Ind. Eng. Chem. Res. 2001, 40, 5719-5723
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Prediction of the Molecular Weight for a Vinylidene Chloride/Vinyl Chloride Resin during Shear-Related Thermal Degradation in Air by Using a Back-Propagation Artificial Neural Network Jiantao Liu* Department of Chemical and Materials Engineering, University of Kentucky, Lexington, Kentucky 40506-0046
Thermal degradation of a copolymer is characterized by time-dependent molecular weight change mainly resulting from syntheses of chain-scission and cross-linking reactions. The kinetics models for thermal degradation have been well established. However, the kinetics mathematical model is limited for shear-related thermal degradation of polymers in air because shear stress has uncertain effects on the degradation process. In this paper, the back-propagation artificial neural network (ANN) is adopted to predict both the number-average molecular weight and the weightaverage molecular weight of vinylidene chloride (VDC)/vinyl chloride (VC) during shear-related thermal degradation in air. The optimum architectures of ANN, 3-9-9-1 and 3-10-1, have been found for predicting the number-average molecular weight and the weight-average molecular weight of VDC/VC during shear-related thermal degradation, respectively. 1. Introduction In past years, thermal degradation of poly(vinyl chloride)1-4 and poly(vinylidene chloride)5-9 has been investigated intensively. The kinetics of thermal degradation has been well developed, and the mechanism of thermal degradation has been explained very well. Unfortunately, polymers usually experience shearrelated thermal degradation during the process of manufacture. Therefore, investigation of the effects of shear on thermal degradation is of great importance. However, information about the thermal degradation of polymer under shear is limited in the literature. Betso et al.10 investigated the shear-related thermal degradation of vinylidene chloride (VDC)/vinyl chloride (VC; ∼15 wt %) in air. The results showed that shearrelated degradation of VDC/VC in air is characterized by early predominant chain scission with cross-linking and latter predominant cross-linking with chain scission. Both chain scission and cross-linking are dependent on the shear rate and temperature. Furthermore, the shear stress dependency of degradation was modeled by a kinetic expression that incorporated shear stress into the Arrhenius preexponential factor. However, the effects of shear rate and temperature on the degradation process are difficult to evaluate directly. Consequently, the effects of shear stress on degradation have to be investigated under constant shear rate and temperature. The main description method of thermal degradation was a kinetic equation or mathematical model of measured data from experiments. However, a lot of mechanisms concerning the effects of various factors on shear-related thermal degradation cannot be accepted to construe and describe currently. As has been mentioned in ref 10, shear has uncertain effects on the processes of thermal degradation: it may have no effect * To whom correspondence should be addressed. Tel.: +001 859 2578089. Fax: +001 859 3231929. E-mail: jliu1@ cs.uky.edu.
on processes of thermal degradation, it may accelerate some or all of the processes, it may decelerate some or all of the processes, or it may affect some processes and not others. Both oxidative chain scission and crosslinking are dependent on the shear rate and temperature. Chain-scission and cross-linking reactions happen simutaneously during the whole shear-related thermal degradation process. There exists a critical time when the statuses of prevalence exchange. At the beginning of thermal degradation, chain scission is dominant and latter cross-linking is dominant. Modeling the critical time is not an easy task. Difficulties lie with the dependency of the critical time on the shear rate and temperature and the poor understanding of the shear rate and temperature for the critical time. Consequently, empirical methods have to be adopted to simulate the effects of shear rate, temperature, and time on the molecular weight of a VDC/VC copolymer. Seeing the difficulty of developing mechanistic models, we attempt an alternative method to model shearrelated thermal degradation of a VDC/VC copolymer based on the artificial neural network (ANN). Unlike the method of regression, ANN does not require a mathematical model. The ANN can perform not only a high accurate nonlinear fit but also memory of data. Besides, the ANN is powerful in handling dispersed data; moreover, it can also adjust the state of the network on the base of the original one in order to adapt to new datasets through training with these new datasets. In this paper, the ANN is used in predicting the number-average molecular weight and the weightaverage molecular weight of VDC/VC under different shear rate, temperature, and time. The optimum architectures of the ANN are also proposed for prediction of the number-average molecular weight and for prediction of the weight-average molecular weight. 2. Theory of the ANN 2.1. Structure of the Neuron and Model of the ANN. The structure of the neuron is shown in Figure
10.1021/ie000200j CCC: $20.00 © 2001 American Chemical Society Published on Web 10/24/2001
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Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001
Figure 1. Structure of the neuron.
