0 1 4 i 4 N k. 2 i.e. | (i - 3)Pi = 2$z~ = K4/-2 * R (19> Inserting (19) into expression, we obtain Kd,s = K2 + K4R + K3K4/2 * r2 (20) For initial mondisperse system, at the early stage of thermal decomposition, dR/dt can be approximately written as; dR/dt = 2k1NPN^0 - k4R2 (21) dR Under the steady state assumption, ^ = 0, R becomes R = (2klA 4 HP ) = (2/K4 NPN,0)!5 (22) To simplify analytical treatment further, by intro ducing the assumption dRn/dt = 0, eq. (15) is rearranged and approximately equal to 20 where dQN-l/dT ~ "'AN-1QN-1 dV dT “ - V n + 2(B0 - K3R/4)Qn+2 (N-n-3) /2 + 2 1 BiQn+2i +2 (23) 4 < n < N - 3 (N-5)/2 dQ2/dT * K2R + 2(B0' - K3R/4)Q4 + 2 J. B / Q ^ . An = (n - 3)(Kf - 1) B. = K [1 - j I1 + j I1+1] + K.R x f 4 4 3 B±'= ~Kf (21-lj/ (1 -I)2[ l - | l 1 + | I1+1] + K2R 1 = K2/Ka,s The above equations have been derived'in Appendix C. To transform eq. (23) into difference equations, we obtain QN-1,T' = QH-1,T'+ ^dQN-l/dT^ t ’ AT Qn,T' = Q„,T + <aV dT)T • AT (24> 4 < n £ H -3 °2,T' = °2,T + <dV dT)T • AT The analytical solution of eq. (23) has the form: 2 21 N-l -A.T Qn(T) = din V n i e 1 '25> 4 _< n N - 1 , i even By introducing two new parameters, kinetic chain length Z and transfer constant C . j . r# eg, (23) has alterna tive form; dQN_l/dT “ *(1 + ctrHN " 4>Qn-1 aQn/4T = -(1 + Ctr) (n - 3)Qn + 2(Bo - ^ > Q n+2 (N-n-3)/2 + 2 ' \ B .Q _ , _ 1 i n+2i+2 where <: (N-5)/2 dQ2/dt = K2B + 2(BQ' - Ctr/4)Q4 + 2 | B±'Q^ + 2i Ctr = K3R (26.1) K2 Z = ---------_---- = i/i _ i (26.2) K4R(1 + -|£) E . = ( C t r + 2 ) t l - i l j f t l i - ± + 1 J + C t r Y = (ctr + 2> (1 - z2>-[1 - 7'zll'1 - l(z7T)1+l1 + ctr i = 0 , 1 , 2 , . . . Eq. (23) is the starting point for a series of Simha's work about thermal decomposition from 1950 to 1969. The 22 work having been done is briefly referred as follows: i) approximate solutions of eqs. (23) and (26) have been developed by neglecting chain transfer effect 2 analytically. ii) a similar result to (i) was obtained by replacing random chain scission initiation with chain-end initiation.^ iii) by using the results from (i) and (ii), the rate of conversion was discussed at two extreme cases, one for small kinetic chain length and the other for large one; and by adjusting the rate con- .stantsy those analytical results were used to match the experimental data of thermal degrada-' 24 tions of PE, PMMA and PS. iv) - ) to avoid the oversimplified situation caused by introducing too many assumptions in analytical method, the similar form as eg. (26) has been 15 solved numerically. v)' thermal decomposition of polystyrene has been Studied by fitting the experimental data with 25 numerical approach. 23 3, Preparation of Computer Simulation. 3.1 Basic Assumptions Made in this Work. i) Thermal decomposition proceeds isothermally. ii) Rate constants are determined by the Arrehnius equation , i.e. k = Ae iii) Rate constants are independent of chain length and activation energy is independent of temper ature, i.v) The volume of sample is unchanged during the reaction, i.e. the density of the sample will decrease as the reaction proceeds. 3.2 Estimation of Rate Constants. 3,2.1 Initiation The first step in the thermal degradation is the dis sociation of a carbon-carbon bond into two radicals. Therefore, the bond dissociation energy plays a very im portant role in determining the rate constant of initiation. Table 1 shows the bond dissociation energies of various C-C and C-H bonds. In a study of the pyrolysis of simple organic compounds, the dissociation of molecules is a 24 Table I. The bond dissociation energies of C-H and C-C (Kcal/mole), 1 Kcal = 4,184 Joules.26 H c. h 3 C2H5 n-C3H7 ch3 104 88,4 C2H5 98 84.5 81.6 n' " C3H7 97.6 84.9 81.4 81.4 n~CAU9 98 84.7 81.4 81.2 unirtiQlecular reaction. The frequency factor, A^, in the 13 rate constant for thrs step is usually of the order of 10 sec""'*' and the activation energy, E^, is nearly equal to the bqnd dissociation energy. As for the chain rupture of polymeric molecules, whether the frequency factor is of the same order or not is still uncertain. Since the vibration al motion of covalent bonds and the subsequent small move ment of radicals in local regions in molten polymers will not differ appreciably from those of small molecules in 27 the liquid state , it seems reasonable to consider that the same relation holds also for polymers. The bond dis sociation energy is strongly dependent on the steric Structure of molecules and substituent groups in the mole cule, Generally, the value of the bond dissociation energy lies in the range of 50-100 Kcal/mole. In this work, we 13 -1 let A^ = 10 sec and lie in the range of 50 to 8 0 Kcal/mole.- respectively. 3,2,2 Depropagation. Depropagation is also a unimolecular reaction. The frequency factor, A .of its rate constant is equally 13 -1 assigned to be of the order of 10 sec . With regards to the activation energy, under equilibrium propagation and depropagation reactions, it can be estimated by summing the activation energy of propagation and the enthalpy of poly merization, in polymerization process. Shown in Table II, 26 Table II. Activation energies of depolymerization of some vinyl monomers (Kcal mol-^)^®. M o n o m e r - A H A E A E ______________________P p_________: Ethylene 21.2 4.4 26 Vinyl acetate 21.2 4.4 25 Propylene 19.5 5.6 25 Methyl acrylate 18.5 7.1 26 Acrylonitrile 18.4 4.1 24.5 Styrene 16.7 7.8 24.5 Butadiene 17.9 9,3 27 Methyl me tha c rylate 13.3 5.8 18 Vinyl Chloride 27 3.7 31 the mean value, 25 Kcal/mole , of activation energy for the depropagation reaction is selected here. 3,2,3 Chain Transfer. For this step of the reaction, it seems impossible to obtain reliable information for the frequency factor and activation energy for any polymer species. Since both are required in order to compute the reaction rate, we have to make reasonable estimates from the limited resources available. The chain transfer reaction k3 R + p .