Article pubs.acs.org/Macromolecules
Quantifying the Copolymerization Kinetics of Ethylene and 1‑Octene Catalyzed with rac-Et(Ind)2ZrCl2 in a Solution Reactor Saeid Mehdiabadi and Joaõ B. P. Soares* Department of Chemical and Materials Engineering University of Alberta Edmonton, Alberta Canada ABSTRACT: We developed a new methodology to estimate the crosspropagation rate constants and their confidence regions for the copolymerization of ethylene and 1-olefins using single site catalysts. We applied the method to a series of ethylene/1-octene copolymers made with rac-Et(Ind)2ZrCl2/ MAO in a solution reactor operated in semibatch mode. The method estimates the reactivity ratios using the Mayo−Lewis equation and the cross-propagation rate constants with combination of the Mayo−Lewis equation and a polymerization kinetics model based on monomer uptake curves. More importantly, our strict statistical treatment of the data allows us to estimate the joint confidence regions for these parameters, which is essential to establish the reliability of these estimates.
■
INTRODUCTION Polymer reaction engineering models usually combine multiple elements, such as polymerization kinetics, thermodynamics, transport properties, and reactor configuration to describe how laboratory-scale and industrial reactors behave.1−6 The level of complexity of these models, besides depending on the polymerization process itself, also depends on the purpose to which we intend to use the model. Is this a process control model, a reactor scale-up model, or a product quality control model? For example, in fluidized-bed polymerization reactors, differences in fluidization regimens expose the growing polymer particles to different velocity fields and thus heat transfer conditions, which may lead to particle overheating and agglomeration, eventually causing the bed to collapse.7 This, in turn, affects intraparticle rates of polymerization and the microstructure of chains made at different radial positions in the polymer particle. For solution polymerization reactors, non ideal flow and residence time distributions will also add more complexity to the reactor model if we are interested in predicting the detailed microstructure of the polymer.8−11 Depending on polymerization conditions and on the purpose we have in mind for the model, we may chose to neglect some of these modeling elements, except the polymerization kinetics component, which remains the key element as it is at the heart of any polymer reaction engineering model. Polymerization kinetics models are exceptional tools to store the knowledge generated in polymerization experiments. They are indispensable to scale-up polymerization conditions from laboratory-scale reactors to industrial conditions. They are also used in the design, optimization, and control of polymerization reactors.12,13 Another important use of polymerization kinetics models is to quantify how polymer microstructure depends on polymerization conditions.14,15 If we know the polymerization kinetics model for a given system, we can predict the polymer © XXXX American Chemical Society
microstructure made in any type of reactor by introducing the thermodynamics, transport phenomena, and mixing characteristics associated with that type of reactor. These models also help elucidate many qualitative questions related to polymerization mechanisms and catalyst behavior in a more definitive way by providing quantitative predictions to different competing hypothesis. Quoting Lord Kelvin: “When you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind.” [Lecture on “Electrical Units of Measurement” (3 May 1883), published in Popular Lectures Vol. I, p. 73.] Polymer reaction engineering models provide the quantification needed to improve our understanding of polymerization processes. Kinetic models can also be used to predict the microstructure of the polymers made using two or more metallocene catalysts.16 We need to know the kinetics of polymerization of both catalysts to make polymers with the proper balance of properties. For instance, a polyolefin with bimodal molecular weight distribution will be produced only if the mass fractions and ratios of molecular weight averages of the polymers made by the two catalysts are within a specified range.17−19 Since these variables are sensitive to polymerization temperature, monomer pressure, catalyst and hydrogen concentrations, a polymerization kinetics model is essential to properly control the properties of polymers made with dual single-site catalysts. Polymerization kinetics models contain unknown parameters whose values are sometimes difficult to estimate, even using specialized equipment and applying sophisticated design of experiment techniques. Estimation of uncertainty associated with the parameters is another problem that requires more Received: December 21, 2015 Revised: March 3, 2016
A
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
using kpAA estimates for the homopolymerization of ethylene in the absence of 1-olefin comonomer. The method we are proposing herein eliminates these technical difficulties by estimating all the four cross propagation constant simultaneously, by extending our ethylene polymerization kinetics study with the rac-Et(Ind)2ZrCl2/MAO to the copolymerization of ethylene and 1-octene, and will show how we can estimate the cross-propagation rate constants and their respective joint confidence regions. To the best of our knowledge, this is the first time the method we are proposing was applied to this or any other metallocene-catalyzed copolymerization.