Figure 4. Sketch map of the Delta rule.
Rumelhart and McClelland.11 The BPA is a kind of generalized form of the least-mean-square algorithm usually used in engineering. By using an algorithmgeneralized gradient descent search technique, BPA adjusts the weights of the network and threshold of each neuron recurrently according to the criterion that the cost function is minimized. The cost function is the mean-squared error between the actual outputs of network and the target outputs. 2.2. Training Algorithm of the ANN. The training rule of the ANN adopts a continuous perceptron training rule that is also called the Delta rule. The training value is
r ) [di - f(w b tib x )]f ′(w b tib x)
Figure 2. Sigmoid function.
(4)
x) is the derivative function of the activawhere f ′(w b tib x. The Delta rule can be tion function f(net), net ) w b tib obtained from the condition that the least-squares error between oi and di is minimized (Figure 4). At first, one needs to compute the gradient of squared error versus wi, i.e., Figure 3. Structure of an MLP with a hidden layer.
1. yj is the output of the neuron: N
∑ i)1
yj ) f(
w b tijb xi
- θj)
(1)
in which f(x) refers to the activation function, being a nonlinear function; the most widely used activation function is the Sigmoid function (Figure 2). θ is the deviation of the neuron. A multilayer perceptron (MLP) is a type of widely used ANN. Between the input and output layers of this kind of network are several hidden layers. Theoretically, an MLP with a hidden layer can achieve very good estimation of the function through appropriate training. An MLP with a hidden layer is shown in Figure 3: L
w b tlmb y l - θm), ∑ l)1
om ) f(
m ) 1-3, ..., M
(2)
l ) 1-3, ..., L
(3)
N
yl ) f(
b v tnlb x n - θl), ∑ n)1
The training of MLP usually adopts a back-propagation algorithm (BPA) which was first proposed by
1 1 b tib x )]2 E ) (di - oi)2 ) [di - f(w 2 2
(5)
b tib x) b x ∇E B ) -(di - oi)f ′(w
(6)
∂E ) -(di - oi)f′(w b tib x ) xj, j ) 1-3, ..., N ∂wij
(7)
To minimize the error, the weights should be adjusted with respect to the direction of the negative gradient; consequently, take
∆w b i ) -η∇E B ) η(di - oi)f ′(neti) b x
(8)
∆wij ) η(di - oi)f ′(neti) xj + R∆wij, j ) 1-3, ..., N (9) where η is the learning rate that determines the training rate of the network. Another term, named the momentum term, is added in eq 9 to improve the convergence of the algorithm in which R is the momentum coefficient. The source code of the ANN was written in standard C language, and training of the ANN is performed on a PC. 3. Prediction of the Molecular Weight of a VDC/VC Resin during Thermal Degradation under Shear 3.1. Kinetics of Degradation under Shear. Shearrelated degradation is characterized by shear-related
Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001 5721 Table 1. Molecular Weight Data for a VDC/VC Resin Degraded under Shear in Aira,10 SD (rpm)
Tmelt (°C)
t (min)
Mn (g/mol)
Mw (g/mol)
Mw/Mn
40
170 172 172 172 172 172 172 171 172 172 173 178 180 181 181 181 181 181 173 173 172 172 171 171 172 172 173 178 179 176 177 177 177 177 177 179 183 186 181 181 180 180 180 181 183 187 184 185 185 186 190 197 173 174 174 173 173 173 175 178 180 181 181 181
0 5 10 15 20 25 30 35 40 45 50 55 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 45 50 55 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 0 5 10 15 20 25