*».p +p + r (3-1) n s+r n s r can be regarded as the overall reaction of k31 Rn + Ps+r Pn + Rs+r (3-2) R , 32+ P + R (3-3) s+r s r According to eq. (3-1), the reaction rate can be expressed as dP S = k_(s + r - 3)R P , (3-4) dt 3 n s+r Alternatively, in view of eqs. (3-2) and (3-3) dP S R„. . (3-5) and dt 3 2 s+r dR = k-v. (s + r - 3)R P ^ - k_„ R (3-6) dt 31 n s+r 32 s+r 28 At steady state dR s+r - k0,(s + r - 3)R P - k_„ R = 0 dt 31 n s+r 32 s+r i.e, k_„ R , = k_- (s + r - 3) R P , /-> -i\ 32 s+r 31 n s+r (3-7) Inserting eg. (3-7)• into eg. (3-5) , we obtain dP~ s - k_T(s + r - 3)R P (3-8) dt 31 n s+r Comparing eg. (3-6) with eg. (3-8), we get k3 = k31 (3-9) Next, we estimate k ^ by using the known value of the chain transfer coefficient C , which is defined as the ratio of s' the rate constant of the hydrogen-abtraction reaction to that of the propagation reaction during polymerization. That is, kha Cs = -jM (3-10) According to the definitions of k31 and for a very long linear polymer molecule with n -CH2- units, k^a is approximately egual to n k ^ by neglecting chain-end effects. Here, we attempt to estimate k ^ from the avail able k^a for cyclohexane. The procedure is shown as follows: 29 The chain transfer constants for the transfer agent, 29 cyclohexane, have been obtained separately at tempera tures 130°C and 200°C, which are shown in Table III. The rate constant of the propagation reaction is calculated by assigning 7 Kcal/mole to the activation energy and 6 " “1 5.x 10 s to the frequency factor. is assumed to be equal to one-sixth of • Thus, k ^ can bo calculated at the temperatures mentioned and both the frequency factor and activation energy may be obtained subsequently. They 5 are A ^ = 8.5 x 10 and = 10.8 Kcal/mole. 3.2,4 Termination. As is well known, for two large polymeric radicals in the molten phase to react, the following three steps are required. i) Translational diffusion: First, the two macro radicals must diffuse together so that it is possible for the two radical chain ends to move into close proximity, ii) Segmental rearrangement: After translational diffusion, the two free-radical sites may still be separated by inert chain segments. In order that the radicals may react they must approach to within the "nearest neighbor" configuration. This process takes place as each chain moves through a variety of conformations by way of 30 Table III. The chain transfer constants for transfer agent, cyclohexane at temperatures 130°C and 200°C. T(°C) 130 200 C (kha/k ' ) 0.0095 0.019 s na. p k (5xl06e"7000/RT) 846 3058 P kha 8.04 58.1 k31 = kha^6 1,34 9,45 31 rotation about the backbone bonds, iii) Chemical reaction; By translational diffusion and segmental rearrangement, the two free radical sites are brought into a position in which chem ical reaction can occur. Usually, the termination reactions of polymer degrada tion are dif fusion-.controlled. This can be deduced from the study of the termination reactions of solution poly merization in which the rate constants are strongly depen dent on the viscosity of the solvent. The diffusion of a macromolecule in the molten phase is more difficult than in a solution. With regard to either translational diffu sion or segmental rotation playing the more important role, this depends on the flexibility of the chemical structure of the chain. Due to the shortage of reliable experimental data and sound theoretical estimates, it is very difficult to pre dict this constant reasonably. The only value 4 - 1 - 1 reported is 4 x 10 mol sec , which was obtained in the work on the photodegradation of PMMA at 167°C by Cowley 30 and Mellville , Here, we assume the activation energy of the termination rate constant is identical to that of the self diffusion constant which lies between 5 and 10 Real/ mole, At a thermal degradation temperature of 400°C, the termination rate constant deduced from above data has the 32 6 8 — 1 — 1 magnitude 10 10 .# mole S 3,2.5 Summary of the Estimation of Rate Constants. The frequency factor, activation energy and the rate constant at a temperature of 400DC for each reaction step * are listed in Table TV, where four different energy levels have been assigned to the activation energy of the initia tion rate constants. 3,3 Procedure for.Numerical Calculation. 1. According to eqs. (16) and (17), program A, shown in Appendix D, has been developed to calculate the radical and polymeric concentrations for each species during the thermal degradation. 2. Assume E^ = 80 Kcal/mole, then choose K2 = 10^, = 10^~* and = 10^ from Table IV, and input them to program A. 3. To speed up the numerical calculation, = 10"*"0, 7 7 = 10 and = 10 are selected to be inputted in pro gram A* 6 4. To simulate the random scission process, = 10 , 3 13 =; 10 and = 10 are the input of program A. 5. According to eq. (24), program B which is shown in Appendix E has been developed to calculate the polymer' concentration for each species during the reaction. Table IV. Frequency factor, activation energy and rate constant at temperature 400°C for each reaction step. initiation depropaga t i'on chain transfer termination A1 E1 kl ' 'A2 E2 k2 A3 E3 k3 k4 i I 1 " 3 10 ^ 80 70 60 50 -io'13 ~10"10 *io-7 ~10~4 io13 25 i : 1 —' * ' O ■ J s . o i —i ■ 11 ~102 io6 ~ io8 units A, , k, = S ^ = S_1 A1 , kx , k„ 2 A3 , k3 A4 ' k4 E = I mole S ^ = I mole ^ S ^ E = Kcal/mole 34 — — — ---------------------------------------------------------------------------------— — 14 6, The set of rate constant ratiosf K2 - 10 , 12 17 = 10 and = 10 are inputted to program A and program B separately in order to compare the result from program A with that from program B. 7V Program C, shown in Appendix F, is obtained by modifying program A by introducing the assumption that the radical concentration for each species will be invariant during a short time interval. To obtain appreciable con version, program A needs to be replaced by program C. 8, Continuing the numerical calculation of step 6 by program C replacing program A, the concentration profiles for polymer species at conversions of 4% and 20% have been obtained. 9. The case in which the ratio of arid is g g 10 /10 /10 is repeated as step 6 and step 8. 