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Diels−Alder Reaction Kinetics for Production of Norbornene Monomers: Evaluation of Parameters and Model Reliability Using Markov Chain Monte Carlo Methods Kari Vahteristo,* Arto Laari, and Antti Solonen Department of Chemical Technology, Lappeenranta University of Technology, P.O.B 20, FIN-53851 Lappeenranta, Finland S Supporting Information *

ABSTRACT: The reaction kinetics of styrene and indene with cyclopentadiene in the presence of tert-butylcatechol as a polymerization inhibitor was studied in a batch reactor in the production of potential monomers for amorphous polymers: 5phenyl-2-norbornene and indenylnorbornene. The experimental results can be explained with a reaction mechanism using four separate reactions: an equilibrium reaction of dicyclopentadiene and cyclopentadiene, a Diels−Alder reaction of cyclopentadiene with a dienophile to produce norbornene, disproportionation of norbornene, and a reaction between norbornene and dicyclopentadiene. Some polymerization could be noticed when an excess of dienophile was used. The reaction kinetic parameters of this mechanism were fitted and their reliabilities tested using Markov chain Monte Carlo (MCMC) methods. The suggested reaction mechanisms explained the experimental results well, but analysis with MCMC methods revealed clear uncertainties in the reliabilities of the estimated parameters. Here, MCMC provided a useful tool for analyzing in detail what the measured data tell about the unknown parameters and how reliably they can be identified.

1. INTRODUCTION Amorphous polymers with high glass transition temperatures and crystalline polymers with high melting temperatures are promising heat-resistant engineering plastics. The polymer from 2,5-norbornadiene (bicyclo[2.2.1]hepta-2,5-diene) is a typical example of a high glass transition temperature amorphous polymer achieved by incorporating a bulky−rigid alicyclic structure into a polymer main chain.1,2 When a norbornene-type monomer having an aromatic ring structure (e.g., 5-phenyl-2-norbornene) is subjected to ring-opening polymerization, a resin can be produced for use in, for example, optical materials, medical tools, and the treatment of electronic parts.3,4 The studied monomers can be produced by Diels−Alder reaction of the corresponding dienes and dienophiles. The Diels−Alder reaction has remained remarkably useful in synthetic organic chemistry, especially for synthesis of polycyclic natural products, and is one of the most important methods available for creation of carbon−carbon bonds and formation of six-membered rings.5 When the diene and dienophile have strongly opposite electronic character, the reaction proceeds under mild conditions with excellent yield.6 Cyclopentadiene (CPD) contains conjugated double bonds and an active methylene group and can thus undergo a Diels− Alder diene addition reaction with almost any unsaturated compound. CPD dimerizes spontaneously and exothermically at ambient temperature to dicyclopentadiene (DCPD).7 As DCPD is solid at ordinary temperatures and is readily reconverted to the monomer, it is a more convenient form in which to handle CPD.8 The prevailing reaction is the dedimerization of DCPD to form CPD at 160−180 °C.9 CPD and vinyl aromatics oligomerize rather easily. The presence of high levels of the oligomers of CPDs in the reaction mixture complicates handling of heat-treated products. Too© 2013 American Chemical Society

high levels of the polymers of vinyl aromatics lead to high viscosity of the heat-treated product and, consequently, functionality issues. The glass transition temperature also decreases. Efforts to inhibit oligomerization can be made using antioxidants such as phenol compounds.2 This work studies the reaction kinetics of styrene (STY) and indene (IND) with CPD in the presence of 4-tert-butylcatechol (4tBC) in the production of 5-phenyl-2-norbornene (PN) and indenylnorbornene (IN). It aims to demonstrate how the reliability of the estimated kinetic parameters can be analyzed using Markov chain Monte Carlo (MCMC) methods. MCMC is a modern random sampling-based parameter estimation approach that can be used to obtain the distribution of possible values of the parameters in nonlinear models, and to support and extend results from classical regression analysis based on linear approximations. Here, MCMC is found to be a useful tool for analyzing, in detail, information about the parameters contained in the experimental data, and for assessing to what extent the parameters can reliably be identified from the available data.