rigorous attention than estimation of the parameters. In our previous publication,20 we described the kinetics of ethylene polymerization with rac-Et(Ind)2ZrCl2/MAO using first-order reactions for polymerization and catalyst deactivation. In another article,21 we refined the statistical treatment of our previous work for propagation rate constants and determined the chain-transfer constants to MAO and ethylene and by βhydride elimination. The activation energies for propagation and for the leading chain-transfer steps were also determined in that investigation. Galland et al.22 estimated the reactivity ratios for ethylene/1octene copolymerization with rac-Et(Ind)2ZrCl2/MAO under different experimental conditions from those studied in the present investigation using a different estimation approach that did not compute the joint confidence interval of the copolymerization parameters. They reported the following polymerization conditions: Al/Zr = 1750, solvent = toluene, temperature 60 °C, ethylene pressure 1.6 bar, and reaction time 30 min. They calculated reactivity ratios using the Fineman− Ross method, and from 13C NMR comonomer sequence data according to a method described by Soga and Uozumi.23 The values reported for rA and rB were 59 ± 10 and 0.004 ± 0.002, respectively, by the 13CNMR method and 45 ± 10 and 0.05 ± 0.03 by the Fineman−Ross method. The reliable estimation of cross-propagation constants for ethylene/1-olefin copolymerization with coordination catalysts has been plagued for a long time with experimental and methodological difficulties, some of which we propose to ameliorate with the method we are proposing in this article. Even in the case of solution polymerization with single-site catalysts, wherein we may ignore intraparticle mass and heat transfer resistances associated with slurry and gas phase polymerizations with supported catalysts, or the even more complex case of multiple-site heterogeneous Ziegler−Natta catalysts, estimation of the cross-propagation constants involving the 1-olefin comonomer (kpAB, kpBA, and especially kpBB, where A = ethylene and B = 1-olefin) is very challenging, which leads to large unknown uncertainties in the estimates for these constants. One of they technical problems associated with this estimation is because the 1-olefin comonomer is usually fed in batch mode, at the beginning of the polymerization; therefore, we do not know the consumption rate of 1-olefin during the polymerization, differently from ethylene, which is customarily measured with an online flow meter. One rather inconvenient partial solution for this problem could conceivably be to run several batch homopolymerizations of the 1-olefin comonomer at different times, recover and weigh the formed poly(1-olefin), and fit the yield versus polymerization time curve to obtain a rough estimate for kpBB. This approach, besides being tedious and time-consuming, would lead to large uncertainties in the kpBB estimates because of batch-to-batch variations, difficulties in separating the low molecular weight, low crystallinity poly(1-olefin) from the solvent, and accurately weighing it due to low yields, and very likely 1-octene concentration drift during the polymerization (needed to obtain yields large enough to be weighed). Perhaps more importantly, due to the well known comonomer effect, even if accurate, these kpBB estimates would not reflect the actual values for kpBB in the presence of ethylene, since we know that addition of 1-olefin comonomers affect the rate of ethylene polymerization substantially in a way that is not predicted by the traditional terminal model. The same reasoning applies for
■
EXPERIMENTAL SECTION
Materials. Methylaluminoxane (MAO, 10 wt % in toluene) from Sigma-Aldrich was used as received. We purified ethylene and nitrogen (Praxair) by flowing these gases through beds packed with molecular sieves (a mixture of 4-Å and 133 sieves) and copper(II) oxide. Toluene (EMD) was purified by refluxing it over an n-butyllithium/styrene/ sodium system for 40 h and then by distilling it under nitrogen atmosphere. We purchased rac-ethylenebis(1-indenyl) zirconium dichloride (rac-Et[Ind]2ZrCl2) as a powder from Sigma-Aldrich, and dissolved it in distilled toluene before polymerization. All air-sensitive compounds were handled in a glovebox under inert atmosphere. Polymer Synthesis. All polymerizations were performed in a semibatch reactor. The reaction medium was mixed using a pitchedblade impeller connected to magneto-driver stirrer, rotating at 2000 rpm. Prior to use, the reactor was heated to 125 °C, evacuated, and refilled with nitrogen six times to reduce the oxygen concentration, before we charged it with 250 mL of toluene and 0.5 g of AliBu3 as a scavenger. The temperature was then increased to 120 °C and kept constant for 20 min. Finally, the reactor contents were blown out under nitrogen pressure. This procedure ensured excellent removal of impurities from the reactor walls. In a typical polymerization run, we charged 200 mL of toluene into the reactor and introduced an appropriate amount of MAO via a 5 mL tube and a 20 mL sampling cylinder connected in series with an ethylene pressure differential of 40 psig. A specified volume of toluene was placed in the sampling cylinder before injection to wash the tube walls from any MAO solution. We used the same method to inject the catalyst solution into the reactor. Monomer was supplied on demand to maintain a constant reactor pressure of 120 psig and monitored with a mass flow meter. With the exception of a 1−2 °C exotherm upon catalyst injection, the online controller kept the temperature at 120 °C ± 0.15 °C throughout the polymerization. After 15 min, we closed the monomer valve to stop the polymerization and immediately blew out the reactor contents into a 1-L beaker filled with 400 mL of ethanol. The polymer produced was kept overnight, filtered, washed with ethanol, dried in air, and further dried under vacuum. Polymer Characterization. Molecular weight distributions (MWD) were determined with a Polymer Char high-temperature gel permeation chromatograph (GPC), operated at 145 °C under a flow rate of trichlorobenzene of 1 mL/min. The GPC was equipped with three detectors in series (infrared, light scattering, and differential viscosimeter) and calibrated with polystyrene narrow standards using the universal calibration curve. We used a Polymer Char CEF-21 unit to determine the copolymer composition by thermal gradient interaction chromatography (TGIC). In a typical analysis, we dissolved the polymer sample in 8 mL of TCB at a concentration of 0.4 mg/mL, heated the polymer solution to 160 °C, held it for 1 h at this temperature to ensure complete dissolution, then decreased the temperature to 40 °C under a cooling rate of 2 °C/ min. The flow rate during the cooling cycle was 0.03 mL/min. The heating rate during elution cycle was 2 °C/min, and the elution pump flow rate was 0.5 mL/min. The 13C NMR spectrum was acquired on a Bruker 500 MHz system. The probe temperature was set at 120 °C. Acquisition parameters were optimized for quantitative NMR, including a 14 μs B
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules 90° pulse, inverse gated proton decoupling and 10 s delay time between pulses. A total of 10 000 scans were used for data averaging. The biggest peak was referenced to 30.0 ppm. Deuterated odichlorobenzene was used to obtain the field-frequency lock.