35 4, Results and Discussion During the execution of program A, the first problem is to choose the time increment. Although a fixed time in crement is commonly used, there are several deficiencies. First, to maintain the accuracy of calculating results throughout the reaction, the time increment must be chosen small enough so that excessive computer time must be. spent inefficiently to achieve certain extents of reaction. In order to speed the calculation, the accuracy of the results is sacrificed by using larger time intervals. The time in crements are chosen dynamically here: the criteria are that time increments are selected during which the variation of some radical species' concentration that is:the most sensi tive to time can not exceed some fraction of its original concentration. Next, the choice of rate constants also plays a very important role in executing program A. Since the frequency factor and activation energy of each rate constant have been discussed in section 3, we choose them from Table IV to test program A. At the start of the calculation, only one polymer species exists, we specify its concentration (in 36 fact, a, reduced concentration) as unity. For other species, whether polymeric or radical, we.assign a small number - m 8 “15 (say, 10 for polymeric and 10 for radicals) instead Of zero to avoid numerical instability in the calculation. » Due to the tiny radical concentrations and the very steep slopes of R vs. time curves, the time increment is too n Small tp obtain appreciable conversion in a reasonable CPU (Central Processing Unit) time. This essential difficulty is even considered indicative of the wrong choice of rate constants or logic errors in program A initially. To facilitate the execution of program A, a set of rate constant ratios which have a tendency to speed up the conversion is chosen. They are = IO"*"0, = 10^& mole 7 - 1 and = 10 % mole , After the numerical calculation, the result is shown in Table V. The final conversion obtained is about 54%. It is high enough to satisfy us. However, a,t such high conversions, the total radical concentration still keeps increasing. Obviously, the rate of formation of radicals is still greater than the rate of disappear ance, so it cannot be expected to provide evidence that the tptal radical concentration reaches steady state. This sit uation may be met in the thermal degradation of polymers at higher temperatures than that with which we are concerned. Different values of the factor, TF, which is used to deter mine time increments also have been tested in this case. 37 They are 0.25, 0.1, 0,01, and 0,001. Comparisons of re action time (T) and total radical concentration at several conversion levels, which are shown in Table V, have been made at different TF's and there exists only slight dis crepancy among them. Therefore, it is not necessary to reduce TF too much to obtain more precise results, after this, the TF was fixed at 0.01. The second case in which the ratio of K^t and 6 3 ' 13 is 10 /10 /10 is used to test the efficiency of calcula tion in program A and to simulate the random scission pro- '1 '” cess. According to eqs. (22),(26.1) and (26.2), the total steady state radical concentration, transfer constant and unzipped length are calculated as follows: R = 4.472 x 10-6 mole/X C = 0.0045 Z = 0.22 Considering such small values of C^. and Z, th'fs case can be regarded as a pseudo random scission process. The numeri cal result is shown in Table V. The data are collected in the interval during which one tenth of PN (0) has decreased. When P„ decreases to half of its initial concentration, N the numerical calculation is stopped. The final conversion is 0,065% and the CPU time spent is about 12 minutes. The total radical concentration, which is smaller than the 38 Table V For K2 = 10 - K3 = 10? S , mole" , K4 = 10^'% mole" , the reaction times and overall radical concentrations at several stages of conversions are calculated by program A with four different TF's which are 0.25, 0.1, 0.01, and 0.001. . ’*exp. pn (t )/pn (o) ' 0.9. 0.8 " 0.7 0.6 0.5 TF = 0.25 Cal. P t(T)/P <0r Tx.10^ N R x 10 (mole/Jl) C(%) 0,901 3.336 0.637 4.22 0.800 4.904 0.920 8.72 0. 702 6.213 1.140 13.36' 0.601 7.491 1.348 18.40 0.502 8. 778 1.539 23.76 TF = 0.1 Cal. pn (t )/pn (0) T x 1CP . R x 10 (mole/fi,) ' C(%) 0. 901 3. 345 0. 639 4.24 0, 800 4. 900 0.920 8.72 0. 701 6. 221 1.145 13.40 0.602 7.480 1.346 18 .36 0.500 8.80 1.542 23.86 TF = 0.01 Cal, P (T)/p-(0) T x 1064 R x 10 (mole/£) C(%) '0.900 3.347 0. 639 4.24 0,802 4.822 0.916 8.64 0,700 6. 234. 1.147 13.42 0.600 7,496 1.348 18 .42 0.501 8.797 1.541 23.84 TF = 0.001 Cal. P„(T)/P„(0) T x l 06N N R x 10 (mole/Ji) C(%) 0.901 3.343 0.638 4.240 0,800 4.890 0.918 8.692 0.700 6. 232 1.146 13.44 0,602 7.482 1.346 18.36 0.500 8.799 1.542 23.84 *exp. means "expected" U> K O steady state value/ increases initially and subsequently decreases with the reaction. Inserting the data from rows 2 and 3 in Table VI into eqs. (6.2) and (5), af ter iteration by trial and error, the rate constant for random scission, -4 -1 k , is obtained as 10 S Based on this rate constant, s ' the dimensionless reaction times, T, are recalculated and shown in row 6 of Table VI. By using eq. (6.1), the con version or the production of monomers is calculated and listed in row 7 of Table VI. From a comparison of the conversions calculated from these two different approaches, we can deduce that the inherent is slightly smaller than k by observing that C increases faster than Ch in Table s ■ rs A VI. Under the same rate constant ratios, program B is also _ g executed by setting R = 4.37 x 10 mole/it. The result of it is shown in the last two rows in Table VI. The reaction times calculated by program B show little discrepancy from those from program A. However, due to the neglect of the induction period in program B, the conversion calculated by it is much larger than that from program A. The result at high conversions is not expected here, since, the rate of monomerproduction is limited in the early stages of random scission. To further study this case, keeping 14 constant and increasing and greatly, we let = 10 , 8 —1 13 “I = 10 % mole ' and = 10 i mole . Based on our judgement, much higher conversions should be obtained by 40 Table VI The dimensionless reaction times and the overall radical concentrations are calculated by program A and program B with K2 = s = lO4"^, K3 = 10+3 I mole'"-'- and K4 = 10+^-3 & mole"! respectively, at several fractions of PN (0). Results are also calculated by using a random process with ks = lO-^1 exp. pn (T)/pn (Q) 0,9 0.8 0.7 0.6 - 0.5 Cal. pn (t)/pn (0) 0,901 0,802 0,704 0.604 0,500 T& K 103 1.07 2,26 3. 