2. EXPERIMENTAL SECTION 2.1. Apparatus. Experiments were carried out in a 300 cm3 stainless steel autoclave reactor (Parr 452HC-2979) equipped with a magnetically driven impeller, an electric heater, and a control and measurement unit (Parr 4843) to control the stirring speed and temperature of the reaction mixture. Experiments were carried out under a nitrogen atmosphere. Received: Revised: Accepted: Published: 6357

December 19, 2012 April 8, 2013 April 8, 2013 April 8, 2013 dx.doi.org/10.1021/ie303529u | Ind. Eng. Chem. Res. 2013, 52, 6357−6365

Industrial & Engineering Chemistry Research

Article

2.2. Reaction Procedure. The reactants were DCPD (95 wt %, Aldrich, 200 ppm inhibitor), STY (99 wt %, Aldrich, 10− 15 ppm inhibitor), and indene (90 wt %, Aldrich). 4tBC (97 wt %, Aldrich) was used to inhibit polymerization of the reactants. The reactants and the inhibitor were weighed and put into the reactor (one-third of the reactor volume), and oxygen was removed with nitrogen for 5 min at a pressure of 5 bar. Over 20 min the temperature of the reaction mixture was raised to the reaction temperature, and sampling was started according to the following timetable: 20, 40, 60, 80, 100, 130, 160, 190, and 270 min. The stirring speed was 500−530 rpm. 2.3. Analysis. Samples without solvent were analyzed using a gas−liquid chromatograph (GLC; HP 5790A) equipped with a capillary column (DB-1). Conditions during the analysis were the following: initial temperature 35 °C, initial time 5 min, heating rate 5 °C/min to final temperature 300 °C, injector temperature 200 °C, and flame ionization detector (FID) temperature 300 °C. The injector was a split/splitless type, and the split ratio was 1:50. The peaks of CPD, DCPD, STY, and IND were identified by comparison with pure components, which were also used for calculation of the response factors. The GC analysis calibration table was prepared based on peak identification results with a gas chromatograph−mass spectrometer (GC−MS; JEOL JMS-AX505WA). The reaction products are depicted in Figures 1 and 2.

Figure 2. Main and side products in Diels−Alder reaction of IND with DCPD.

isomer of OHBF. At the beginning of the experiments, a small amount of CPD was also detected. 3.3. Kinetic Model. On the basis of the results of GLC and GC/MS analyses, the kinetic models depicted in Figures 1 and 2 were assumed. At the beginning of the reaction process, DCPD dedimerizes to CPD, which acts as a diene in the Diels− Alder reaction. In the second step, STY or IND act as a dienophile to form the desired norbornene compound with CPD. In the third reaction, two norbornene molecules react further to PTeCDD (OHBF) or its isomer and dienophile. In the fourth reaction, DCPD reacts with norbornene to form TCPD and dienophile. The existence of reactions 3 and 4 (Figures 1 and 2) was assumed, because it was discovered that the reactions started when a necessary amount of the main product (norbornene) had formed. As a consequence of these reactions, the amount of dienophile started even to increase. The reactions were confirmed also experimentally using PN or PN and DCPD as reactants. 3.4. Kinetic Modeling. Based on the reaction scheme depicted in Figure 1, the reaction rate for each component rj can be presented using the following equations: rDCPD =

Figure 1. Main and side products in Diels−Alder reaction of STY with DCPD.

dC DCPD dt

= −k1C DCPD + k −1CCPD2 − k4C DCPDC PN rCPD =

3. RESULTS AND ESTIMATION OF KINETIC PARAMETERS 3.1. Product Mixture of STY/DCPD: Experiments. The main product of the Diels−Alder reaction of STY and DCPD was PN, which consisted of both exo and endo forms (see Figure 3a). Detected side products were tricyclopentadiene (TCPD), phenyltetracyclododecene (PTeCDD), and an isomer of PTeCDD, the structure of which was not determined. At the beginning of the experiments, a small amount of CPD was detected (see Figure 3b). 3.2. Product Mixture of IND/DCPD: Experiments. Correspondingly, the main product of the Diels−Alder reaction of IND and DCPD was IN, which also consisted of both exo and endo forms. Detected side products were TCPD, 1,2,5,6diendomethyleneoctahydrobenzofluorene (OHBF), and an