■
RESULTS AND DISCUSSION We made 12 homopolymers and copolymers, in random order, at seven equally spaced 1-octene concentrations to study the copolymerization kinetics of ethylene and 1-octene using Et(Ind)2ZrCl2/MAO. Except for the concentration of 1-octene, we kept all reaction variables constant during the polymerizations. Table 1 lists the experimental conditions for these polymerizations, their polymer yields, and molecular weight averages. Table 1. Polymerization Conditions and Polymer Average Propertiesa run
1-octene in feed (g)
polymer yield (g)
Mw
Mn
PDI
H-0A H-4A H-12A H-0B H-8A H-12B H-8B H-4B H-8C H-4C H-0C H-16A
0 4 12 0 8 12 8 4 8 4 0 16
4.4 3.78 3.34 4.05 3.2 3.3 3.49 3.75 3.65 3.7 4.5 2.98
24250 23300 22200 25300 22800 23200 23300 22400 23300 23600 24800 22100
51700 51000 48400 52000 50400 50200 48700 50000 50600 49600 50900 47600
2.13 2.18 2.18 2.05 2.2 2.16 2.1 2.22 2.17 2.1 2.05 2.16
Figure 2. TGIC profiles of the samples listed in Table 1.
[Zr] = 13.3 nmol/L, [Al] = 15.5 mmol/L, T = 120 °C, P = 120 psig, tp = 15 min, Total solvent volume =223 mL.
a
Figure 1 shows ethylene uptake curves for all polymerizations listed in Table 1. Increasing the concentration of 1-octene
Figure 3. TGIC peak temperature as a function of 1-octene loading.
The reaction mechanism proposed in our recent study21 for ethylene homopolymerization with rac-Et(Ind)2ZrCl2 /MAO explains the monomer uptake profiles shown in Figure 1. In this model, ethylene propagation and catalyst deactivation are first order reactions with respect to monomer and catalyst concentration, respectively. We also found out that chain transfer to ethylene is the main transfer reaction for generation of vinyl-terminated polymers, and that β-hydride elimination is negligible. Chain transfer to cocatalyst was found to be a first order reaction on cocatalyst concentration. Considering these facts, we used the terminal model (Table 2) to describe the copolymerization of ethylene and 1-octene with Et(Ind)2ZrCl2/ MAO. Site activation (the reaction between catalyst and cocatalyst to produce active sites) was assumed to be instantaneous. We have also assumed that deactivation of living chains, bearing catalyst sites on one end, did not depend on the type of monomer last added to the chain; therefore, we assumed that all deactivation reactions had the same rate constant. In contrast to the Bernoullian model, which assumes that the value of the propagation rate constant depends only on the type of comonomer coordinating to the active site, the terminal model assumes that the type of comonomer last inserted into the polymer chain also influences the propagation rate constant. The terminal model is a closer depiction of olefin copolymerizations with metallocene catalysts as the type of
Figure 1. Ethylene uptake curves for different 1-octene concentrations. The legends show 1-octene concentrations in the liquid phase.
decreases monomer uptake rate. The TGIC profiles of all samples were unimodal (Figure 2), indicating that the catalyst had single-site behavior. The TGIC peak temperature decreased when we increased the concentration of 1-octene because more comonomer was incorporated in the copolymer (Figure 3). C
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules Table 2. Copolymerization Modela initiation
k iA
(1)
k iB
(2)
C* + A ⎯→ ⎯ P1A C* + B ⎯→ ⎯ P1B
propagation
PrA+ 1
(3)
PrB+ 1
(4)
PrB + A ⎯⎯⎯⎯→ PrA+ 1
k pBA
(5)
k pBB
(6)
PrA PrA
k pAA
+ A ⎯⎯⎯⎯→ k pAB
+ B ⎯⎯⎯⎯→
PrB + B ⎯⎯⎯⎯→ PrB+ 1 deactivation
kd
(7)
kd
(8)
kd
(9)
PrA → Cd + Dr
PrB → Cd + Dr
C* → Cd a A Pr
Since f B = 1 - fA and ϕB = 1 − ϕA, we conclude that for a binary copolymer, ϕA =
kpBAfA kp AB(1 − fA ) + kpBAfA
(15)
Hence, the value of ϕA will be constant if the molar fraction fA is kept constant during polymerization. The pseudopropagation rate constant kp̂ will also be constant in this case. The molar balance for living polymer chains is d[Y0] = −kd[Y0] + ki A[C*][A] + ki B[C*][B] dt
(16)
d[Y0] = −kd[Y0] + kî [C*][M] dt
(17)
or,
DAr
and are living and dead chains ending with monomer A (ethylene), respectively; Ci, are monomer-free active sites; A and B are ethylene and 1-octene, respectively; and Cd are deactivated catalyst sites.