60 5.18 7.11 \ x 106 4.396 4.385 4.374 4 ,361 4.345 CA x 102(%) 0,968 2.054 3.272 4,708 6.48 T . . x 103 rs. 1.05 2.23 3.62 5.18 7.01 c x 102 (%) rS 0.884 1,992 3.368 5.100 7.04 Td x 103 B 1.07 . 2.2 6 3.62 5.16 7.09 Cb (P2 + P3) X 102 (%) 2,56 5.40 ’8.6 6 12 .4 17.2 t± spending the same CPU time as that in the second case. As a matter of fact, the much larger values of and than those in second case have made the slope of the curve versus T become much steeper. Due to this, the order of quasi-steady state time increment for this case is about 6 10 times of that for last case and it is difficult to simulate it by program A. 14 The third set of rate constant ratios, = 10 , = 10"^ % mole - and = 10"^ % mole ^ has been se lected from Table IV by assuming an activation energy for the initiation rate constant of 70 Kcal mole-1. The over all radical concentration, transfer constant and unzipped length are calculated as follow: R = 4.47 x 10"8 mole iT1 Ctr “ 4*47 x 1C)4 Z = 1.006 The value of R is quite reasonable compared with those re ported in polymerization reactions ^P',, which lie between -7 -1 -9 -1 10 mole & and 10 mole & , and the large transfer coefficient means that chain transfer scission is dominant in the scission process. Since this set of rate constant ratios was chosen by a method which is more reliable than before, this case has been investigated in more detail. 42 After the results (dimensionless reaction time and polymeric concentration for each species) at several tiny conversions have been obtained by program A, extensive calculation to higher conversion of the reaction is ex^ pected. Due to the problem of the steep slope of the curve of R versus T, however, it is more difficult to obtain n higher"conversions. On the average, 20 min. CPU time is needed to continue the calculation for an increase of 0.008% conversion. In view of this, it is not expected to obtain much higher conversion. Before program A is modified, a file is created to control the execution of program A. It possessess the property of executing program A batch by batch recursively, the input data for each batch are read from the output file of the last batch, the; conversion increasing 0.008% between two consecutive batch jobs. By. using this method, we proceeded with the calculation of extents of reaction from 0.032% to 0.272% by spending about 15-hour CPU time. The CPU time spent for executing each batch job has no tendency to decrease as the extent of reaction increases. We believe that at this low conversion, the pseudo steady state radical concentration for each species is reached in this case. Therefore, program A should be modified before any appreciable conversion can be obtained. Observing the variations of radical concentra tion for any of R50r R100 anC^ R respect to 43 reaction time, as shown in Table VII,'.we .find '.thaf the . rate of change of any radical concentration is very small. This phenomena is in contradication to that observed in each cycle of numerical calculations by executing program A. The only reason to account for the difference is that the smooth curve (R versus t)7 shown in Fig. 2(a), consists of many very steep small segments, which is shown in Fig. 2(b), in numerical calculation. The steep slopes of these small segments will prohibit the larger time increment. In view Of Fig. 3(a), the variation of radical concentration can be neglected during a period which is much larger than the time increment used in program A. Therefore, to facilitate the calculation of the extent of reaction but not to lose the accuracy of the calculation, program A has been modi- f ied, Program C, shown in Appendix F, is modified from pro gram A. In this program, each radical concentration will be modified for m^ cycles after variations of polymeric concentration for each species are calculated for m ■ cycles by regarding each radical concentration as a constant. Program C is identical to program A when m is adjusted to zero? that is, program A is one extreme case of program C. Therefore, by reducing gradually and letting m^ be a constant, the decomposition behavior simulated by program C will approach that by program A. In executing program C, 44 Table VII The concentrations for Rio, R50/ &90 R at different conyersions are calculated by program A with K2 = 1014, i K3 •= lO^-2 £ mole~i and = 10-^ £ mole-- * - . cA (%) t(sec) Rioxl°10 R50K1010 R90Xl°10 Rx 10 0.032 .38.3 4. 650 4.537 4.328 4.404 0.136 157.6 4.770 4,519 4.137 4,399 0.152 176.0 4.775 4.516 4.109 4.398 0.160 185.1 4.823 4.514 4.094 4 .397 0.168 194.2 4.888 4.513 4. 080 4.397 0.176 202.4 4.776 4.512 4.066 4.397 0.184 212.6 4.665 4.510 4.052 4.396 0.200 230.9 4.748 4.507 4.024 4.395 0.216 249.3 4.840 4.505 3.996 4.394 0.24 8 277.1 4.939 4.498 3.954 4.392 0,256 286.3 4.953 4.497 3.940 4.392 0. 264 295.5 4.966 4.495 3.926 4.392 0,272 304,8 4.979 4.493 3.913 4.391 45 R90x1°10 (mole/£) .3 2 .1 .0 .9 30 60 90 120 150 180 210 240 270 300 t(sec) Fig, 2(a), The smooth curve, R versus t, drawn through thirteen discrete data points which are listed on Table VII. 4* a\ R90X1° 10 (mole/ii) 60 t(sec) Fig. 2(b). The same curve as in Fig. 2(a) consisting of many very steep small segments, which are shown as very large dR^/dt in each cycle of the numerical calculation of program A. we set m^ ~ 100 and - 200 initially, then keep m^ constant and decrease m2« At conversion 4% and 20%, the molecular weight fractions for several selected species and reaction times, shown in Tables VIII and IX respective ly, have been obtained with different ratios of m2 to m^, which are 2, 1, and 1/4 respectively. For a further reduced ratio of m^ to m^, program C has also been executed, but no result is obtained. The reason for this is that the CPU time becomes too large when m2 is less than 25. As we can see from Tables VIII and IX respectively, the results in the first three columns, which are the weight fractions for several selected species, are almost identical for each degree of polymerization? but the reaction time will in crease when m2 decreases. This tells us that the small variation in radical concentration for each species has little effect on the molecular weight fractions, and the apparent reaction-time, which is calculated by program A or program C, is larger than the actural one because of the local fluctuation of radical concentration during the re action. At the same conversions mentioned above, 4% and 20%, with a steady state concentration of each radical species, the molecular weight fractions and the reaction times are calculated by program B, The input data comes from numer ically calculated results at conversion of 0.192% and 4%, 48 Table VIII The molecular weight fractions for eleven selected species and reaction times cal culated by program C with K2 = 10+14, K3 = 10+12 i t mole"1 and K4 = 10+17 i t mole-1 at three different ratios of nu to m, when conversion is 4%. 