(1)

dCCPD = 2k1C DCPD − 2k −1CCPD2 − k 2CCPDCSTY dt (2)

rSTY =

dCSTY = −k 2CCPDCSTY + k 3C PN 2 + k4C DCPDC PN dt (3)

rPN =

dC PN = k 2CCPDCSTY − 2k 3C PN 2 − k4C DCPDC PN dt (4)

dC PTeCDD = k 3C PN 2 dt

(5)

dC TCPD = k4C DCPDC PN dt

(6)

rPTeCDD =

rTCPD = 6358

dx.doi.org/10.1021/ie303529u | Ind. Eng. Chem. Res. 2013, 52, 6357−6365


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⎛ Ej ⎛ 1 1 ⎞⎞ kj = kj ,mean exp⎜⎜ − ⎜ − ⎟⎟⎟ Tmean ⎠⎠ ⎝ R ⎝T

(7)

Here, kj,mean is an apparent reaction rate constant of reaction j at some “mean” temperature value between the minimum and maximum used in the experiments, and E is the activation energy. When adapting the monomolecular reaction mechanism (eqs 1−6 and 7) 10 parameters are needed: k1,mean, k−1,mean, k2,mean, k3,mean, k4,mean, E1, E−1, E2, E3, and E4. 3.5. Modeling of Polymerization. In experiments using an excess of STY or IND, polymerization reactions were also noted. The radical polymerization of STY is a chain reaction which comprises the stages of initiation, propagation, and termination (Figure 4).

Figure 4. Radical polymerization of STY.

The formation of radicals (M*) is typical for the polymerization reaction of STY. These radicals are extraordinarily reactive and readily react with monomers containing double bonds.10 STY is a monomer that polymerizes in the absence of initiators, by thermal initiation.11 The reaction rate for the initiation can be given as (reaction 5, Figure 4): r5 = k5CSTY

(8)

The propagation step involves the addition of further monomer molecules to the active end of the chain which carries an unpaired electron (reaction 7, Figure 4). The reaction rate for the propagation can be calculated accordingly:

Figure 3. (a−c). Calculated uncertainties from MCMC analysis for concentrations of reactants and products in Diels−Alder reaction of STY with DCPD at 463 K.

r7 = k 7CSTYCM * Here, kj is the apparent reaction rate constant of the reaction j, Ci is the concentration of component i, and t is the reaction time. The temperature dependence of the reaction rate coefficients is expressed by the Arrhenius law:

(9)

The growth of the polymer chain is stopped through termination in which the radicals disappear through bimolecular reaction (reaction 8, Figure 4): r8 = −2k 8CM *2 6359

(10)



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If eq 20 is multiplied by mL

The inhibitor reacts with the radicals to yield inactive products that do not participate in further polymerization. After the inhibitor is consumed, polymerization continues at the normal rate. The reaction rate by which the inhibitor (4tBC) decreases the amount of the radicals can be calculated using the equation (reaction 6, Figure 4) r6 = −2k6CM *2C4tBC

dn TP = k 8′mL CM *2 dt

Here nTP is the amount of TP. The differential mass change of TP can be calculated using eq 22 (Figure 4):

(11)

dm TP = MSTY dn1 + ... + nMSTY dnn + (n + 1)MSTY

The amount of the products of reaction 6 (Figure 4) is too low to be detected. Therefore, the existence of these products is an assumption.12 The reaction rate of the polymer radicals (Figure 4) can be calculated using eqs 8−11: rM * =

dnn + 1 + ... + (n + m)MSTY dnn + m

dn1 ≈ ... ≈ dnn ≈ dnn + 1 ≈ ... ≈ dnn + m = dn TP

dm TP = [1 + ... + n + (n + 1) + ... + (n + m)]MSTY dn TP

dm TP = [1 + ... + n + (n + 1) + ... + (n + m)]MSTY dt dn TP dt

dC M * = k5CSTY − 2k6C4tBCCM *2 − 2k 8CM *2 = 0 dt

= k 8MSTY mL CM *2

k5CSTY 2(k6C4tBC + k 8)

k 8 = k 8′[1 + ... + n + (n + 1) + ... + (n + m)]

dmL = −k 8mL MSTY CM *2 dt

rDCPD = (16)

+ k 8MSTY CM *2C DCPD

(28)

The production rate for other compounds can be calculated in a similar way: rCPD =

dCCPD dt

= 2k1C DCPD − 2k −1CCPD2 − k 2CCPDCSTY

(17)