where we defined the pseudoinitiation constant kî as kî = (ki AfA + ki BfB )
(18)
Similarly, the molar balance for the activated catalyst sites is, d[C*] = −kî [C*][M] − kd[C*] dt
monomer added to the chain influences the value of propagation rate constant for the next propagation step. This makes the estimation problem more involved because the terminal model needs four propagation rate constants instead of two (Table 2). Higher order models, such as the penultimate model, have also been proposed for some copolymerizations, but due to little supporting experimental data they are rarely used, particularly for olefin copolymerization. The rate of polymerization with the terminal model is expressed as,
Solving eqs 17 and 19 with the initial conditions [C*] = [C*]0 and [Y0] = 0 yields ̂
[Y0] = [C*]0 (e−kdt − e−(kd + ki[M])t )
[Y0] = [C*]0 e−kdt
FA, in d[A] = − (kp AA[Y0A ] + kpBA[Y0B])[A] dt VR
(10)
FA, in
(11)
VR
where ϕA and ϕB are the molar fractions of living polymer chains terminated in monomer A and B, respectively, fA and f B are the mole fractions of comonomers A and B in the liquid phase, [M] = [A] + [B] is the total comonomer concentration at the active sites, and [Y0] = [Y0A] + [Y0B]. eq 11 is more conveniently expressed as,
FA, in VR
(23)
= (kp AA(1 − ϕB) + kpBAϕB)[Y0][A]
(24)
Substituting eq 21 in eq 24 and rearranging FA, in
(12)
VR
= (kp AA − (kp AA − kpBA )ϕB)[C*]0 e−kdt [A]
(25)
eq 25 can be rewritten into the more convenient form, (13)
⎛ FA, in ⎞ ln⎜ ⎟ = ln((kp AA − (kp AA − kpBA )φB)[C*]0 [A]) − kdt ⎝ VR ⎠
It is easy to calculate the parameters ϕA and ϕB using the long chain approximation, kp ABϕA fB = kpBAϕBfA
= (kp AA[Y0A ] + kpBA[Y0B])[A]
or
where we defined the pseudo kinetic constant kp̂ as kp̂ = kp AAϕA fA + kp ABϕA fB + kpBAϕBfA + kpBBϕBfB
(22)
where VR is the volume of the reaction medium, FA,in is the molar flow rate of ethylene to the reactor, and [A] is the ethylene concentration. Since ethylene concentration is kept constant during polymerization in a semibatch reactor, eq 22 simplifies to
RP = (kp AAϕA fA + kp ABϕA fB + kpBAϕBfA + kpBBϕBfB )[M]
RP = kp̂ [M][Y0]
(21)
The molar balance for ethylene, A, in the reactor is
where Y0A and Y0B are the number of moles of living polymer chains terminated in monomer A and B, respectively, and kpIJ is the propagation rate constant when monomer I reacts with polymer chains ending with monomer J. eq 10 can be written more conveniently as,
[Y0]
(20)
The second exponential term in eq 20 may be neglected because it contains the large term kî [M]. Thus, eq 20 simplifies to
RP = (kp AA[A] + kp AB[B])[Y0A ] + (kpBA[A] + kpBB[B]) [Y0B]
(19)
(26)
(14)
or, D
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules ⎛ FA, in ⎞ ln⎜ ⎟ = ln(kp̃ ϕB[C*]0 [A]) − kdt ⎝ VR ⎠
Table 3. Pseudo-Kinetic Parameters (27)
where we defined the apparent propagation rate constant kp̃ as kp̃ = (kp AA − (kp AA − kpBA )ϕB)
(28)
Since ϕB is a function of f B, we can also express kp̃ as a function of f B. Substituting eq 15 in eq 28 results in the expression, kp̃ = kp AA − (kp AA − kpBA )
kp ABfB kpBA(1 − fB ) + kp ABfB
(29)
This equation is still not suitable for curve fitting because it has three adjustable parameters. Equations having three parameters have large degrees of freedom. Therefore, we substituted the reactivity ratio rA into eq 29 to reduce its number of adjustable parameters kp AA
kp̃ = kp AA − (kp AA − kpBA )
rA
run
kd [s−1]
kp̃ [L·mol−1·s−1]
H-0A H-4A H-12A H-0B H-8A H-12B H-8B H-4B H-8C H-4C H-0C H-16A
0.001 08 0.001 17 0.001 21 0.001 08 0.001 21 0.001 19 0.001 24 0.001 18 0.001 18 0.001 15 0.001 12 0.001 21
204636 181170 160792 188799 154707 157781 170840 181152 174107 177693 211269 142857
fB
kpBA(1 − fB ) +
kp AA rA
fB
(30)
Even though it may seem that the number of adjustable parameters in eq 30 remained the same as in eq 29, rA can be estimated independently with the Mayo−Lewis equation using data on comonomer concentration in the reactor versus comonomer molar fraction in the polymer chains. FA, in
( ) versus time should be a
According to eq 27, a plot of ln
VR
straight line with slope − kd, and intercept ln(kp̃ ϕB[C*]0 [A]). We replotted the monomer uptake curves in Figure 1 using eq 27. All curves follow the expected linear trend, as illustrated in Figure 4, indicating that our model describes the ethylene uptake in these polymerizations well.