28 945 27 867 29 899 30 199 28 544 29 255 26 876 28 076 29 415 28 975 25 543 32 064 29 871 24 396 26 764 23 795 22 543 25 158 28 882 27 899 34 921 36 660 34 949 31 879 33 236 30 593 30 400 30 114 30 271 30 621 31 535 27 949 48 880 39 253 38 700 37 669 41 538 36 014 35 452 35 426 36 739 11 481 38 636 37 095 35 116 32 969 32 385 31 893 32 513 32 222 18 275 40 124 38 989 35 880 35 023 33 675 33 698 33 652 36 504 33 449 31 354 33 003 30 118 27 022 26 825 29 987 36 274 28 614 28 192 29 204 29 304 27 027
100 978 92 671 97 596 91 299 90 356 88 273 84 258 94 763 93 262 104 459 98 951 128 220 87 014 72 675 75 037 68 356 68 557 93 791 100 646 119 715 98 218 95 061 90 744 91 471 83 874 85 846 93 445 93 198 100 981 121 634 151 643 140 191 101 079 98 470 95 033 91 570 92 825 98 850 108 322 128 854 159 597 121 328 97 171 93 228 85 566 82 605 84 830 97 516 119 840 140 439 124 182 96 691 90 600 83 281 85 584 93 949 115 540 148 381 95 562 95 530 90 264 84 866 82 116 82 126 90 819 120 667 96 894 88 388 80 440 81 084 89 008 101 619
3.49 3.33 3.26 3.02 3.17 3.02 3.14 3.38 3.17 3.61 3.87 4.00 2.91 2.98 2.80 2.87 3.04 3.73 3.48 4.29 2.81 2.59 2.60 2.87 2.52 2.81 3.07 3.09 3.34 3.97 4.81 5.02 2.07 2.51 2.46 2.43 2.23 2.74 3.06 3.64 4.34 10.57 2.52 2.51 2.44 2.51 2.62 3.06 3.69 4.36 6.80 2.41 2.32 2.32 2.44 2.79 3.43 4.41 2.62 2.86 2.88 2.57 2.73 3.04 3.39 4.02 2.62 3.09 2.85 2.78 3.04 3.76
40
60
60
60
60
80
80
a The data used for prediction are in italics. The data for the 0 time in each group have not been used because the Tmelt values of these data are not available. Anyway, the loss of these data should not affect the results because at the 0 time the resin has not been given the shear.
components of chain-scission and cross-linking processes. The kinetic equations of chain-scission and cross-
Figure 5. Optimum architecture of ANN for simulating (a) Mn and (b) Mw.
linking processes have been derived by Betso et al.10 A brief description of the kinetics of chain scission and cross-linking is included as follows. The kinetic equation for chain scission is
[
( )]
EAs dc (FO2)q ) ks(FO2)q ) τmAs exp dt RT
(10)
where dc/dt is the rate of chain scission, τ is the shear stress, m is a constant, As is the nonshear component of the preexponential factor, EAs is the activation energy of chain scission, R is the gas constant, T is the temperature, (FO2) is the partial pressure of oxygen available for the chain-scission process, and q is a constant. The kinetic equation for cross-linking is
[
( )]
EAx dx ) kx(X - x)(NEP)2(1 + (FO2)) ) τnAx exp dt RT (X - x)(NEP)2(1 + PO) (11) where dx/dt is the rate of cross-linking, τ is the shear stress, n is a constant, Ax is the nonshear component of the preexponential factor, EAx is the activation energy of cross-linking, R is the gas constant, T is the temperature, NEP is the n-ene polyenes, X is the maximum level of attainable cross-links, x is the number of crosslinks at time t, and PO ) f(O2). Chain scission and cross-linking happen simultaneously during the shear degradation. Therefore, the net rate of bond formation db/dt is
db dx dc ) dt dt dt
(12)
The experiment of thermal degradation was performed under shear in a Brabender Plasti-Corder torque rheometer. The details about the experiment were illustrated in ref 10.