+ k 8MSTY CM *2CCPD

The mass of the liquid phase of the reaction mixture as a function of time can be calculated using the equation mL = mL0 − m TP (18)

rSTY =

Here mL0 is the mass of the reaction mixture of the liquid phase at the beginning of the experiment and mTP is the mass of TP. After differentiation of eq 18 one obtains

(29)

dCSTY dt

= −k 2CCPDCSTY + k 3C PN 2 + k4C DCPDC PN − k5CSTY − k 7CSTYCM * + k 8MSTY CM *2CSTY

(19)

rPN =

The rate of reaction for TP production can be written as

dC TP = k 8′CM *2 dt

dC DCPD dt

= −k1C DCPD + k −1CCPD2 − k4C DCPDC PN

Here mL is the mass of the liquid phase of the reaction mixture. After derivation of the left-hand side of eq 16, the reaction rate of DCPD can be calculated using the equation:

dmL dm = − TP dt dt

(27)

Here MSTY is the molar mass of STY. When inserting eq 27 into eq 17

d(C DCPDmL ) = −k1mL C DCPD + k −1mL CCPD2 dt

dC DCPD = −k1C DCPD + k −1CCPD2 − k4C DCPDC PN dt C dmL − DCPD mL dt

(26)

and taking into account eq 19:

(15)

As a consequence of polymerization, the mass of the reaction mixture changed because the terminated polymer (TP) separated into its own phase. In that case, the following equation can be written for the rate of reaction of DCPD:

− k4mL C DCPDC PN

(25)

here

(14)

CM * =

(24)

If eq 21 is taken into account

(13)

If the same number of radicals disappear per unit of time as polymer radicals are formed, then the system is said to be in a stationary state. Based on this stationary state assumption, eqs 14 and 15 can be written as rM * =

(23)

Equation 22 can be written as

dC M * = r5 + r6 + r7 − r7 + r8 = r5 + r6 + r8 dt dC M * = k5CSTY − 2k6C4tBCCM *2 − 2k 8CM *2 dt

(22)

If it is assumed that

(12)

rM * =

(21)

(30)

dC PN dt

= k 2CCPDCSTY − 2k 3C PN 2 − k4C DCPDC PN + k 8MSTY CM *2C PN

(20) 6360

(31)



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Table 1. Apparent Reaction Rate Constants of Different Reactions (Figures 1 and 2) at Different Temperatures When Using Different Reactant Mixtures k reactants DCPD, STY

PN DCPD, PN DCPD, IN

a

1, min−1

temp, K 443 std error, 453 std error, 463 std error, 473 std error, 453 std error, 453 std error, 443 std error, 453 std error, 463 std error, 473 std error,

% % % %

−1a

0.0128 28.0 0.020 16.8 0.0537 32.4 0.11 103

2a

1.94 403.5 0.106 28945 4.25 188.6 5.56 273

3a 38.5 × 10 22.6 64.9 × 10−6 21.6 255 × 10−6 6.2 522 × 10−6 6.1 68.8 × 10−6 5.1 202 × 10−6 5.2 16.5 × 10−6 46.1 51.9 × 10−6 202 260 × 10−6 12.9 700 × 10−6 19.2

0.256 190.3 1.52 321.8 0.509 74.0 0.543 70.9

% % % % % %

0.023 4878 0.00812 5.6 0.0125 62.0 0.0542 113 0.101 543

4.76 5534 1.16 297.5 3.13 1617 22.8 512 5.66 976

4a −6

6.04 184.4 0.511 182.6 0.626 458 0.555 222 0.196 166

21.0 × 10−6 304.8 76.0 × 10−6 126.9 49.1 × 10−6 224.9 121 × 10−6 186

coeff of determination, % 98.1 99.3 99.6 99.5 99.2b

223 × 10−6 20.2 26.5 × 10−6 153 50.2 × 10−6 474 143 × 10−6 195.2 400 × 10−6 214.7

99.7 99.5 98.5 99.0 95.7

−1 b

Units: kg mol−1 min . Correlation coefficient.