Figure 5. Plot of kp̃ and kd versus 1-octene concentration in the liquid phase.
decreases as the concentration of 1-octene increases, as expected since 1-octene is less reactive than ethylene. eq 29 quantifies the decrease in k p̃ , and we can estimate the values of kpAA and kpBA by fitting the k p̃ data with eq 30, but first we need to estimate the comonomer reactivity ratios for this catalyst. Reactivity Ratios. To determine the reactivity ratios for ethylene/1-octene using rac-Et[Ind]2ZrCl2/MAO, we used the TGIC elution peak temperature data, a TGIC calibration curve, the Mayo−Lewis equation, and a method to estimate 1-octene and ethylene concentrations in the liquid phase at polymerization conditions. Figure 6 shows the TGIC profiles of a set of ethylene/1-octene copolymers with known 1-octene fractions. We made these calibrant copolymers using a constrained geometry catalyst (CGC-Ti) and analyzed them with 13C NMR to determine their commoner contents.24 Figure 7 illustrates the TGIC calibration curve determined with these copolymer samples. eq 31 is the TGIC calibration curve, relating TGIC peak temperature to 1-octene fraction in the copolymer,
Figure 4. Plot of ln(F/VR) versus polymerization time.
We estimated the kd and k p̃ from the slopes and intercepts of the lines in Figure 4. Table 3 summarizes the values of these parameters. Figure 5 shows how kd and kp̃ depend on the concentration of 1-octene in the feed. The value of kd increases slightly as the concentration of 1-octene increases, perhaps because 1-octene carries impurities that poison the catalyst. The values for k p̃
1‐octene mol % = −0.20327 × Tpeak + 27.5434
(31)
The Mayo−Lewis equation, relating mole fraction of ethylene in the copolymer, FA, to ethylene mole fraction in the feed, fA, is given by the expression, E
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
Robison equation of state. Figure 8 shows that the relationship between the mass of 1-octene added to the reactor and the resulting 1-octene concentration in the liquid phase is almost linear.
Figure 6. TGIC profiles of the copolymer samples used for calibration. Legends show 1-octene incorporated in the copolymer in mole percent.
Figure 8. Relationship between 1-octene concentration in the liquid phase and the amount of 1-octene added to the reaction mixture (Aspen Plus and Peng−Robinson equation of state).
Table 4 summarizes TGIC peak temperatures and their corresponding 1-octene contents, calculated using the TGIC Table 4. Summary of Copolymerization Runs for Samples Made Using rac-Et[Ind]2ZrCl2/MAO
Figure 7. TGIC calibration curve.
FA =
rAfA 2 + fA fB rAfA 2 + 2fA fB + rBfB 2
(32)
The variable fA is calculated using the ethylene and 1-octene molar concentrations in the reaction medium, [A] and [B], respectively, fA = 1 − fB =
[A] [A] + [B]
(33)
kAA kAB
(34)
rB =
kBB kBA
(35)
1-octene (mol/L)
Tpeak (°C)
fA
1-octene in polymer (mol %)
FA
H-0A H-4A H-12A H-0B H-8A H-12B H-8B H-4B H-8C H-4C H-0C H-16A
0.0 0.1339 0.3823 0.0 0.2611 0.3823 0.2611 0.1339 0.2611 0.1339 0.0 0.4979
135.41 133.1 128.5 135.41 130.79 128.57 130.81 133.07 130.95 132.96 135.41 126.28
1 0.7699 0.5393 1 0.6316 0.5393 0.6316 0.7699 0.6316 0.7699 1 0.4733
0.000 0.488 1.423 0.000 0.958 1.409 0.954 0.4944 0.925 0.517 0.000 1.874
1.0000 0.9951 0.9858 1.0000 0.9904 0.9859 0.9905 0.9951 0.9907 0.9948 1 0.9813
calibration curve, and their FA and fA values. Since the 1-octene conversions for all runs were low, we assumed that changes in their concentration were negligible during the polymerizations (negligible composition drift). Figure 9 plots average 1-hexene mole percent in the copolymer versus 1-hexene concentration in the liquid phase inside the reactor. We fitted eq 32, the Mayo−Lewis equation, to the experimental data to estimate rA and rB. The estimates were obtained by minimization of the sum of the squares of the residuals between model predictions and experimental data. The confidence region for rA and rB was calculated using the equation below,25
where rA and rB are the reactivity ratios, given by, rA =
run
and kAA and kBA are propagation rate constants when ethylene reacts with polymer chains ending with ethylene or 1-octene, respectively, and kAB and kBB are the corresponding propagation rate constants when 1-octene reacts with polymer chains ending with ethylene or 1-octene, respectively. We estimated the concentrations of ethylene, 1-octene, and toluene in the liquid phase using Aspen Plus and the Peng-
ri ± tα /2, n − ps (X ′X )ii−1 F
(36) DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
Figure 10. Copolymer composition versus liquid phase composition for ethylene/1-octene copolymers made with rac-Et(Ind)2ZrCl2/ MAO. The dashed curve shows FA values calculated with eq 33.
Figure 9. Average 1-octene mole percent in copolymer versus 1octene concentration in the liquid phase for copolymers made with rac-Et[Ind]2ZrCl2/MAO.