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Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001
Figure 6. Training and the prediction results of Mn.
Figure 7. Training and the prediction results of Mw.
The experimental data are shown in Table 1. It should be pointed out that the average shear rate (γ˘ ) that the copolymer experiences is a weight average of the shear rates of the drive (SD) and slave rotor blades (SS), and the drive rotor blade rotates three revolutions for every two revolutions of slave rotor blade:
are used to verify the prediction results. ANN requires that the range of both input data and output data should be 0-1; consequently, data must be unified. The widely used method of unification is
SD 3 ) SS 2
(13)
3.2. Design and Training of the ANN. The purpose of the ANN is to predict the number-average molecular weight and the weight-average molecular weight of a VDC/VC resin under different shear rates, temperatures of the resin, and times. Therefore, the input mode should include the tachometer of the drive blade (SD), the temperature of the resin (Tmelt), and the time (t): X ) (SD, Tmelt, t). The output modes are the numberaverage molecular weight (Mn; Y1 ) Mn) and the weightaverage molecular weight (Mw; Y2 ) Mw). The training of the ANN is carried out with momentum term R ) 0.8 and learning rate η ) 0.9. Training was carried out for 30 000 epochs or until the average sum square errors over all training epochs was under 1 × 10-6. The progress of the ANN was monitored by observing the average sum square error after each training epoch. A total of 46 groups of data are selected randomly to form the training set, and 18 groups of data
Z′ )
Z - 0.95Zmin 1.05Zmax - 0.95Zmin
(14)
where Z, the original data, could be SD, Tmelt, t, Mn, and Mw, Zmin is the minimum value of Z, Zmax is the maximum value of Z, and Z′ is the unified data of the corresponding Z. 4. Results and Discussion The optimum architectures of the ANN adopted for predicting the number-average molecular weight (Mn) and the weight-average molecular weight (Mw) are found to be 3-9-9-1 (Figure 5a) and 3-10-1 (Figure 5b), respectively. The training results and prediction results of the number-average molecular weight and the weight-average molecular weight are shown in Figures 6 and 7, respectively. Figure 8 shows the training results and prediction results of Mw/Mn. The mean error
Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001 5723 Table 2. Errors of Training and Prediction by Using Different Architectures of ANN Mn
Mw
architecture of the ANN
mean error of the training (%)
mean error of the prediction (%)
sum (%)
mean error of the training (%)
mean error of the prediction (%)
sum (%)
3-9-10-1 3-10-1 3-9-1 3-11-1 3-11-11-1 3-10-10-1 3-10-10-10-1 3-11-11-11-1 3-9-9-9-1 3-10-10-10-10-1 3-10-15-1 3-10-11-1 3-9-9-1
0.76 3.31 1.26 0.49 0.31 1.31
4.64 6.5 3.95 4.38 5.5 3.42
5.4 9.81 5.21 4.87 5.81 4.73
0.49 0.08 1.87 1.53 0.17 2.42 0.07 0.18 1.66 0.22 0.44 0.58
6.56 4.42 7.41 6.18 6.49 5.07 5.91 10.57 7.15 9.03 4.88 4.82
7.05 4.5 9.28 7.71 6.66 7.49 5.98 10.75 8.81 9.25 5.32 5.4
neurons in each hidden layer. Anyway, different architectures of the ANN have great effects on the prediction results. In general, the accuracy of the prediction results will be improved by increasing the hidden layer and neurons in the hidden layer. However, this is not always the case. 5. Conclusions BP ANN can be used to simulate the effects of shear rate, temperature, and time on the thermal degradation of a VDC/VC resin in air. The optimum architectures of ANN for modeling the number-average molecular weight and the weight-average molecular weight are 3-9-9-1 and 3-10-1, respectively. Literature Cited
Figure 8. Experimental results and prediction results of Mw/Mn.
is defined as
mean error )
K
pk - m k
k)1
mk
∑|
k
|
, k ) 1-3, ..., K
(15)
where pk is a predicted value and mk is a measured value. The optimum architecture of the ANN is defined as the architecture that makes the sum of the prediction mean error and the training mean error the minimum. The prediction mean error and training mean error by the same ANN with different architectures are shown in Table 2. The unavailable data means the minimum average square errors (1 × 10-6) cannot be reached after 30,000 epochs of training. It can be seen that 3-9-9-1 ANN give the best fitting of the data for prediction of the number-average molecular weight while 3-10-1 ANN give the best fitting of the data for prediction of the weight-average molecular weight. After being retrained by using new data from the experiment, these two optimal ANN can also be used for predicting the number-average molecular weight and the weightaverage molecular weight during degradation under shear of other resins, respectively. The more the new data is used, the more the accuracy of prediction will be reached. Then, the retrained ANN can be adopted for prediction of more than one resin. There is not yet the theoretical guidance for the determination of the number of hidden layers and
(1) Montaudo, G.; Puglisi, C. Evolution of Aromatics in the Thermal Degradation of Poly(Vinyl Chloride): A Mechanistic Study. Polym. Degrad. Stab. 1991, 33, 229. (2) Knu¨mann, R.; Bockhorn, H. Investigation of the Kinetics of Pyrolysis of PVC by TG-MS-Analysis. Combust. Sci. Technol. 1994, 101, 285 and references therein. (3) Patel, K.; Velazquez, A.; Calderon, H. S.; Brown, G. R. Studies of the Solid-State Thermal Degradation of PVC. I. Autocatalysis by Hydrogen Chloride. J. Appl. Polym. Sci. 1992, 46, 179 and references therein. (4) Jime´nez, A.; Berenguer, V.; Lo´pez, J.; Sa´nchez, A. Thermal Degradation Study of Poly(Vinyl Chloride): Kinetic Analysis of Thermogravimetric Data. J. Appl. Polym. Sci. 1993, 50, 1565 and references therein. (5) Bohme, R. D.; Wessling, R. A. The Thermal Decomposition of Poly(Vinylidene Chloride) in the Solid State. J. Appl. Polym. Sci. 1972, 16, 1761. (6) Ballistreri, A.; Foti, S.; Montaudo, G.; Scamporrino, E. Mechanism of Thermal Decomposition of Poly(Vinylidene Chloride). Polymer 1981, 22, 131. (7) Montaudo, G.; Puglisi, C.; Scamporrino, E.; Vitalini, D. Correlation of Thermal Degradation Mechanisms: Polyacetylene and Vinyl and Vinylidene Polymers. J. Polym. Sci., Polym. Chem. Ed. 1986, 24, 310. (8) Howell, B. A. The Application of Thermogravimetry for the Study of Polymer Degradation. Thermochim. Acta 1989, 148, 375. (9) Simon, G. Kinetics of Polymer Degradation Involving the Splitting off of small Molecules: Part 10sThermal Degradation of Poly(Vinylidene Chloride). Polym. Degrad. Stab. 1994, 43, 125. (10) Betso, S. R.; Berdasco, J. A.; Debney, M. F.; Murphy, G. L.; Rome, N. P.; Richards, S. G.; Howell, B. A. Shear Degradation of Vinylidene Chloride Copolymers. J. Appl. Polym. Sci. 1994, 51, 781. (11) Rumelhart, D. E.; McClelland, J. L. PDP Group Parallel Distribution Processing; MIT Press: Cambridge, MA, 1986; Vols. I and II.
Received for review February 4, 2000 Accepted August 1, 2001 IE000200J