1 200/100 100/ioo 25/ioo m! = °* n Wf (Pn) 0 , * o 2 4 . 000 4 .000 4.000 4. 000 10 0.608 0. 608 0.607 0. 606 20 0.939 0.939 0.936 0.939 30 1. 080 1,080 1.080 1,090 40 1,110 1.100 1,100 0:il0 50 1,060 1,060 1.050 1,070 60 0.962 0,062 0.962 0.974 70 0. 847 0,847 0.847 0.862 80 0.727 0,726 0,725 0. 742 90 0,599 0. 597 0.597 0.616 100 1.380 0.139 0,142 0.141 t(sec) 4693 4706 4720 4 63 7 RxlO8 •4.34 4.31 4.25 4.396 * The results in this column are calculated from program B. Table IX The: molecular weight fractions for eleven selected species and reaction times are calculated by program C with K£ =10 K3 = 10^ £ mole-- * - and K4 = 1CTI '17£ mole-- * - at three different ratios of m2 to m-^ when conversion reaches 20%. m2/ml 200/ioo 100/100 25/AL00 = 0* t . n Wf (P ) n % 2 20 20 20 20 10 3.417 3.425 3.428 3.580 20 2. 010 2.020 2. 030 2.250 30 0.882 0. 886 0.889 1.050 40 0.339 0.341 0.342 0.430 50 0.120 0.121 0.122 0.163 60 0. 040 0.040 0.041 0.058 70 0.013 0.013 0.013 0.020 80 0.004 0. 004 0.004 0.00 6 90 0. 001 0.001 0.001 0.002 100 0. 028 0.021 0.020 0.0 35 t (sec) 27,724 28,308 29,300 24,920 RxlO8 3.662 3.516 3.400 4 .250 *The results in B this column are calculated from prograir 50 respectively, these results, shown in the last column in Table VIII and IX respectively, are compared to those cal culated by program C. At a lower conversion of 4%, for molecular weight fraction, the discrepancies are not large enough to be detected, the reaction time calculated from program B is slightly shorten than those from program C. This shows that whether or :not the concentration of each radical species can be assumed constant has little effect on the final results at this early stage of reaction; at the higher stage of reaction (20%), based on the original weight of' sample, the weight fraction for each polymeric species obtained by executing program B is slightly greater than that by executing program C. This is because the overall mass of polymeric species for the degraded sample calculated from program B increases during the reaction. The reason for the increase of the overall mass is that the law of conservation of mass does not precisely hold in eq. (26), which forms the main part of program B. As a result, the molecular weight fractions calculated from program B are slightly greater than the actual ones, that is, the actual conversion, shown in Table IX, is less than 20%. Should this case be modified, the results obtained from program C.would be much closer to those from program B. Therefore, after the induction period, a small variation Of radical concentration has no significant influence on 51 the calculation of molecular weight fractions. As for reaction time, the one obtained from program B is shorter than those from program C. The difference between them becomes larger and larger with increase of conversion. The reason for this is that in program B, we kept the overall radical concentration constant, while in fact it will de crease as the conversion increases. As is known, the higher the radical concentration, the faster the reaction will be. Lowering the activation energy of initiation from 70 Kcal/mole to 50 Kca1/mole, the fourth set of rate con- 8 6 —1 i] Stant ratios, K2 = 10 , =10 I mole and = 10 % mole ^ was also selected from Table IV. Due to the significant decrease in the energy barrier for chain scis sion, the radical concentration for each species has in creased large, enough to allow the simulation of thermal decomposition by program A to occur in a reasonable cal culation time. In this case, the highest conversion cal culated by program A is 25.8%. The overall radical concen trations at several selected conversions are shown in Table X. After a conversion of 0.32%, the overall radical con centration decreases, slowly. Therefore, after the induc tion period, the. ''steady state" radical concentration should be modified intermittently to cope with the actual Variation, A similar analysis like the above case has also 52 Table X Overall radical concentrations and reaction times calculated by program A with Kg = 10®, Kg = 10® £' mole--*- and K4 = 10-^ £ rnole’ *^- at several conversions; c (%} 0. 32 4 '10 20 25.8 t(S) 0.37 4.77 12 .7-5 29.36 41.99 R x lO5 4.39 4.18 3 .86 3.34 3. 04 53 been made. The molecular weight fractions for eleven se lected species and the reaction times calculated by using program A and B separately at conversions of 4% and 25.8% are shown in Table XI. In the execution of program B, it is "observed’ ' that the overall mass will increase as the reaction proceeds, and the reaction times is underestimated by assuming a fixed overall radical concentration which is calculated by eq. (22). The distortion of the molecular weight distribution and the shorter reaction times obtained by executing program B are unavoidable. Therefore, in this case, Simha's method provides us only with an approximate approach to describe the thermal degradation of linear polymers. Table XI The molecular weight fractions for eleven selected species and reaction times cal culated by program A and program C sepa rately with Ko = 10+8, K3 = 10+6 I mole , and K4 = 10+-*-~ £ mole-- * - at conversion of 4% and 25.8%. c Wf(P ) % at C = 4% n Wf(P ) % at C = 25.8%' n n Prog. A Prog. B Prog. A Prog. B 2 4.0 4.00 25.8 25.8 10 0.625 0.63 3.72 4.23 20 0.960 0.968 1.40 1.91 30 1.100 1.130 0.393 0.639 40 1.120 1.130 0.048 0.18 7 50 1. 060 1.080 0.022 0. 051 60 0.961 0.978 0.005 0. 013 70 0. 840 0.859 - - 80 0.718 0.734 - - 90 0. 588 0.606 - - 100 13.10 13. 