where n and p are the number of data points and parameters, respectively. The term tα/2,n−p is the upper 100 × α/2 percentage point of the t-distribution with n − p degrees of freedom.25 X is the Jacobian evaluated at the optimum value of parameters ⎡ ∂g (f , r ) ⎤ A ⎥(i , j)th element X=⎢ ⎢⎣ ∂(rj) ⎥⎦
(JCR). Joint confidence region, also called joint confidence interval, include all combinations of values for the parameters that are simultaneously acceptable at the specified level of confidence. Therefore, the JCR of the parameters gives more information about their accuracy. The JCR of rA and rB having the correct shape, but approximate probability content, was constructed by solving the equation below25 ⎡ ⎤ p Fp , n − p , α ⎥ S(r ) ≤ S(r )̂ ⎢1 + n−p ⎣ ⎦
(37)
and g is the function acting as the coefficient for the parameter rj, (X′X)ii−1 is the ith diagonal element of the (X′X)−1 matrix, and s can be calculated as s2 =
S(r )̂ n−p
(39)
where n and p are the number of data points and parameters, respectively, Fp,n‑p,α is the upper critical value of the Fp,n‑p distribution, S(r) is the sum of the squares of the residuals at (100 − α)% confidence level, which is a function of the parameters, and S(r )̂ is the corresponding value at the estimated values of the parameters r̂. We used Excel to solve eq 39 iteratively and construct the JCR shown in Figure 11. The JCR stretches from 59.5 to 65.9 for rA and from 0.02 to 0.144 for rB covering a wider range of reactivity ratios than the corresponding confidence intervals estimated through asymptotic regression,(the two perpendicular lines intersecting at
(38)
where S(r )̂ is the sum of squares of residuals calculated at the optimum value of the parameters. An estimate of the standard deviation (estimated standard error) for parameter ri was obtained by calculating (X ′X )−ii 1s 2 . Table 5 summarizes these estimates and their confidence intervals. The confidence intervals for rA and rB do not includes zero, proving that the estimates are statistically significant. Table 5. Reactivity Ratios for Ethylene/1-Octene Copolymerization with rac-Et[Ind]2ZrCl2/MAOa parameter
value
lower confidence limit
higher confidence limit
rA rB
62.85 0.085 05
61.06 0.049 86
64.65 0.1202
a Sum of the squares of the residuals = 1.254 × 10−7, R2 = 0.9997, level of confidence 95%
Figure 10 demonstrates that the Mayo−Lewis equation fits well the composition curve for these copolymers. Joint Confidence Region (JCR). The individual confidence intervals in Table 5 are the asymptotic estimations of rA and rB using approximate t test statistics, and hence may be underestimated. This is a general trend when we face models with more than one parameter and the individual confidence interval for each parameter does not provide full statistical information. A more elaborate way of expressing uncertainty in parameter estimates is through the joint confidence region
Figure 11. Plot of 95% joint confidence region and individual confidence intervals for rA and rB. G
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules Table 6. Nonlinear Regression Estimates of kpAA and kpBA and Their Corresponding 95% Confidence Intervalsa
a
parameter
value (L·mol‑1·s‑1)
lower confidence limit (L·mol‑1·s‑1)
higher confidence limit (L·mol‑1·s‑1)
kpAA kpBA
200500 8858
192100 6562
209000 11150
Sum of the squares of the residuals = 5.48 × 108, R2 = 0.86
point O in Figure 11).This indicates that estimation of confidence intervals through asymptotic regression is underestimated as expected and it is recommended to use JCR to have a more precise information about reactivity ratios. Moreover, the orientation of the ellipse of the JCR is tilted, indicating that the reactivity ratios are correlated, possibly due to the high nonlinearity of the function. We substituted the estimated value of rA = 62.85 into eq 30 and fitted kp̃ versus f B to estimate kpAA and kpBA. Table 6 lists these estimates and their corresponding asymptotic confidence intervals. Figure 12 shows the JCR for kpBA and kpAA when rA = 62.85. Since rA can vary in the envelope of the JCR shown in Figure
⎛ kpBB ⎞ kp̃ = kp AA − ⎜kp AA − ⎟ rB ⎠ ⎝
kp AA rA kpBB rB
fB
(1 − fB ) +
kp AA rA
fB
(40)
According to eq 40, nonlinear regression estimates for kpBB and kpAA depend on the values of rA and rB. Since we know the range of variation in rA and rB (see Figure 11), we can construct the joint confidence region for kpBB and kpAA by fitting the curve of kp̃ versus f B at the specified values of rA and rB and then solving eq 39. Figure 13 shows the JCR for kpBB and kpAA for different values of rA and rB. The top and bottom JCRs were calculated using rA
Figure 13. 95% joint confidence region for kpBB and kpAA. Figure 12. 95% joint confidence region for kpBA and kpAA for different values of rA, as indicated in the legends.