Table 2. Activation Energies and “Mean” Apparent Reaction Rate Constants of Different Reactions (Figures 1 and 2) Fitted Using MCMC Method (All Temperatures) and Linearization (kj Fitted Separately) reaction reactants

parameter

DCPD, STY

all temperatures: Ej, kJ mol−1 std error, % kj,meana or b std error, % kj fitted separately: Ej, kJ mol−1 kj,meana or b correln coeff all temperatures: Ej, kJ mol−1 std error, % kj,meana or b std error, % kj fitted separately: Ej, kJ mol−1 kj,meana or b correln coeff

DCPD, IN

a

1

−1

2

3b

4b

126 23.3 0.0357 38

54.5 859 5.07 294

36.7 622 0.462 137

346 21.3 32.4 × 10−6 64

79.3 298 59.5 × 10−6 212

129 0.0359 0.989

62.1 3.41 0.998

46.0 0.399 0.971

160 138 × 10−6 0.985

84.2 56.2 × 10−6 0.837

136 24 0.0321 31

50.4 2580 3.65 763

52.6 1051 0.771 331

250 21 141 × 10−6 39

167 241 112 × 10−6 270

157 0.028 0.977

87.0 3.03 0.960

7.25 0.574 0.416

224 116 × 10−6 0.996

160 95.8 × 10−6 0.992

a

b

b

coeff of determination, %

96.5

−

93.5

−

Units: min−1. bUnits: kg mol−1 min−1.

rPTeCDD =

dC PTeCDD = k 3C PN 2 + k 8MSTY CM *2C PTeCDD dt

3.6. Modeling Experiments of PN and IN Production. At the beginning of the modeling procedure, the results of the STY/DCPD and IND/DCPD experiments at 443, 453, 463, and 473 K were modeled separately in order to get values for the apparent reaction rate constants of the separate reactions (see eqs 1−6 and Table 1). The estimated reaction kinetic parameters and traditional error estimates are presented in Table 1. The coefficient of determination was over 95%, indicating good agreement between the measured and calculated concentrations.

(32)

rTCPD =

dC TCPD = k4C DCPDC PN + k 8MSTY CM *2C TCPD dt (33)

r4tBC =

dC4tBC = −k6CM *2C4tBC + k 8MSTY CM *2C4tBC dt (34) 6361



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With the benefit of these values, approximations for the values of activation energies and apparent reaction rate constants at some “mean” temperature were calculated (Table 2). These approximations were used as initial guesses at the beginning of the parameter fitting procedure. The existence of reaction 3 was confirmed using PN as a reactant (Figure 1). The apparent reaction rate constant (k3) could be approximated using the results of this experiment because the differential equation can be solved analytically. The kinetics for this reaction can be presented as

dC PN = −2k 3C PN 2 dt

(35)

When eq 35 is integrated, one obtains 1 1 − = −2k 3t C PN C PN0

(36)

Here CPN0 is the concentration of PN at the reaction time t = 0 min. The apparent reaction rate constant was determined using linear regression (Table 1). Reaction 4 (Figure 1) was also confirmed using DCPD and PN as reactants (Table 1). Following this preliminary parameter determination, activation energies and apparent reaction rate constants at some “mean” temperature were determined using all experiments at temperatures of 443, 453, 463, and 473 K (Table 2). Additionally, in order to test the polymerization kinetics, two experiments were performed using an excess of dienophile (STY/DCPD 86/14, IND/DCPD 84 mol/16 mol). When the polymerization kinetics (eqs 28−34) was included in the kinetic model, the fitting results were about the same as without the polymerization reactions (eqs 1−6). The amounts of inhibitor (4tBC) and its assumed reaction products (reaction 6 in Figure 4) were below detection limits.

Figure 5. One-dimensional (1D) marginal posterior distribution plots for the apparent reaction rate constants of reactions 1, −1, 2, 3, and 4 when DCPD reacts with STY at 453 K.