30 - - t(sec) 4.7 4.55 42 30.4 55 5, Conclusion Strictly speaking, it is impossible for a degrading polymer to attain a steady state concentration of each radical species. Too many radical species are involved in the reacting system and some small molecules will leave the system during the reaction'. In this computer simula tion of thermal degradation, a steady state is assumed when the rate of change of the overall radical concentration becomes relatively small, Without the steady state assumption, an appreciable extent of reaction can be calculated by computer only when each radical concentration reaches some steady state value, which is large enough to allow the time increment to be long enough. In the final case discussed in section 4 for example, the magnitude of the overall radical concentration -5 has the order of 10 mole/£ , and it becomes possible to calculate an appreciable extent of reaction by executing program A in a reasonable time. However, in the third case/ due to the high energy barrier for chain scission, the production of radicals is greatly reduced. The overall radical concentration is also correspondingly reduced. 56 Therefore, results at appreciable conversion cannot be obtained by executing program A» For other casesf in which the activation energy of initiation lies between those in the last two cases, the difficulty with which the molecular weight distribution and reaction time are calcu lated by program A will increase as the activation energy of initiation increases. In a comparison of the results calculated by executing the programs with and without the steady state radical con centration assumption for the last two cases, we did not find that any large discrepancy exists. It is believed that after the induction period, the steady state radical concentration assumption can be suitably used in the ther mal degradation system. In this work, the steady state overall'radical concentration is assigned the value at an eariy stage of the reaction or is calculated by eq. (22). Usually it is higher than the actual value in every stage of reaction. As a result of this, the rate of reaction is larger and the reaction time is shorter. This phenomena can be.improved by selecting a steady state overall radical concentration carefully. For a linear polymer system, the ratio of the initia tion to the termination rate constants(k^/k^) will increase as; the reaction temperature is higher. To reach the steady state, the radical concentration for each species must be 57 larger at higher temperatures. This is because the initia tion constant increases faster than the termination con-: stant. Therefore, the radical concentration must be larger to make the termination rate equal to the initiation rate. As we know, larger radical concentrations can be reached at higher conversion of the reaction, and Simha's assumption, which has already been verified in this work, can be ap-. plied correctly for the steady state system. Therefore, if we want to use his approach to simulate the behavior of thermal degradation, we will risk the larger possibility of errors caused by neglecting the induction period at higher temp era tur e. The numerical conclusions of this research are summa rized in Tables V to XI. We conclude that Simha’s method applied to the thermal degradation of polymers involving the characterization of the reaction as a free radical process with initiation, depropagation, chain transfer and termination steps yields reasonable numerical values for MWD and reaction time. 58 APPENDIX A As is well known, the "most probable" molecular size distribution is expressed mathematically by N = t (1 - zi-)11-1 (A-l) ntz DPt DPt where Nn t : T^e num^er of the molecules with degree of polymerization n at time t. N4 - *.’ • The total number of molecules at time t. tot, t n; Degree of polymerization. DP^_; Average degree of polymerization at time t. We will check the assumption that the "most probable" molecular size distribution is maintained during the random 1 21 chain scission process as made in MacCallum's work ' We substitute eq. (A-l) at t = 0 into eq. (19) and obtain the relationship between molecular size distribution and reaction time/ then calculate the number average degree of polymerization further. The procedure is shown as follows: 59 'tot,0 00 CO • N - A = • S.'l i=n+l 1,0 i=n+l Dp DP tot, 0 i=n+l DP DP DP to t , 0 DP DP i-1 tot, 0 i=ri+l DP DP 1-1 (A-3) tot, 0 00 . z i=n+l DP DP (A-3) becomes Let K = 1 DP tot, 0 i=fi+l 1 i=n+l DP tot,0 d „ ” — dK i=n+l tot,0 d DP. dK (A-4) DP tot, 0 DP DP DP Inserting egs.(A-3) and (A-4.1) into eq. (9), we ob tain tot, 0 n-1 n, t DP DP n-1 tot, 0 (1 - DP DP n-1 a [ (n + l)a DP tot DP. DP n-1 DP 1) } ot£ (n + 1) a - 2] (DP n-1 ) (1 DP (A-5) i=l DP CO • £-1 X = 1 61 J - t i - I d - ~ ~ ) (1 - c O 1 " 1 DP0 __ „ = : T— = I [ (1 - -i-) (1 - a) ]X 1 DP0 (A-6) Let K - (1 - — )(1 - a) , eq. (A-6) becomes DP0 00 . ir^l * i l l -1-1 _ 1/(1 - K)2 “ 1/(1 - K) 1 1 - K (A-6.1) i.e. DP. = i - (i _ _^L.) (i _ a) DP0 (A-6.2) *"k t Inserting a = 1 - e s into eq. (A-6.2), we get DP. = t 1 - (1 - -i-) e"kst DP0 ekst TCt (1 - -±-) DP e S ,n 1 0 (A-7) The number average degree of polymerization at time t deduced by MacCallum under same mechanism is^ 62 DPt + 1 + “kit------ G ‘ S- C (A-8) where a; The maximum degree of polymerization with with which the molecules will volatile from the degraded system. DP - a - 1 C = ---- : ---------— For a closed system a = 0, eq. (A-8) becomes ■“ o - 1 + — -------- _— — 1 1 _ -±- DP0 ekst »kst - (1 — ) dPq (A-9) The expression for DPt in eq. (A-9) is identical to that ini eq. (A-7) , this confirms . that the "most probable" molecular size distribution is maintained during the random scission process. The above assumption still holds for the system in which molecules whose chain lengths are equal to or less than a will volatilize from the degraded system. 63 APPENDIX B Equation (18) is derived as follow ; To sum each differential equation in eq. (12) , we ob- tain i = kf CPN + 2 p + N-l i * ‘ ’ N f j ? P. + 5 i 4 Pi] kdfRN-2 + ^N-3 + * * ' * + r3 + r2] + k2 lRN-2 + ,RN-3 + • • * * + r3 + R2] (B-l) where N N N P„T + P...+ N N-1 i . + 2 p, + 5 i Z p 4 l + S5 CU I I P + P N N- P + P N N- ■1 + + P + 1 N-2 (N-3) expression P + P N • N- + P + 1 N-2 * * ‘ * + P4 N = .2 -(i - 3)P . x=4 l " (B-2) N R = . Z0 R. 1-2 x (B-3) 6:4 inserting egs. ( B-2.) and ( B-3 ) into (B-l) we get dR dt k_)R f i=4 [(k 3) P . i=4 3 1=4 (B-4) 3) P . 