and rB values corresponding to points A and B in Figure 11, respectively. Interestingly, the JCRs for points O, E, and F overlap, showing that changes in rA have no effect on the variance of kpBB as long as rB is constant. All JCRs corresponding to the points on the paths connecting A to B are located between the JCRs for points A and B. For instance, the JCR for point C lies below the JCR for point O. If we increase rB from 0.021 to 0.144, the JCR moves upward until it matches the JCR for point B. The solid curve encompassing all the JCRs in Figure 13 is, in fact, the overall JCR for kpBB and kpAA considering all variation in rA and rB, that is kpBB varies from 190 to 1840 L·mol−1·s−1, while kpAA varies from 190 000 to 212 000 L·mol−1·s−1. We can also transform eq 40 to the form below to obtain the JCR for kpAB and kpBA,
11, we need to construct the JCRs for k pBA and kpAA corresponding to all those rA values. To simplify this, we also constructed JCRs for the two limiting values of rA, 59.5 and 65.9 (points A and B in Figure 11). The JCRs of all other rA values lie in region between the JCRs of these two limiting values of rA.Therefore, the true JCR for kpAA and kpBA is the union of these two regions, and covers a wider range of kpAA and kpBA combinations than the individual regions. The JCR stretches from 6200 to 14 000 L·mol−1·s−1 for kpBA, and from 190 000 to 212 000 L·mol−1·s−1 for kpAA. Our previous estimate for kpAA21 shows that the its value vary between 206 500 to 212 100 L·mol−1·s−1 which agrees with our current estimate for kpAA. Uncertainty in determination of rA led to larger uncertainty in kpBA. Interestingly, the variability in rA has no effect on the confidence interval of kpAA, which is reasonable because we can estimate kpAA using only homopolymerization data, independently of rA. If we combine eq 35 with eq 30 we get,
kp̃ = rAkp AB − (rAkp AB − kpBA )
kp ABfB kpBA(1 − fB ) + kp ABfB (41)
Since rA varies from 59.5 to 65.9 with the optimum value of 62.85, we constructed JCRs at these values to estimate the overall JCR for kpAB and kpBA taking into account the H
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules uncertainty coming from determination of rA value. The procedure was the same to what we used to construct JCRs in Figures 12 and 13. Figure 14 depicts the JCRs corresponding to three different values of rA. The solid curve enclosing these three curves is the
Figure 15. Comparison of experimental polymer yield versus polymer yield predicted using eq 43.
copolymerizations. The proposed method is statistically rigorous, yet easy to implement, enabling us not only to get excellent estimates for the cross-propagation rate constants but also to establish their statistical significance under a given set of experimental conditions. We believe this technique could be useful to estimate the propagation rate constants for copolymerizations of ethylene and 1-olefins with other singlesite catalysts. Our approach shows that, even using carefully designed experiments and applying a rigorous statistical analysis procedure, variations in the cross-propagation constants and reactivity ratios, particularly rB, are to be expected. These uncertainties will pose upper and lower bounds on reactor design and scale up studies. This information is extremely useful for polymer reactor engineers, because it helps one to plan sensitivity analysis studies within these confidence regions and test the effect of these variations on reactor operation conditions and product quality. This information is also essential to help them design other experiments to narrow these confidence regions, if required for their particular applications. We applied our novel method to estimate the crosspropagation constants for the solution polymerization of ethylene/1-octene using a single-site catalyst. Polyolefins made in solution processes by Dow and Nova are among the most successful high-performance single-site polyolefin grades made today, making our method immediately applicable for the scale-up and optimization of catalysts for these processes. Many other leading polyolefin producers such as ExxonMobil, Chevron-Phillips, Borealis, and Total make important singlesite polyolefin grades using supported metallocenes. When combined with an adequate model for intraparticle mass and heat transfer resistance, our method could easily be extended to estimate kinetic parameters for these catalyst systems. The case of multiple-site heterogeneous catalysts, such as Ziegler−Natta
Figure 14. 95% joint confidence region for kpAB and kpBA.
overall JCR for kpAB and kpBA. The optimum values and confidence intervals for kpAB and kpBA are 3190 (2900−3600) mol−1·L·s−1 and 8858 (6200−14 000) mol−1·L·s−1, respectively. As expected, kpBA is greater than kpAB indicating that insertion of ethylene into polymer chains ending with 1-octene is easier than when 1-octene reacts with polymer chains ending with ethylene. Table 7 summarizes the values estimated for the four propagation rate constants and their corresponding range of variations as extracted from their relevant JCRs. Polymer yield, mp, can be obtained by integration of the rate of polymerization, g ⎞ ⎛ ⎟ × mp = ⎜28 ⎝ mol ⎠ mp = 28
kp̂ kd
∫0
t
R p dt
(42)
Zr[M](1 − e−kdt )
(43)
Figure 15 compares measured polymer yields with those calculated using eq 43 with the parameter estimates listed in Table 7, showing that model predictions agree well with the experimental values.
■
CONCLUSIONS We have shown that that terminal model, including first order propagation and deactivation steps, quantifies well the copolymerization kinetics of ethylene and 1-octene with racEt(Ind)2ZrCl2/MAO. We estimated four propagation rate constants and their joint confidence regions by combining reactivity ratio estimations with ethylene uptake rates of a set of
Table 7. Estimates for Propagation Rate Constants and Their Range of Variations at the 95% Confidence Limit parameter
estimated value (L·mol−1·s−1)
lower limit (L·mol−1·s−1)
higher limit (L·mol−1·s−1)
kpAA kpAB kpBB kpBA
200500 3190 753.4 8858
192000 2900 190 6200
212000 3600 1840 14000
I
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
(16) Mehdiabadi, S.; Soares, J. B. P.; Dekmezian, A. H. Production of Long-Chain Branched Polyolefins with Two Single-Site Catalysts: Comparing CSTR and Semi-Batch Performance. Macromol. React. Eng. 2008, 2 (6), 529−550. (17) Soares, J. B. P.; Kim, J. D. Copolymerization of ethylene and alpha-olefins with combined metallocene catalysts. I. A formal criterion for molecular weight bimodality. J. Polym. Sci., Part A: Polym. Chem. 2000, 38 (9), 1408−1416. (18) Kim, J. D.; Soares, J. B. P. Copolymerization of ethylene and alpha-olefins with combined metallocene catalysts. II. Mathematical modeling of polymerization with single metallocene catalysts. J. Polym. Sci., Part A: Polym. Chem. 2000, 38 (9), 1417−1426. (19) Kim, L. D.; Soares, J. B. P. Copolymerization of ethylene and alpha-olefins with combined metallocene catalysts. III. Production of polyolefins with controlled microstructures. J. Polym. Sci., Part A: Polym. Chem. 2000, 38 (9), 1427−1432. (20) Mehdiabadi, S.; Soares, J. B. P. Influence of Metallocene Type on the Order of Ethylene Polymerization and Catalyst Deactivation Rate in a Solution Reactor. Macromol. Symp. 2009, 285, 101−114. (21) Mehdiabadi, S.; Soares, J. B. P. In-Depth Investigation of Ethylene Solution Polymerization Kinetics With rac-Et(Ind)2ZrCl2/ MAO. Macromol. Chem. Phys. 2013, 214 (2), 246−262. (22) Galland, G. B.; Quijada, P.; Mauler, R. S.; de Menezes, S. C. Determination of reactivity ratios for ethylene/α-olefin copolymerization catalysed by the C2H4[Ind]2ZrCl2/methylaluminoxane system. Macromol. Rapid Commun. 1996, 17 (9), 607−613. (23) Uozumi, T.; Soga, K. Copolymerization of Olefins with Kaminsky-Sinn-Type Catalysts. Makromol. Chem. 1992, 193 (4), 823−831. (24) ASTM, Standard Test Method for Determination of Linear Low Density Polyethylene (LLDPE) Composition by Carbon-13 Nuclear Magnetic Resonance 1; 2003. (25) Seber, G. A.; Wild, C. J. Nonlinear Regression; John Wiley & Sons: New York, 1989.