4. MCMC ANALYSIS The equations for the reaction kinetic model form a system of ordinary differential equations which can be integrated starting from the initial conditions. The reaction kinetic parameters were first estimated with standard least-squares fitting by minimizing the squared difference between the measured and calculated concentrations. To evaluate the accuracy of estimated parameters in a nonlinear multiparameter model, it is important to consider possible cross-correlation and the identifiability of the parameters. Classical statistical analysis that gives optimal parameter values, their error estimates, and correlations between them is approximate (based on linearization of the model) and may sometimes be quite misleading, especially if the available data are limited and the parameters are poorly identified. Moreover, the question of the reliability of the model predictions remains unaddressed, i.e., how the uncertainty in the model parameters is reflected in the model response (Figures 5, 6, 7, and 8). Both these problems may be treated by MCMC methods. Using MCMC methods, estimation of model parameters and model predictions are performed according to a Bayesian paradigm. All uncertainties in the data and the modeling results are treated as random variables that have statistical distributions. Instead of a single fit to the data, “all” the parametrizations of the model that statistically fit the data “equally well” are determined. A distribution of the unknown parameters is generated using available prior information (e.g., results obtained from previous studies or bound constraints for

Figure 6. 1D marginal posterior distribution plots for the apparent reaction rate constants of reactions 1, −1, 2, 3, and 4 when DCPD reacts with PN at 453 K.

the parameters) and statistical knowledge of the observation noise. Computationally, the distribution is generated using the MCMC sampling approach. Up-to-date adaptive computational schemes are employed in order to make the simulations as effective as possible.13,14 In this study, a FORTRAN 90 software package MODEST15 was used for both the least squares and the MCMC estimation. The referred to methods are also implemented in a MATLAB package.14,16 For 6362



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Figure 7. 1D marginal posterior distribution plots for Arrhenius parameters when using DCPD and STY as reactants (see Table 2 and Figure 1).

CPD + STY → PN

completeness, a short introduction to Bayesian parameter estimation and MCMC is given in the Supporting Information. Here, MCMC was found to be a useful tool for studying the identifiability of parameters, i.e., which parameters can be reliably determined from the data. In many experiments, the available data were rather limited, and some parameters remained unidentified (see Figures 5−8). In such cases, classical nonlinear regression analysis that relies on local linearizations and Gaussian approximations can give misleading results. With MCMC, one can study in detail which parameter values are supported by the data, and obtain valuable information to extend and support the classical analysis, which usually only gives approximations of the standard deviations of the parameters. Moreover, one can study how the parameter uncertainty is translated into the model response and model predictions, which cannot be easily done with classical regression analysis (see Figure 3).

2PN → PTeCDD or its isomer + STY DCPD + PN → TCPD + STY

The formation of CPD and the formation of PN were much faster reactions than the formation of PTeCDD and TCPD. The formation of PTeCDD was verified using only PN as a reactant. Respectively, the formation of TCPD was verified using PN and DCPD as reactants. A similar kind of mechanism was assumed when IND was used as a reactant. The apparent reaction rate constants and Arrhenius parameters were estimated for the pseudohomogeneous model, and their reliability was studied using MCMC analysis in addition to classical regression analysis. When the apparent reaction rate constants for the different reactions were determined in separate experiments, the coefficient of determination was almost always about 99%. The calculated concentration curves using the determined parameters explained the experimental results rather well (see Figure 3a,b) with high reliability. However, when the concentration of the compound in the reaction mixture was very low, the calculated reliability for model predictions was not as good (see Figure 3c). Although a good fit between the model and the data was obtained, many of the parameters remained unidentified, as seen from the high standard errors (see Table 1) and the parameter histograms computed from the MCMC analysis (see Figure 5). This means that there are many parameter combinations with which the model gives a good fit to the data. The MCMC analysis reveals that the parameters k1

5. SUMMARY AND CONCLUSIONS The Diels−Alder reaction kinetics of CPD with STY and IND was studied experimentally using DCPD, STY, and IND as reactants. In these reactions, CPD reacts as diene, whereas STY and IND are dienophiles. At the beginning of the reaction mechanism, DCPD reacts according to an equilibrium reaction to produce CPD: DCPD ⇆ 2CPD

When STY was used as a reactant, the following reaction mechanism was assumed: 6363



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Figure 8. 1D marginal posterior distribution plots for the apparent reaction rate constants when polymerization has been taken into account (Figures 2 and 4) using DCPD and IND as reactants at 453 K.

and k3 seem to have a unique optimum, whereas the rest of the parameters cannot be reliably estimated. Some information was obtained for k2 and k4: k2 seems to prefer larger values and k4 prefers small values. The distribution for k−1 is flat, which means that all parameter values within the bounds yield a good fit to the data, and the data have essentially no information about k−1. The identifiabilities of the apparent reaction rate constants are slightly different when PN and DCPD are used as reactants (see Figure 6). Now, k3 and k4 seem to have unique optima, and the rest of the parameters have identifiability problems. When the parameters of the modified Arrhenius law were determined using experiments performed at all temperatures, the coefficient of determination was over 93%. First, the Arrhenius law parameters were determined by linearizing the reaction rate constant fitted at different temperatures. Then, the Arrhenius law parameters were determined using the experimental results at all temperatures in one parameter determination procedure. The separately determined parameters corresponded to each other rather well, with certain exceptions (see Table 1). For example, when DCPD and IN were used as reactants, a reaction rate constant k2 of 0.511 kg mol−1 min−1 was obtained at 443 K. At the temperature of 473 K, the corresponding value was lower, 0.196 kg mol−1 min−1. Because the value of the apparent reaction rate constant decreased substantially at higher temperature, it has been omitted from the Arrhenius plot (see Table 1). The MCMC analysis revealed how well the parameters are identified when using experiments at all temperatures (see Figure 7): 7 out of the 10 parameters are somewhat identified, but three parameters (k−1,mean, E−1, and E2) cannot be reliably determined from the data. The identifiability problems are also indicated by

high standard errors in classical regression analysis (see Table 2), which mostly agree with the MCMC results. However, there are some differences: for instance, the standard error for k2,mean is overestimated compared to MCMC. When excess of dienophile (STY or IND) was used as a reactant, the polymerization reaction took place (see Figure 4). Although the mass of the polymer phase was rather small, polymerization was taken into account when modeling experiments of excess dienophile. When polymerization kinetics was taken into account, the obtained fit was rather good, but, according to the MCMC analysis, the apparent reaction rate constants were poorly identified (see Figure 8). In fact, the data seem to contain information only about the reaction rate constant k1, whereas the other parameters cannot be reliably estimated. In this experiment, there were too many parameters and a lack of essential data concerning the amount of inhibitor (4tBC) and the mass of the polymer phase as a function of reaction time to permit the parameters to be identified. In this paper, we demonstrated how MCMC methods can be used to study what the experimental data tell about kinetic parameters. MCMC can be used to compute the underlying distribution of the parameters in nonlinear models without having to use the linear approximations on which classical regression analysis is based. The approach can give valuable, detailed information about the identifiability of the fitted parameters.

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ASSOCIATED CONTENT

S Supporting Information *

Appendix: Bayesian Inference and MCMC. A short introduction to Bayesian parameter estimation and MCMC methods. 6364



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This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: kari.vahteristo@lut.fi. Fax: +358 05 6212199. Notes

The authors declare no competing financial interest.

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Article

ACKNOWLEDGMENTS

The authors are grateful to Matti Lindström, Erkki Halme, and Mika Perälä17 for their advice and help performing the experiments. Neste Oy is gratefully acknowledged for financial support.

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NOMENCLATURE C = concentration, mol kg−1 CPD = cyclopentadiene E = activation energy, kJ mol−1 DCPD = dicyclopentadiene IND = indene IN = indenylnorbornene k = apparent reaction rate constant, min−1, kg mol−1 min−1 kj,mean = parameter in eq 7, min−1, kg mol−1 min−1 M = molar mass, kg mol−1 M* = radicals m = mass, g m = integer MCMC = Markov chain Monte Carlo n = amount, mol n = integer PN = 5-phenyl-2-norbornene PTeCDD = phenyltetracyclododecene OHBF = 1,2,5,6-diendomethyleneoctahydrobenzofluorene R = gas constant, J K−1 mol−1 r = reaction rate, mol kg−1 min−1 STY = styrene T = temperature, K Tmean = average temperature, 456 K TCPD = tricyclopentadiene TP = polymer t = reaction time, min 4tBC = inhibitor, 4-tert-butylcatechol

Subscripts and Superscripts

CPD = cyclopentadiene DCPD = dicyclopentadiene i = component j = reaction L = liquid phase L0 = liquid phase at the beginning of the experiment M* = radicals PN = 5-phenyl-2-norbornene PTeCDD = phenyltetracyclododecene STY = styrene TCPD = tricyclopentadiene TP = polymer 4tBC = inhibitor, 4-tert-butylcatechol −1, 1, 2, 3, 4 = reactions in Figures 1 and 2 5, 6, 7, 8 = reactions in Figure 4 6365


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