2k i=4 APPENDIX C dQi The expression — — • , under the assumption of steady- state concentration for radical species, are derived as follows: dR. Since -g— ■ = 0 , egs. (14) become 0 = K_P ~ K, R„ . f N d,s N-2 N 0 = K. .t£, P. - K, R__ _ f N"1 x d, s N-3 N 0 = K- r P- “ K0R - K , R f n+2 x 2 n+2 d,s n (C-l) 4 < n < N - 4 N 0 * Kf i P-i + K2R5 - <Ka,s - W N 0 = Kf |P. + K2R4 - (Kdig - K2)R3 Eg. (C-l) rearranged can be written as _ Kf ^N^2 K, _ PN d, s K N „ . f JL. P. V-3 = K. , 1 d, s (C-2) • • Kf ? K2 Rn K n+2Pi + K, Rn+2 d,s d,s . . . > , 4 < n < N - 4 66 K, N • K , R3 " K Rr E P + — t „ A » - K 0 5 1 “ K 0 5 a,s 2 d, s 2 K, N Kj R2 K P. + d,s " K2 4 1 Kd,s “2 - K„ R4 Here, letting N be odd, Rn can be simplified further: Per an eyen number , n , K, N K_ K. N K_ r = - — ' E p + r— £_ E p + r n K , n+2 i K, lK, n+4 *i K, n+4] d,s d,s d,s d,s K_ N K0 N K. _ •CjoR + + (s ) R K, n+2 i K, n+4 i n+4 d, s d, s d, s Kf N K N k2 2 ? — =— r E p + (— £_) E p + (__£_\z E pi K, n+2 x K , • n+4 1 ' n+6 iJ d, s d, s d, s K 2 + ( ~ - ) R K, n+6 d, s K N K N K, — f E p + (— — ) E p. + (— - K, n+2 x n+4 1 VK d,s d,s K, N-5-n N ) ?c P- + ' • • • n+6 1 N K, d, s N-3-n + <K r r ) 2 - (k 2 d, s d, s rn - K - N K0 N r— f E p 4. (_____) E p + K, n+2 1 4-, n+4 1 *•* d, s d,s K, N-3-n + (^- ) 2 d, s N+3-n N P- w-1 1 K. 2 K_ N „ — — ) ^ E P K , j=0 K , ' i=n+2 . + 2 i d,s J d,s 3 (C—3) 67 Similarly- K, (N-n-3)/2 K_ . N p = — — r (- £_) j z p (c-4) n+l K, e j=0 [K . } i=n+2.+3 Pi 4' d,s J d,s 3 After rearrangement, eq. (C-4) becomes K (N-n-3)/2 K _ N Rn+1 = KZ j=0 (iT )3 (i=n+2 . + 3 Pi" Pn+2 . + 2} d, s d, s j j Kf (N-n-3)/2 * Rn ~ K , j=0 Pn+2.+2 (C-5) d, s J 3 The Siam of R and R - is n n+l Kf (N-n^3)/2 Rn + Rn+1 “ 2Rn " i c ~ jl0 Pn+2.+2 (C'6) d.s ] To let 0 = P + P N n n n+l and P - = Qr/2 , then inserting them into eq. (C-6), we obtain 2K_ (N-3-n) /2 K . R + R .t. — .Zn ( - — ) D . , Q . n n+l K, 3 = 0 K, i=n+2.+2,2 i d,s J d,s 3 1 Kf (N-3-n)/2 K2 )JQ 2 Ka,s 3=° Kd,s n+2j+2 (c-7) N-3-n;. R - 2 . 2 .j N-l ^ - To expand T (=----)J % Q . and j—0 d,s i=n+2 .+2,2 1 3 (N-3-n)/2 K2 . i=0 Qn+2 +2 seParatelY/ we 9et d,s j 68 N-3-n j=o. n+4 n+2 n+6 N-l n+4 n+6 •N-l n+ 6 N-3-n N-l n+2 n+4 N-3-n [1 + N-l d, s d, s (C-8) and n+2 .+2 N-l n+2 n+4 (C-9) Inserting eqs. (C-8) and (C-9) into eg. (C-7), we obtain n+4 2 vn+2 n+l f2(l n+6 N-3-n 2 [1 ’ 4 1s d,s 69 K N-3-n 2 (K, ) *"*' QN-1 (C-10) d, s For Q2n+2i+2 term' - * - ts coefficient can be written as ^2 ^2 i-1 3 ^2 i 2[1 + + .... + ( , , + < 2 ) d,s d, s d,s : K9 ‘ i - (rJ- ) X k = 2 • • s 3,__2_x i K-, 2lK, } 1 - (j-?-) d's d, s 2 + (------- — + h (=^-)1 K_ K_ 2 K, , ■ 2 . 2 d,s d, s d, s 2.K, (K, + 3 k J K„ . d, s_________ d, s_______2 (__2_j i K, - K„ 2 (K, - K_ ) K , d, s 2 d, s 2 d, s (C—11) Inserting eq. (C-ll) into eq. (C-10), Rn + can expressed as K (N-3-n)/2 2 K K + 3 K *p r -p — •* “ y r Q / S vj.f & £ - n . n+l K, i=0 lK, - K. ~ H kT - K0) d, s d,s 2 d, s 2 7 K2 >i, (K J ] ■ n+2i+2 d,s 2K_ (N-3-n)/2 K, + 3 K0 c ,s„ n :l,!' 2 Ka,s - k2 1=0 4 Ka,s ( fi.il, „ K, ' 1 Qn+2i+2 d, s (C-12) dQ Starting from eq. (13), can be obtained by adding dP. dP. 1 ~ w and ~ ~ d~f~~ tQgether^ 70 dP dT dT 3) (K 1)P 4) (K 1)P. N-l 1) (P 3] (K N-l (23) -[ (n - 3) (K 2) (K 1)P n+l dT n+l 3) (K 'd,'s (N-3-n)/2 • E„ i=0 n+2i+2 d, s (N-3-n)/2 iSo n+2 n+2i+2 -(n - 3) (K d, s (N-3-n)/2 .Ei {'K [ 1 i=l a d, s n+2 i n+2i+2 (23) Similarly 71 dQ~ K R (N-5)/2 — = K2R + 2 (Bq ' - -j~) Q4 + 2 . £ r Bj[Q4 + 2 i . 4 <_ n _< N - 3 (23) where An = (n - 3)(Kf - 1) K . - 3 K K„ . Bi = V 1 - - T X - <k— > > + k3R d, s d, s = Kf [1 - Jl1 + |l1+1] + K3R B .' = -K- X f K, (2 K« -"K, ) K, + 3 K„ # d, s 2 d, s j - - ^ _ d , s 2 (Kd,s " K2} k9 . (j^-^-)1) + k3r d,s.' 4 K ~ ( 21 - 1) n “■ JNjp f t ^-1 - * (1 - I) 72 APPENDIX D c Program A c This program is written to calculate the variations c of concentrations of radical and polymeric species c during the thermal decomposition, DOUBLE PRECISION P(101),DP(100),DR(100),SR/Cl,C2 REAL K2,K3,K4 OPEN (UNTT=35,FILE='Z2JML.DAT1) K2=10.0** (14.0) K3-10.0**(12.0) K4=10,0**(17.0) SR=0,0 C0NV=/ . 9 Pmin=10.0**(-8.0) Rmin=10. 0** (-15. 0) READ (21,100)N 100 FORMAT (IX,14) DO 1 1=2 ,N READ(21,110)P (1),R(I) 110 FORMAT(IX,E22.16/1X,E22.16) SR=SR+R(I) 1 CONTINUE READ(21,120) T 120 FORMAT(1X,E14.8) DQ 2 1=1,N-3 IP3=1+3 SP1=SP1+I*P(IP3) 2 CONTINUE SRM2 3=SR-R(2)—R (3) 40 C1=2+K3*SR C2=K2+K4*SR+K3*SP1 SP=P(N) DR(N^2)=C1*SP”C2*r ;(N-2) RATIO=ABS (DR(N-2) /R(N-2) ) ' DR (N-3) =C1*SP-C2*R (N-3) DROR=ABS(DR(N-3)/R(N-3)) IF (DROR.LT.RATIO)GO TO 10 RAT10=DR0R 73 10 CONTINUE DO 3 1=1,N-7 IX=N-3-I IXP2=IX+2 SP=SP+P(IXP2) DR(IX)=C1* SP+K2 *R (IXP 2)-C2 * R (IX} DROR=ABS(DR(IX)/R)IX)) IP(DROR.LT.RATIO) GO TO 3 RATIO=DROR 3 CONTINUE SP=SP+P (5) C2=C2-K2 DR(3)=C1*SP+K2*R(5)-C2*R(3) DROR=ABS (DR (3)/R(3) ) IF(DROR.LT.RATIO) GO TO 20 RATIO=DROR 20 SP=SP+P(4) DR(2)=C1*SP+K2*R(4)-C2*R(2) DROR=ABS(DR(2)/R(2)) IP(DROR.LT.RATIO) GO TO 30 30 DT=.01/RATIO T=T+DT C3=1+K3*SR DP(2)=K2*SRM2 3+C2*R(2)+K3*SR*SP C4=K3*SP1 DO 4 1=1,N-4 IP2=I+2 IP3=1+3 SP=SP~P(IP3) DP(IP2)=-(IP2-3)*C3*P(IP2)+K4*SR*R(IP2)+C4*R(IP2) 1 +K3*SR*SP 4 CONTINUE DP(N-l)=-(N-4)*C3*P(N-l) DP(N)=-(N-3)*C3*P(N) P (N-l) =P (N-l) +DP (N-l) *DT P (N) =P (N) +DP (N) *DT SR=0.0 spi=o,o DO 5 1=1,N-3 IP1=I+1 IP3=I+3 R (IPl) =R (1P1) +DR (IPl) *DT P(IPl)=P(IP1)+DP(IPl)*DT SR=SR+R(IP1) SP1=SP1+I*P(IP3) 5 CONTINUE N0=N 300 IE(P(N).GT.Pmin) GO TO 310 P(N)—0.0 74 310 320 330 210 NbNt- , 1 GO TO 300 N=N 0 N0=N IF (R (N) , GT, Rjnin) GQ TO 330 R(N)=Rmih NsN-,1 ' GO TO 320 N=N0 SRM23=SRr-R(2) -R(3) IF(P (N) .GT. CONV) GO TO 40 WRITE(20,210) P(N), P (2),SR,T FORMAT (IX,E22 .16,1X,E22.16 , IX,E14 . 8 , IX,E14 . 8) CLOSE(UNIT=35,FILE='Z2JML.DAT') STOP END APPENDIX E c Program B. c This program is written to calculate the varia-' c tions of concentrations of polymeric species by c keeping the radical concentration of each species c constant during the thermal decomposition. REAL K ,KB,MDQ,K2,K3,K4,KO,KB 0 DIMENSION K( 5 0) ,KB(5 0) , AX (100) DOUBLE PRECISION Q (100) ,DQ (100):, P (101) OPEN(UNIT-32,FILE='Kl.DAT') K2=10**(14.0) K3=10**(12.0) K4=10**(17.0) R=.4472E-7 READ(24,100) N 100 FORMAT(IX,I4) DO 1 1=1,N-l IP1=1+1 READ ' (24,110)P(IPl) 110 FORMAT(IX,E22.16) 1 continue ALFA=K2+K4 *R+K3*R4/2.0 * R* * 2 A=l.0+K3*R B=2,0+K3*R C=(ALFA+3,0*K2)/4.0/ALFA D=K2/ALFA E2=K2*R E3=K3*R F=ALFA*(2. 