and Phillips catalyst, poses a more severe challenge but, in principle, our method could be combined with molecular weight and chemical composition deconvolution techniques to estimate cross-propagation rate constants and their joint confidence intervals for these systems as well.
■
AUTHOR INFORMATION
Corresponding Author
*(J.B.P.S.) E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
REFERENCES
(1) Mueller, P. A.; Richards, J. R.; Congalidis, J. P. Polymerization Reactor Modeling in Industry. Macromol. React. Eng. 2011, 5 (7−8), 261−277. (2) Hatzantonis, H.; Yiannoulakis, H.; Yiagopoulos, A.; Kiparissides, C. Recent developments in modeling gas-phase catalyzed olefin polymerization fluidized-bed reactors: The effect of bubble size variation on the reactor’s performance. Chem. Eng. Sci. 2000, 55 (16), 3237−3259. (3) Kiparissides, C. Polymerization reactor modeling: A review of recent developments and future directions. Chem. Eng. Sci. 1996, 51 (10), 1637−1659. (4) Touloupidis, V. Catalytic Olefin Polymerization Process Modeling: Multi-Scale Approach and Modeling Guidelines for Micro-Scale/Kinetic Modeling. Macromol. React. Eng. 2014, 8 (7), 508−527. (5) Debling, J. A.; Ray, W. H. Heat and Mass-Transfer Effects in Multistage Polymerization Processes - Impact Polypropylene. Ind. Eng. Chem. Res. 1995, 34 (10), 3466−3480. (6) Xu, Z. G.; Chakravarti, S.; Ray, W. H. Kinetic study of olefin polymerization with a supported metallocene catalyst. I. Ethylene/ propylene copolymerization in gas phase. J. Appl. Polym. Sci. 2001, 80 (1), 81−114. (7) Soares, J. B. P.; McKenna, T. F. L. Polyolefin Reaction Engineering; Wiley-VCH: 2012. (8) Zacca, J. J.; Debling, J. A.; Ray, W. H. Reactor residence time distribution effects on the multistage polymerization of olefins. Dech Monogr. 1995, 131, 213−222. (9) Zacca, J. J.; Debling, J. A.; Ray, W. H. Reactor residence time distribution effects on the multistage polymerization of olefins 0.1. Basic principles and illustrative examples, polypropylene. Chem. Eng. Sci. 1996, 51 (21), 4859−4886. (10) Zacca, J. J.; Debling, J. A.; Ray, W. H. Reactor residence-time distribution effects on the multistage polymerization of olefins 0.2. Polymer properties: Bimodal polypropylene and linear low-density polyethylene. Chem. Eng. Sci. 1997, 52 (12), 1941−1967. (11) Debling, J. A.; Zacca, J. J.; Ray, W. H. Reactor residence-time distribution effects on the multistage polymerization of olefins 0.3. Multi layered products: Impact polypropylene. Chem. Eng. Sci. 1997, 52 (12), 1969−2001. (12) Richards, J. R.; Congalidis, J. P. Measurement and control of polymerization reactors. Comput. Chem. Eng. 2006, 30 (10−12), 1447−1463. (13) Ray, W. H.; Soares, J. B. P.; Hutchinson, R. A. Polymerization reaction engineering: Past, present and future. Macromol. Symp. 2004, 206, 1−13. (14) Pladis, P.; Kiparissides, C. A comprehensive model for the calculation of molecular weight−long-chain branching distribution in free-radical polymerizations. Chem. Eng. Sci. 1998, 53 (18), 3315− 3333. (15) Krallis, A.; Kiparissides, C. Mathematical modeling of the bivariate molecular weight - Long chain branching distribution of highly branched polymers. A population balance approach. Chem. Eng. Sci. 2007, 62 (18−20), 5304−5311. J
DOI: 10.1021/acs.macromol.5b02755 Macromolecules XXXX, XXX, XXX−XXX