0 *K2-ALFA)/ (ALFA-K2)* * 2 READ(24,120) T 120 FORMAT(IX,E14,8) GX-11.39 N=101 K0=B*(1.0-C)+E3 H=2.0*(K0-K3*R/4.0) KB0=-F*B*(1.0-C)+E3 G=2,0*(KB0-K3*R/4.0) BC=B*C 76 FBC=F*BC BPE=B+E3 EMFB=E3~F*B DO 2 1-2,N-l,2 IP1=1+1 Q (I)=P(IP1)+P(I) AX(I)=A*(1-3.0) 2 CONTINUE DO 3 1=1, (N-l)/2 > BC=BC*D FBC=EBC*D K (I)=BPE~BC KB(I)=EMFB+FBC 3 CONTINUE 10 SKBQ=0.0 DO 4 1=1, (N-5)/2 SKBQ=SBKQ+KB(I)*Q(4+2*1) 4 CONTINUE DQ(2)=E2+G*Q(4)+2.0*SKBQ MDQ=ABS (DQ(2))/Q(2) DO 5 1=4,N-5,5 DQ(I)=-AX(I)*Q(I)+H*Q(1+2) DO 6 J=l, (N-3-D/2 DQ(I)=DQ(I)+2.0*K(J)*Q(1+2+2*J) 6 CONTINUE ADQ=ABS(DQ(I))/Q(I) IF(ADQ.LT.MDQ) GO TO 5 MDQ=ADQ 5 CONTINUE DQ(N-3)=-AX(N-3)*Q(N-3)+H*Q(N-l) ADQ=ABS(DQ(N-3))/Q(N-3) IF(ADQ.LT.MDQ) GO TO 7 MDQ=ADQ 7 DQ (N-l) =-AX (N-l) *Q (N-l) ADQ=ABS (DQ (N-l) ) /Q (N-l) IF(ADQ.LT.MDQ) GO TO 8 MDQ=ADQ 8 DT=0.01/MDQ T=T+DT DO 9 1=2,N-l,2 Q CD =q(i)+dq:(i) *dt 9 continue IF (Q ( 2) ,LT, QX): GO TO 10 WRITE (33,*)T DO 11 J=2,N-1,2 WRITE(33,*) Q (J) 11 CONTINUE 77 CLOSE(UNIT-33,FXLE='K1,DAT *) STQP END 78 APPENDIX F c Program C c This program is written by combining the programs c A and B to modify the steady state value of each c radical concentration interminttantly. DOUBLE PRECISION P (101),DP(100),R (100),DR(100),SR REAL: K2,/K3,K4 OPEN(UNIT=36,FILE= *Z 4JML.DAT!) K2==10, 0** (14 . . 0) K3—10,0**(12.0) K4=10,0**(17.0) SR-0,0 0X=2,0 READ (24,100) N 100 FORMAT(IX,14) DO 1 1=1,N-l IP1=I+1 READ(24,110) P (IPl) ,R(IP1) 110 FORMAT(IX,E22.16,1X,E22.16) SR=SR+R(IPl) 1 CONTINUE READ(24,12 0) T 120 FORMAT(1X,E14,8) DO 2.1=1,N-3 IP3=I+3 SP1=SP1+I*P (I.P3) SP=SP+P(IP3) 2 Continue SRM23=SR-R(2)-R(3) 50 DO 5 1=1,200 C 2 —K4 * S R+K3 * S P1 C3=1+K3*SR DP (2)=K2 * S RM23+C2*R(2)+K3* SR* SP CV=A.BS (DP ( 2 ) ) /P (2) C4=K3*SP1 DO 3 K=1,N-4 KP2=K+2 KP3=K+3 79 SP=SP-P(KP3) DP(KP2)- - (KP2-3)*C3*P(KP2)+K4*SR*R(KP2)+C4*R(KP2) 1 +K3*SR*SP DPQP=ABS (DP (KP2) )'/p (KP2) IF(DPOP.LT.CV) GO TO 3 CV=DPOP 3 CONTINUE DP (N—1) =- (N-4) *C3*P (N-l) DP (N) ==- (N-3) *C3*P (N) SP1=0.0 SP=0.0 DT=.01/CV T=T+DT DO 4 J=l,N-3 JP1=J+1 JP3=J+3 P (JP1)=P(JPl)+DP(JP1)*DT SP1=SP1+J*P(JP3) SP=SP+P(JP3) 4 CONTINUE IF (P(2) .GT.QX) GO TO 60 5 CONTINUE DO 6 1=1,100 SP=0.0 C1=2+K3*SR C2=K2+K4*SR+K3*SP1 SP=P(N) DR (N-2) =C1*SP-C2*R (N-2) BV=ABS(DR(N^2)/R(N-2)) DR (N-3) =C1*!pP-C2*R(N-3) DR0R=ABS(DR(N-3)/R(N-3)) IF(DRQR,LT.BV) GO TO 20 BV=DROR 20 CONTINUE DQ 7 J=l,N-7 JX=N-3-J JXP2=JX+2 SP=SP+P(JXP2) DR(JX)=C1*SP+K2*R(JXP2)-C2*R(JX) DRQR=ABS(DRO(JX)/R(JX)) IF(DROR.LT,BV) GO TO 7' BV=DROR .7 CONTINUE SP=SP+P(5) C2=C2-K2 DR(3)=Cl*SP+K2*R(5)-C2*R(3) . DROR=ABS (DR (3) /R (3) ) IF(DRQR.LT.BV) GO TO 30 80 BV=DROR 30 SP=SP+P(4) DR (2)'=C1*SP+K2*R(,4;).«C2*.-R (2) DRQR=ABS(DR(2)/R(2)) IF(DROR.LT.BV) GO TO 40 BV=DROR 40 DT=0.01/BV T-T+DT C3=1+K3*SR DP(2)=K2*SRM2 3+C2 *R(2)+K3*SR*SP C4=K3*SPl DO 8 K=l,N-4 KP2=K+2 KP3=K+3 SP=SP~P(KP3) DP(KP2)=-(KP2-3)*C3*P(KP2)+K4*SR*R(KP2)+C4*R(KP2) 1 +K3*SR*SP 8 CONTINUE DP(N-l)=-(N-4)*C3*P(N-l) DP (N) =- (Nr-3) *C3*P (N) P (N-l) =P (N-l) +DP (N-l) *DT P (N) =P (N) +DP (N) *DT S R= 0 . 0 SP1=0.0 DO 9 J=l,N-3 JP1=J+1 JP3-J+3 R (JPl) =R ( JPl) +DR (JPl) *DT P (JPl)=P(JPl)+DP(JPl)*DT SR=SR+R(JPl) SP1=SP1+J*P(JP3) 9 CONTINUE SRM23=SR-R(2) -R'(3) IF(P(2).GT.QX) GO TO 60 6 CONTINUE GO TO 50 60 WRITE(36,*) T,S R DO 10 M=2,N,2 MP1=M+1 Q=P (M) +P (MP1) WRITE(36,*) Q,R(M),R(MPl) 10 CONTINUE DO 11 1=1,N-l IP1=1+1 WRITE(1,200)P(IPl),R(IPl) 200 FORMAT(lXfE22.16,1X,E22.16) 11 CONTINUE CLOSE(UNIT=36,FILE— ■Z4JML.DAT *) STOP END ’ 81 REFERENCES 1, J. Atkinson and J. R. MacCallura, J. Macromol. Sci.- Che. A5(4), 945 (1971) i 2, R, Simha and L, A, Wall, J. Polymer Sci. 5, 615 (1950). 3, N« Grassie, "Chemistry of Vinyl Polymer Degradation Processes", Butterworths, London, 1956. 4, S. L. Hadorsky, J, Polymer Sci. 9, 133 (1952). 5, S. L. Madorsky, J. Polymer Sci. 11, 491 (1953). 6, S. Bywater, J, Phys. Chem. 57, 879 (1953). 7, N, Grassie and H. W. Melville, Proc. Royal Soc. A199, 1 (1949). 8, P. R. E. J. Cowley and H. W. Cowley, ibid. A210, 461 (1952) . 9, H, H. G. Jellinek, J. Polymer Sci. 4, 13 (1949). 10, N, Grassie and W. W. Kerr, Trans. Faraday Soc. 53, 234 (1957). 11, N, Grassie and W, W, Kerr, ibid. 55, 105 (1959). 12, G, G, Cameron and N. Grassie, Polymer 2, 367, (1961). 13, N. Grassie and R. McGuchan, Eur. Polym. J. 6, 1277 (1970) . 14, N, Grassie and A. Heaney, Rubber Chem. & Technol. 48, 678 (1975), 15, R. Simha, L. A. Wall and J. Bram, J. Chem. Phys., 29, 894, (1958) . 82 16, R, H, Boyd, J, Chem, Phys,, 31, 321 (1959). 17, R, H, Boyd and T. P, Lin, ibid, 45, 773 (1966), 18, R, H, Boyd and T. P, Lin, ibid. 45, 778 (1966). 19, R, H, Boyd, J, Polymer Sci,, (A-l) (5) 1573 (1967). 20, J, R, MacCallum, : Eur. Polymer J, 2, 413 (1966). 21, J. Atkinson and J. R. MacCallum, J. Polymer Sci., (A-2) (10), 811 (1972). 22, H. H, G, Jellinek, Degradation of Vinyl Polymers, Academic Press, New York, 1955. 23, R. Simha, J, Applied Phys., 12, 569 (1941). 24, R. Simha and L. A. Wall, J. Phys. Chem., 56, 707 (1952). 25, L. A, Wadi/ S. Straus, J. H. Flynn and D. McIntyre, J, Phys. Chem,, 70, 53 (1966). 26, H, H, G. Jellinek "Aspects of Degradation and Stabilization of Polymers", p. 251. 27, .H, H, G. Jellinek, ibid. , p. 252. 28, H, H, G. Jellinek, ibid., p. 258. 29, G. A. Mortimer, J. Polym. Sci. A-l, 8, 1535 (1970). 30, P. R, E. J. Cowley and H, W. Melville, Proc. R. Soc., London, Ser, A, 210, 461 (1952) . 83

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Asset Metadata

Creator Liao, Jae-Min (author)

Core Title Evaluation of Simha's approach to thermal decomposition of a linear polymer system

Degree Master of Science

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag chemistry, polymer,engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Salovey, Ronald (committee chair), Chang, Wenji Victor (committee member), Yortsos, Yanis C. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-313573

Unique identifier UC11260005

Identifier EP41811.pdf (filename),usctheses-c20-313573 (legacy record id)

Legacy Identifier EP41811.pdf

Dmrecord 313573

Document Type Thesis

Rights Liao, Jae-Min

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA