FULL PAPER Bimodal Polyethylenes
www.mre-journal.de
Mapping the Structure–Property Space of Bimodal Polyethylenes Using Response Surface Methods. Part 1: Digital Data Investigation Paul J. DesLauriers,* Jeff S. Fodor, João B. P. Soares, and Saeid Mehdiabadi Polyethylenes with reverse CCD cannot be produced in a single reactor using Ziegler–Natta, Phillips, or metallocene catalysts. The average 1-olefin fraction decreases with increasing molecular weight when Ziegler–Natta and Phillips catalysts are used; by contrast, the average 1-olefin fraction is constant, or nearly constant, across the MWD when the polymer is made with metallocenes displaying true single-site behavior (when metallocenes are supported on silica and other inorganic carriers, they may lose their single-site behavior and make polyolefins with broader, and perhaps reverse CCD, but in this case they are not behaving like single-site catalysts any longer). One typically relies on one of two approaches to make bimodal poly ethylenes with reverse CCD: 1) use two or more reactors in series with a single catalyst, or 2) use two or more catalysts (typically metallocenes) in a single reactor. The latter approach is the focus of this work. Much effort goes into designing bimodal polyolefins with emphasis on reducing both the amount of material and the time needed for testing their properties and performance. One approach to aid designing polyolefins more efficiently is to apply mathematical models linking polymerization kinetics to polymer microstructure, and polymer microstructure to polymer physical properties (mechanical, rheological, and optical). The goal is to provide a simulation tool that predicts polyolefin properties from polymerization conditions. In Part 1 of our investigation, we explored the feasibility of using response surface methods (RSMs) to estimate the microstructure and physical properties of bimodal ethylene/1-hexene copolymers using simulated digital data. The basic premise of RSMs is that variations within a set of input factors (reactor conditions and catalyst type) can be correlated to changes in a particular set of responses (polymer microstructures and properties). In the present study, we used a fundamental polymerization model to generate digital MWD and short chain branching (SCB) data under various polymerization conditions using different catalyst types. We then estimated the physical properties of the resulting digital polymer samples using semiempirical algorithms based on the digital MWD-SCB data. The fundamental polymerization model serves as a proxy for actual
A new method is proposed to study the multidimensional structure– property space of bimodal ethylene/1-olefin copolymers using fundamental polymerization models and response surface methods. The fundamental polymerization models describe how the polymer microstructure depends on polymerization conditions, in particular how the short chain branching frequency varies across the molecular weight distribution. Statistical design of experiment techniques combined with response surface methods generates the minimum number of digital experiments needed to estimate the parameters for a forward model. The polymer microstructural data are combined with empirical equations to calculate polymer density and the primary structural parameter to quantify the presence of tie molecules in the polymer. This procedure links polymerization conditions to polymer microstructure and properties using the forward model. More importantly, the forward model can be reversed to determine which polymerization conditions are needed to make polymers with a given set of properties.
1. Introduction Polyethylene with bimodal molecular weight distribution (MWD) and reverse comonomer composition distribution (CCD) has excellent mechanical properties. In these polymers, the average 1-olefin fraction in the population with high molecular weight is higher than in the population with low molecular weight (reverse CCD). This microstructural arrangement increases the number of effective tie molecules, which improves the environmental stress crack resistance, impact strength, and Young’s modulus of the polymer. Polyethylenes in which the average 1-olefin fraction decreases with increasing molecular weight generally do not perform as well as those with reverse CCD. Dr. P. J. DesLauriers, Dr. J. S. Fodor Chevron Phillips Chemical Company LP Bartlesville Research Center HWYS 60 & 123, Bartlesville, 74004-0001 OK, USA E-mail:
[email protected] Prof. J. B. P. Soares, Dr. S. Mehdiabadi Department of Chemical and Materials Engineering University of Alberta Edmonton, T6G 1H9 AB, Canada The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/mren.201700066.
DOI: 10.1002/mren.201700066
Macromol. React. Eng. 2018, 1700066
1700066 (1 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de ka
C + Al → Po kp
Pr + M → Pr +1 (1) kd
Pr → C d + Dr
Figure 1. Overall process used to map the reaction-polymer structure– property space of bimodal polyethylenes.
polymerization experiments, and the generated MWD-SCB data replace the results that would be obtained by analyzing actual polymer samples with a high-temperature size exclusion chromatograph (SEC) coupled with an infrared (IR) or Fourier transform infrared detector (FTIR). By using properly designed data sets and RSM models, we showed that the forward model could predict the microstructure and physical properties of the digital samples generated with the fundamental polymerization model, adequately mapping the structure–property space of bimodal polyethylenes. Furthermore, reverse RSM models predicted the poly merization conditions and catalyst properties needed to make the polymer architecture that met a set of application requirements. Figure 1 illustrates this modeling approach. Part 2 of this work will focus on the application of these methods to experimental systems, and will be the subject of a future publication.
2. Mathematical Modeling
d [C ] = −ka [ Al ] [C ] (2) dt If we assume that the concentration of cocatalyst, [Al], is much larger than [C], then we can make the approximation Ka = ka [Al] = constant. With this approximation, and the initial condition [C] = [C0] at t = 0, we can solve Equation (2)
[C ] = [C0 ] exp (−K a t ) (3) The rate of change for the total molar concentration of living ∞ chains, ∑ Pr = [Y0 ], is given by the difference between the rate r =0 of precatalyst activation and the rate of catalyst deactivation d [Y0 ] = K a [C ] − kd [Y0 ] = K a [C0 ]exp ( −K a t ) − kd [Y0 ] (4) dt The rate of polymerization equals the rate of monomer consumption, given by the expression
2.1. Polymerization Rate We will apply the term digital catalyst to characterize the fundamental polymerization kinetics steps, and its associated kinetic constants, used to generate copolymerization rates and ethylene/1-olefin copolymer microstructures. Our digital catalysts behave like actual catalysts, but we do not need to describe the behavior of any actual catalyst. This approach ensured that the model described the general trends of actual polymerization catalysts, but did not require spending large amounts of time estimating polymerization kinetics parameters. Ziegler–Natta and Phillips catalysts have more than one site type, making polymers with nonuniform properties. Several metallocene and post-metallocene catalysts, contrarily, are molecular catalysts that have only one type of active site, producing polymers with uniform (within the expected statistical variation) properties, although interactions with other chemical compounds (such as cocatalysts and scavengers) and catalyst supports may make them deviate from the expected ideal single-site behavior. In this paper, our digital catalysts exhibit true single-site behavior, but the methods we propose are not limited to these systems, and can be easily extended to multiple-site catalysts. A basic model for the homopolymerization rate of a digital catalyst includes steps of activation, propagation, and deactivation[1]
Macromol. React. Eng. 2018, 1700066
where C is the precatalyst, Al is the cocatalyst, Po is a monomer-free active site, Pr is a growing polymer chain of length r, M is the monomer, Cd is a deactivated site, Dr is a dead polymer chain of length r, ka is the activation rate constant, kp is the propagation rate constant, and kd is the deactivation rate constant. Chain transfer steps are not needed at this stage because they do not affect the propagation rate constant of the catalyst. In a batch or semibatch reactor, the molar balance for the concentration of precatalyst, [C], in the reactor is
Rp =
d [M ] = kp [M ][Y0 ] (5) dt
Equation (4) can be solved analytically for [Y0], and the result substituted into Equation (5) to give an expression for the polymerization rate
(
)
1 − exp −K a (1 − kd/K a ) t R p = kp [M ] [C0 ]exp (− kd t ) (6) (1 − kd/K a ) Equations (2)–(6) have been used regularly in the literature to describe ethylene polymerization kinetics with a variety of catalyst types. We will show how they can be extended to ethylene/1-olefin copolymerization below. The most common models for copolymerization are the Bernoullian and the Terminal models. The Bernoullian model assumes that the type of monomer at the end of the polymer chain bonded to the active center does not influence propagation rates: only the type of monomer being inserted into the chain affects the value of kp. Therefore, there are two propagation rates constants, kpA and kpB, for a binary copolymerization of monomers A and B,
1700066 (2 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
kpA
Pr + A → Pr +1 kpB
Pr + B → Pr +1
(7)
The Terminal model assumes that the type of monomer at the end of the chain and the type of monomer being inserted into the growing chain affect the propagation rate constant. Therefore, four possibilities are viable, generating four propagation rate constants: kpAA, kpAB, kpBA, and kpBB. For simplicity’s sake, we will use the Bernoullian model in the present investigation. Using the Terminal model would increase the number of required model parameters, but would not add any relevant new finding to our study. However, when modeling actual, not digital catalysts, the Terminal model is likely to be required to adequately describe the polymerization system under consideration. For the Bernoullian model, the overall Rp is simply the linear addition of the two individual propagation rates for each monomer. After some manipulation, one can show that Equation (6) can be used to model Bernoullian copolymerization by replacing kp with the pseudopropagation constant kˆp
where Xi represents all the transfer species, and ktXi their respective chain transfer constants. The MWD can be obtained from the CLD by multiplying the latter by the average molar mass of the repeating unit.
2.3. SCB across the MWD Along with the MWD, we also need to calculate the SCB (or comonomer fraction) across the MWD. For single-site catalysts, the average SCB does not depend on polymer molecular weight and can be calculated with the Mayo–Lewis equation[8] FA =
(rA − 1) f A2 + f A (12) (rA + rB − 2) f A2 + 2(1 − rB ) f A + rB
where FA is the average molar fraction of comonomer A in the copolymer, and rA and rB are reactivity ratios. For the Bernoullian model, reactivity ratios are defined as
kˆp = kpA f A + kpB f B (8)
rA =
kpA (13) kpB
where fA and fB are the molar fractions of monomers A and B in the reactor, respectively, and kpA and kpB are their respective propagation rate constants.[1]
rB =
kpB (14) kpA
2.2. MWD
In this particular case, the product rArB is equal to 1. With FA known, the SCB for ethylene/1-hexene copolymers is calculated using the expression
So far, we have not included chain transfer steps in the poly merization model because they do not affect the rate of poly merization, only polymer molecular weight averages. The main chain transfer reactions for the Bernoullian model are given below
SCB 1 − FA = 1000 (15) 1000 C 6 (1 − FA ) + 2FA where the subscript A represents ethylene.
ktA
Pr + A → P1 + Dr
2.4. Temperature Dependence of Kinetic Constants
ktB
Pr + B → P1 + Dr ktH
Pr + H → Po + Dr (9) ktAl
Pr + Al → Po + Dr ktβ
Pr → Po + Dr where ktA and ktB are monomer transfer constants, ktH is the hydrogen transfer constant, ktAl is the cocatalyst transfer constant, and ktβ is the beta-hydride elimination transfer constant. It is not hard to show that the instantaneous chain length distribution (CLD) for copolymer made via this mechanism follow Flory’s most probable distribution[1–7] w log r = 2.3026r 2τ 2e −rτ (10) where the single parameter τ is defined as 1 ∑ ktX i [ X i ] (11) τ = = i=1 rn kˆp [M ] n
Macromol. React. Eng. 2018, 1700066
Arrhenius law describes the temperature dependence for each of the kinetic constants introduced above. We used either the general form, with the pre-exponential factor, Ai, and the activation energy, Ei, or the reduced form with a kinetic constant at a reference temperature, kref,i, ki = A i e
E − i RT
Ei 1 1 − T
or ki = kref ,i e R Tref
(16)
2.5. Multiple Catalyst Systems The development above assumes only one type of active site, but it is straightforward to expand it to multiple sites, as for the cases when more than one metallocene, Ziegler–Natta, or Phillips catalysts are used in the polymerization.[1,9] The same equations developed above remain applicable, with the only difference that each catalyst site type will have its own set of polymerization kinetics parameters.
1700066 (3 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
Table 1. Polymerization kinetics parameters for the digital catalysts. Cat 1
Cat 2
A
Units
E
Units
A
Units
E
Units
Activation
1.03 × 1020
L mol−1 min−1
24.90
kcal mol−1
Activation
2.47 × 1020
L mol−1 min−1
25.53
kcal mol−1
Deactivation
3.82 × 1011
min−1
23.02
kcal mol−1
Deactivation
3.16 × 106
min−1
12.35
kcal mol−1
ethylene
4.21 × 1011
L mol−1 min−1
14.11
kcal mol−1
ethylene
3.79 × 1012
L mol−1 min−1
15.40
kcal mol−1
1-hexene
1.28 × 1010
L mol−1 min−1
14.79
kcal mol−1
1-hexene
2.49 × 109
L mol−1 min−1
11.57
kcal mol−1
to ethylene
7.56 × 1010
L mol−1 min−1
16.79
kcal mol−1
to ethylene
2.84
L mol−1 min−1
0.93
kcal mol−1
to 1-hexene
4.43 × 108
L mol−1 min−1
12.57
kcal mol−1
to 1-hexene
1.72 × 108
L mol−1 min−1
13.05
kcal mol−1
b-hydride
5.83 × 10
50.00
mol−1
b-hydride
8.80 × 10
27.73
kcal mol−1
Propagation
Propagation
Transfer
Transfer
24
−1
min
kcal
15
−1
min
For example, consider a system of n single-site catalysts. The rate of polymerization is
2.7. Physical Property Models: Density and Tie Molecule Content
kdi t 1 − exp −K ai 1 − K ai R p = ∑ R pi = ∑ kpi [M ] (17) kdi i =1 i =1 1 − K ai [C0 i ]exp ( −kdit )
Structure–property relationship models estimate how physical properties depend on the MWD-SCB profile. All the structure–property models used in this paper were derived from microstructural data measured by SEC-FTIR or SEC-IR. Both methods provide the molecular weight distribution (MWD) as well as the average short chain branching (SCB) frequency across the MWD (Figure 3). For experimental resins, the acquired SCB data across the MWD may need to be corrected at the tails of the MWD because of poor signal-to-noise ratios and breakdown of spectral integrity.[11,12] For digital bimodal samples, this problem is thankfully absent. Our structure–property relationship model uses correlations based on molecular weight (MW) and SCB at each MW slice. For example, in Figure 3 a property value (density, for instance) calculated at log M = 4 will be different from that calculated at log M = 5. Considering the weight fraction of polymer in each slice, property values for each slice can be summed over the whole MWD to estimate the property for the bulk resin. In this work, two resin properties were selected: polymer density and lamella connectivity, as captured by the primary structure parameter (PSP2).[13]
n
n
The fraction of polymer made by catalyst type i is mi =
∫
tR 0
R pi
∑ ∫ n
i =1
tR 0
R p1
(18)
Finally, the MWD, average copolymer composition, and SCB-MWD can be calculated as follows n
wr = ∑ m i wri (19) i =1 n
FA = ∑ m i FAi (20) i =1
n
FA (r ) = ∑ m i wriFAi (r ) (21) i =1
2.6. Model Parameters for the Digital Catalysts Since the emphasis in this article is on the digital design of experiments and their subsequent analysis, any set of realistic polymerization kinetics constants would be adequate. We based our choice on an actual dual catalyst system for which we had a limited number of data points. A quick fit of the experimental data was performed to provide a reasonable set of initial polymerization kinetics parameters using the Bernoullian model. Table 1 lists the polymerization kinetics constants for the two digital catalysts used in our investigation. Figure 2 shows ethylene uptake rates for each catalyst and for the dual catalyst system for Run 7 in Table 3.
Macromol. React. Eng. 2018, 1700066
Figure 2. Combined ethylene uptake curves for DoE Run 7 in Table 3. The rate of polymerization is plotted for the overall catalyst system (purple -x-), and each of the individual catalyst contributions, Cat 1 (blue –o-), and Cat 2 (red -).
1700066 (4 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
ρ polymer = ∑ w i ρi (25) i
Figure 3. MWD and average SCB across the MWD for a polymer made with two metallocenes.[10]
Density is among the most important properties of poly ethylene, as it dictates its stiffness, impact strength, chemical and heat resistance.[14] Moreover, the density of polyethylene can be directly correlated to its tensile properties.[15] Many factors determine the density of polyethylene, but density itself is not a fundamental property, although it is often treated as such. Rather, the density of a final fabricated article results from the combination of polymer microstructural properties, processing variables, heating and cooling rates. When density is reported as a “material property,” these additional factors are usually controlled by specifying molding information, heating and cooling rates, per some standard, such as ASTM D4703. Only in this sense can density be considered a property of the polymer itself. We will correlate the density of ethylene/1-hexene copolymers using a three-step procedure, as described in a previous publication.[13] In the first step, the effect of molecular weight on the density of ethylene homopolymer is described with the equations
ρH = 1.0748 − 0.0241log M , M > 760 g mol −1 ρH = 1.006, M ≤ 760 g mol −1
(22)
In the second step, the homopolymer density is corrected for the presence of comonomer using the following expression
where wi is the molar fraction of polymer in each MWD slice, and ρi the estimated density per slice. Typically, this approach estimates density to within ±0.002 g cm−3.[13] Many polyethylene performance properties, such as slow crack grow (SCG) resistance, have been linked to their lamella connectivity.[16–18] Tertiary structures, which link lamella together (such as bridging entanglements and bridging tie molecules) act as stress transmitters that increase the resistance to crack propagation.[19,20] One measure of tie molecule levels in polyethylene is the PSP2 defined by DesLauriers and Rohlfing.[13] PSP2 quantifies the effects of the primary structure parameters SCB-MWD into a single dimensionless value that can be correlated to various tests used to assess SCG resistance. PSP2 is calculated using several semiempirical relationships. Starting with the densities calculated as described above, the melting point, lamella thickness, and probability of tie molecule formation are calculated for each MW slice, after which the values are summed up over the entire MWD. The melting point temperature is calculated from the density, estimated with Equation (25), using the following expression Tm ( o C) = 2.06 × 10 4 × ρ 3 + 6.38 × 10 4 × ρ 2 (26) + 6.06 × 10 4 × ρ + 2.26 × 10 4
Crystalline lamella thickness is estimated using the equation lc ( nm ) =
0.626 nm × 417.7 K (27) 417.7 K − (Tm + 273.15 K )
The weight fraction of the crystalline phase is given by the expression wc =
ρc ρ − ρa (28) ρ ρc − ρa
where the densities of perfectly crystalline and amorphous polyethylene are ρc = 1.006 g cm3 and ρc = 0.852 g cm3, respectively. Finally, the thickness of the amorphous phase is estimated with the equation
ρ × la × ( 1 − w c ) (29) ρa × w c
ρ = ρH − ∆ρ (23)
la ( nm ) =
The correction factor Δρ accounts for the nonlinear decrease in density caused by the presence of short chain branches in the copolymer. It also considers the polymer polydispersity, according to the empirical equation
The probability for a molecule to span two lamellae and form a tie molecule (PTM) can be expressed as
0.496
SCB ∆ρ = 1.24 × 10 −2 PDI0.319
−0.781
SCB + 3.46 × 10 −4 PDI0.319
(24)
We used Equations (22)–(24) to estimate the polymer density on a slice-by-slice MW basis. The final polymer density (ρpolymer) was calculated by summing the products of the sliceby-slice densities times their weight fractions across the MWD
Macromol. React. Eng. 2018, 1700066
1 2bL exp ( − b2L2 ) (30) PTM = 1 − erf ( bL ) + 3 π where L = 2lc + la (31) b2 =
3 (32) 2r 2
1700066 (5 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
M r 2 = 0.159 slice (33) 14 Finally, as in the density calculation, the PSP2 value for the polymer is calculated by summing the PSP2 calculated for each slice times the weight fraction of polymer in that MW slice over the whole MWD PSP2 = ∑ w i PTMi × 100 (34) i 2.8. Response Surface Methods RSMs use statistical and mathematical techniques to regress a response variable against a set of independent input variables. In this study, we used RSM to relate polymerization conditions to polymer microstructures and properties. Our focus was to find the appropriate forward model that could predict the microstructure and properties of polymers made with a pair of metallocenes under a set of polymerization conditions. We also wanted to generate a reverse model linking polymer properties and microstructure to polymerization conditions (Figure 1). In RSM, the variability in the response variables is fitted to the variability in the input data using simple polynomials and statistical tools such as analysis of variance (ANOVA). These tools are used to judge the degree of fit, variable significance, and prediction ability of the generated models. Figure 4 summarizes the RSM process. We used the Stat Ease software to generate a five-factor (see Table 2) optimal design of experiments (DoE) and resulting response surface methodology models (forward and reverse). Optimal designs are a class of DoEs that seek to minimize some statistical criterion. In our case, the prediction variance was chosen as the criterion. The information contained in the DoE (values for each input variable) are written in matrix form (referred to as X) and algebraically treated so that some minimum value is obtained. For example, one common criterion used is G-optimality,[21] which seeks to minimize the maximum prediction variance, which results from minimizing the maximum entry in the diagonal of the hat matrix X(XTX)−1XT. In integrated variance (IV) optimal DoE,[21] the average prediction variance over the design space is minimized by
Figure 4. Relationship between properties estimated from semiempirical algorithms derived from SEC-FTIR data and those estimated from RSM using reactor and catalyst input variables.
randomly generating a DoE and then substituting values via a coordinate exchange algorithm until no further improvements to the optimal design can be made. Experimental designs built in this fashion typically result in less sample runs than other design types, since not every point has multiple levels. Equation (35) describes the general approach behind IV optimal designs used in this study. In this equation, xi is a p × 1 vector, 1 element for each term in the DoE model. We used a quadratic model, so xi is a 21 × 1 vector. The product xi × xiT gives a 21 × 21 matrix. This is done for each of the N design points, the N of the p × p matrices are summed, and then divided by each element in the matrix sum by N −1 IVoptimal = trace ( X T X ) M (35)
where M=
1 N ∑ x i x Ti (36) N i =N
Table 2 summarizes the model input variables and their ranges. These values were chosen based on common laboratory polymerization conditions so that a reasonable structure– property space could be explored.
Table 2. Digital domain investigated (factors and respective levels). Study type
Response surface
Runs
31
Design type
IV-optimal
Coordinate exchange
Blocks
Design model
Quadratic
Build time [ms]
2901
Name
Abbreviation
Type
Subtype
Min
Max
Mean
Std. dev.
Factor
No blocks
A
Ethylene [mol]
CE
Numeric
Continuous
0.2
0.75
0.49
0.22
B
1-Hexene/ethylene
C6/CE
Numeric
Continuous
0
0.8
0.41
0.32
C
Catalyst 1/Catalyst 2
CAT/CAT2
Numeric
Continuous
0.6
4
2.33
1.3
D
Temp [°C]
Temp
Numeric
Continuous
120
140
131.02
7.95
E
Time [min]
Time
Numeric
Continuous
2
10
6.08
3.11
Macromol. React. Eng. 2018, 1700066
1700066 (6 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
The primary response variables included: 1) number (Mn) and weight average molecular weight (Mw) of polymer made on each metallocene (Mw = 2 Mn for single-site catalysts); 2) SCB frequency of polymer made on each metallocene; and 3) weight fraction of polymer made in metallocene catalyst 1. The derivative responses (responses calculated from the primary responses) for the respective blends were also calculated and analyzed. These responses include Mw, Mn, PDI = Mw/Mn, density, and PSP2 for the whole polymer. The molecular weight averages for the binary blends were calculated with the classical equations M w ,blend = m1M w1 + (1 − m1 )M w 2 (37)
M n ,blend
1 − m1 m = 1 + M n1 M n 2
−1
(38)
An initial 31 sample DoE was constructed and analyzed. Subsequent validation and design augmentation runs were also conducted, as will be described below. Random error was assigned to all replicate response values. A 3% error was assigned to Mw of polymers made in catalyst 1 (C1) and catalyst 2 (C2) (both with PDI = 2), and a 5% error was assigned to the Mw of the blends (PDI ≈ 7.8). These assigned errors are in line with the errors cited in round-robin studies reported by D’Agnillo et al.[22] for various linear polymers considering the Mw and PDI values of our digital polymers. Although we recognize that due to baseline errors, experimentally determined Mn values can have greater than twice the error of Mw values, for our digital polymers at a fixed PDI the assigned Mn error equaled that assigned to Mw. SCB error was assigned ±0.1 SCB/1000 C. Error for the weight fraction of polymer made in C1 (±0.004) and PSP2 (±0.1) were based on the minimum error needed to obtain nonsignificant lack of fit values for the selected models. Lastly, density values calculated with Equations (22)–(25) were assigned errors of ±0.002 g cm−3 based on our previously reported results.[13] All responses were modeled independently. ANOVA tables and DoE equations used to predict both the primary and derivative responses are the provided in Tables S1–S10 of the Supporting Information. When exploring conditions needed to make targeted resins using the reverse RSM model, optimization conditions were determined by calculating the desirability function[21] provided in the Stat Ease software. The overall desirability function (D) is calculated by first assigning each response Yi(x), a desirability function di(Yi) that ranges between di(Yi) = 0 (undesirable response) to di(Yi) = 1 (highly desirable response). The response transform used by the software to generate the desirability function depends on the desired goal to minimize, maximize, or target a response value. In this work, we targeted a response value within a specific range. In this case, the response transform is based on the NominalThe-Best as outlined by Derringer and Suich.[23] The calculation involves the predicted value ( yˆ), the targeted response (T), the lower (Timin), and upper (Timax) limits of the targeted response
Macromol. React. Eng. 2018, 1700066
d i ( yˆ ( x )) =
0 si
yˆ ( x ) − Timin T − T min i i ti Timax − yˆ ( x ) T max − T i i 1
(39)
where d = 0, if
yˆ ( x ) ≤ Timin
d = 1,
Timin < yˆ ( x ) ≤ Timax
if
or yˆ ( x ) ≥ Timax
(40)
or y i ( x ) = Ti (41)
The values of s and t in Equation (39) are user-specified parameters (s, t > 0) that weight the importance of hitting the target. These weight exponents change the shape of d as it approaches the target. These values are assigned 1 (linear approach), 0.1 (concave curve), or 10 (convex curve) and give added emphasis to the upper/lower bounds. We set s = t = 1 (desirability function increases linearly toward Ti) in the present investigation. The overall desirability objective function, D calculated as the geometric mean of all transformed responses, is shown in Equation (42)
(
D = ( d1 )
w1
w (d2 ) … (d p ) 2
)
∑ wi
w p 1/
(42)
where dp is the desirability range for each response, and w assigns the relative importance to each response. In the Design Expert software, the importance values range from 1 to 5 and are centered at an importance value of 3 (i.e., no importance emphasized and w = 1). In the case of multiple solutions, we selected solutions that had D values greater than 0.999 for further analysis.
3. Results and Discussion 3.1. RSM Models The main goal of the proposed methodology was to use RSM in the laboratory to minimize the number of experiments in the DoE, while retaining an acceptable predictive ability. Typically, IV-optimal designs require fewer experiments than those needed by other commonly used DoE models. For example, using the five catalyst and reactor input variables listed in Table 2, the IV-optimal DoE requires 31 samples (including five replicate samples), whereas a center composite design would require 50 samples. If six factors were used, the IV-optimal DoE requirements increases to 38 samples, but the center composite DoE requirements increases to 86 samples. Table 3 lists the polymerization conditions and primary responses for the initial IV-optimal DoE (calibration set). Table S1 of the Supporting Information lists the predictive ability and ANOVA output for this initial design. We found
1700066 (7 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
Table 3. Initial DoE set for catalyst and reactor conditions. Reactor and catalyst input
Primary response values
E
C6/E
Cat1/Cat2
Temp
Time
C1 Mw
C2 Mw
SCB C1
SCB C2
1
0.2
0
2.9
120
2.0
9 649
105 471
0.0
0.0
0.219
2
0.63
0.8
2.79
124
2.0
8 956
118 261
5.0
28.1
0.221
Run
Wt frac C1
3
0.44
0.32
0.77
131
2.0
9 631
131 895
2.1
11.6
0.527
3R
0.44
0.32
0.77
131
2.0
9 920
135 852
2.0
11.5
0.531
4
0.75
0
4
137.06
2.0
8 356
199 421
0.0
0.0
0.185
5
0.75
0.8
0.6
140
2.0
8 646
118 087
5.2
24.0
0.595
6
0.75
0.8
3.58
140
2.0
8 646
118 087
5.2
24.0
0.198
7
0.2
0.58
3.66
140
2.2
9 014
66 628
3.8
18.1
0.201
8
0.2
0.8
1.18
136.4
3.3
8 718
70 889
5.1
24.9
0.480
9
0.2
0.8
0.6
120
3.6
9 023
89 266
4.9
29.3
0.617
10
0.75
0
0.6
120
4.0
9 650
275 138
0.0
0.0
0.643
11
0.45
0
2.15
140
5.2
8 159
114 987
0.0
0.0
0.402 0.406
11R
0.45
0
2.15
140
5.2
7 914
111 537
0.1
0.1
12
0.51
0.34
4
120
5.3
10 130
151 698
2.1
13.8
0.235
12R
0.51
0.34
4
120
5.3
10 434
156 249
2.2
13.9
0.231 0.404
13
0.75
0.42
2.31
132
6.4
9 509
153 063
2.7
14.7
13R1
0.75
0.42
2.31
132
6.4
9 794
157 655
2.6
14.8
0.408
13R2
0.75
0.42
2.31
132
6.4
9 224
148 471
2.8
14.8
0.400
14
0.2
0
0.6
129.5
7.6
8 891
77 579
0.0
0.0
0.744
15
0.2
0.8
3.83
127.3
7.6
8 896
82 297
5.0
27.2
0.299
16
0.6
0.76
4
140
8.4
8 719
111 951
4.9
23.0
0.293
17
0.62
0.8
0.6
128
8.6
8 884
117 694
5.0
27.0
0.746
18
0.2
0
4
120
9.2
9 649
105 471
0.0
0.0
0.309
19
0.2
0.74
1.37
140
9.6
8 752
66 273
4.8
22.5
0.557
20
0.32
0.44
1.81
120
10.0
9 943
121 825
2.8
17.5
0.503 0.309
21
0.75
0.8
4
120
10.0
9 024
120 303
4.9
29.3
22
0.62
0
3.27
126.5
10.0
9 121
221 822
0.0
0.0
0.381
23
0.44
0.29
1.62
135
10.0
9 441
125 197
1.9
10.1
0.543
24
0.75
0
0.6
140
10.0
8 160
183 031
0.0
0.0
0.751
25
0.75
0.77
1.73
140
10.0
8 701
119 714
5.0
23.2
0.500
26
0.2
0.13
4
140
10.0
8 890
60 090
0.9
4.4
0.310
Wt frac C1
Table 4. Validation sample set A. Reactor and catalyst input
Primary response values
Run
E
C6/E
Cat1/Cat2
Temp
Time
C1 Mw
C2 Mw
SCB C1
SCB C2
V-1A
0.34
0.2
3.15
132.0
6.0
9528
116 130
1.3
7.3
0.326
V-2A
0.34
0.7
0.6
125.0
3.0
9184
106 529
4.4
24.9
0.612
V-3A
0.34
0
4
140.0
8.0
8158
88 404
0.0
0.0
0.299
V-4A
0.2
0.35
2.3
125.0
6.0
9886
93 999
2.2
13.4
0.381
V-5A
0.47
0.6
3.15
126.5
7.6
9380
122 288
3.8
21.4
0.342
V-6A
0.61
0.4
1.73
135.0
5.2
9407
139 906
2.6
13.6
0.449
V-7A
0.61
0.2
1.35
125.0
6.0
9935
178 183
1.3
8.0
0.514
V-8A
0.47
0.8
3
123.0
4.0
8973
111 741
5.0
28.4
0.261
V-9A
0.68
0.7
0.95
133.0
4.0
8999
124 921
4.5
22.9
0.555
Macromol. React. Eng. 2018, 1700066
1700066 (8 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de Table 5. Results from evaluation of the initial DoE design space. V-1A Input
DoE Pred.
95% PI low
C1 Mw (g mol−1)
9 528
9 325
9 031
9 628
C2 Mw (g mol−1)
116 130
103 521
97 111
110 354
SCB C1 (/1000 TC)
1.3
1.3
1.2
1.4
SCB C2 (/1000 TC)
7.3
7.3
7.1
7.5
0.326
0.309
0.299
0.319
Input
DoE pred.
95% PI low
95% PI high
C1 Mw (g mol−1)
9 886
9 898
9 577
10 230
mol−1)
Wt frac C1
95% PI high
V-4A
Figure 5. Selection of validation experiments (open circles) from the initial set of experiments (solid circles).
R2
(R2 =
that reasonable predictive values 0.851 to 0.999) for the all the primary response variables were obtained by the model. More importantly, this approach is not restricted to the catalyst system adopted in this investigation: any digital or experimental set of polymerization kinetics parameters with two (or more) single-site catalysts could have been used to generate and test the general concept of using RSM models. A set of validation samples (Table 4) was used to further test the predictability of the initial RSM forward models generated from the design described in Table 3. These validation samples were chosen by visually inspecting the design space and selecting samples that lie in the gaps of the design space (samples not included in the calibration set). This process is illustrated in Figure 5 for a plot of the ethylene flow rate and C6/E input variables. Table S2 of the Supporting Information shows that the model fitted the data well both primary and derivative responses (within the 95% predictive interval limits) for 7 out of the 9 samples. For samples V-1A and V-4A, however, a few predicted responses (Mw and weight fraction of polymer made on catalyst 1) fell slightly out of the 95% predictive interval limits, as highlighted in Table 5. Considering these results, the initial DoE given in Table 3 was augmented with the two outlier samples. The final RSM
93 999
85 261
79 949
90 924
SCB C1 (/1000 TC)
2.2
2.2
2.1
2.3
SCB C2 (/1000 TC)
13.4
13.4
13.2
13.6
Wt frac C1
0.381
0.383
0.369
0.398
C2 Mw (g
forward model generated using the augment DoE (new calibration set) fitted all primary responses well, as shown in Table 6. The final RSM forward model was also reevaluated using the remaining 7 validation samples from Table 4 and the additional 9 validation samples show in Table 7. Once again, the second set of validation conditions were chosen from gaps in 33-sample set. The results for the reevaluation of the augmented DoE using the 16 validation samples show excellent predictive fits for all primary responses (Table S3, Supporting Information). Figure 6a,b illustrates the predictive fits for Mw and weight fraction of polymer made on catalyst 1, respectively, for both sets of validation samples using the augmented DoE.
3.2. Derivative Responses Derivative responses are those calculated from the primary responses. For example, the values for the Mw, Mn, and PDI for the polymer reactor blend are the weighted average of these same responses for each polymer population. Similarly,
Table 6. Primary response variable ranges, assigned errors, and model fits for a 33-sample set. Response
C1 Mw
C2 Mw
SCB C1
SCB C2
Units
g mol−1
g mol−1
SCB/1000 TC
SCB/1000 TC
Minimum
7 996
60 090
0
0
Maximum
10 130
275 138
5.2
29.3
0.751
3%
3%
0.1
0.1
0.004
Transformation
Base 10 log
Base 10 log
None
None
Inverse Sqrt
Model
RQuadratic
Assigned error (±)
Wt frac C1
0.185
RQuadratic
RQuadratic
Linear
RQuadratic
Adjusted R2
0.9508
0.9872
0.9992
0.9999
0.9981
Predicted R2
0.9403
0.9776
0.999
0.9999
0.9969
Macromol. React. Eng. 2018, 1700066
1700066 (9 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
Table 7. Validation sample set B. Reactor and catalyst input
Primary response values
Run
E
C6/E
Cat1/Cat2
Temp
Time
C1 Mw
C2 Mw
SCB C1
SCB C2
V-1B
0.27
0.5
1.45
122.0
4.0
9 740
109 174
3.1
19.2
Wt frac C1 0.424
V-2B
0.27
0.2
3
133.0
8.0
9 471
96 554
1.3
7.3
0.372 0.473
V-3B
0.68
0.5
1.2
123.0
4.0
9 707
144 408
3.1
19.0
V-4B
0.68
0.2
2.3
130.0
7.0
9 643
178 368
1.3
7.5
0.417
V-5B
0.47
0.8
3
123.0
4.0
8 973
111 741
5.0
28.4
0.261
V-6B
0.68
0.6
3.15
137.0
9.0
9 078
128 610
3.9
19.3
0.361
V-7B
0.34
0.5
0.9
123.0
8.0
9 706
118 205
3.1
19.0
0.646
V-8B
0.47
0.23
1
123.0
4.0
10 063
157 835
1.5
9.3
0.524
V-9B
0.47
0.12
3.74
123.0
9.5
9 942
172 685
0.8
5.0
0.335
density and PSP2 values are calculated directly from the primary response values, and can be combined to give the respective blend response. Therefore, there is no need to develop a forward RSM model to estimate derivative responses from reactor input data. However, to build the reverse model (that is, a model that will predict the polymerization conditions needed to obtain a targeted set of blend properties that include derivative responses) forward models for derivative responses must be constructed so that optimization equations can be applied to solve the reverse problem. Using the primary responses given in Table 3 and samples V-1A and V-4A to calculate the derivative responses, we constructed RSM forward models for the blend Mw, Mn, density, and PSP2 responses for the 33 sample DoE. Figure 7 shows that the model fitted the data very well for all cases (predicted R2 > 0.95).
3.3. Explanatory Validation Another benefit in using the RSM approach is how easy explanatory validations can be conducted. In other words, do these results make sense? Such validations can be done by constructing contour plots and noting the effects of changing input variables on the selected response, or by simply assessing the directional effects of variable coefficients of the chosen factors. For example, Figure 8a shows how the Mw predicted for the polymer blend varies with the ratio of the two catalysts
(Cat 1/Cat 2) and ethylene concentration in the reactor, while the other model parameters (C6/C1 = 0.4, T = 130 °C, t = 4 min) remain the same. The forward RSM model predicts that the Mw for the blend increases as both ethylene concentration and Cat 1/Cat 2 ratio increases. This makes sense, since Cat 1 makes polymer populations with higher molecular weights, and because the molecular weight of polyethylene increases with increasing ethylene concentration in the reactor. Both effects are directionally correct. Likewise, at the same reactor settings, the predicted resin densities (Figure 8b) are affected primarily by the catalyst ratio. One again, as the molecular weight goes up, density should decrease at a set C6/C2 ratio. Plotted response data can also help validate the overall directional predictability for the forward model. Figure 9 shows that as the polymer density increases, the PSP2 decreases. This is what one would expect, since the lamella thickness generally decreases as the polymer density is lowered, making it more likely that the polymer chains become tie molecules bridging two (or more) lamellae.[16–19] These complex responses, and other known structural effects can be predicted by the forward RSM model and the input polymerization variables.
3.4. Reverse Models One of the strengths of the RSM approach is that a reverse model can be generated, allowing one to select response targets and solve for what polymerization conditions are needed to
Figure 6. Comparison between DoE predicted values and digital data: a) Mw values for C2; b) weight fraction of polymer made by catalyst 1.
Macromol. React. Eng. 2018, 1700066
1700066 (10 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
Figure 7. Comparison plots for blend responses log Mw, log Mn, density, and PSP2 calculated from primary responses and those predicted by RSM forward models.
meet those targets using the numerical optimization function contained in the RSM software. The strategy we used was to first define the resin target in terms of its density and molecular weight average. These targets are typically used in industry and dictate the mechanical and processing properties of the resin. Although no rheological responses are included in this current DoE model, responses such as Carreau–Yasuda parameters could also be easily added. The results given in Table 8 demonstrate the optimization process. Table 8 first shows the catalyst and reactor input variable ranges. These values can be specifically set to a desired value or range of values. In this example, the entire DoE range for each input variable is considered. Moreover, each input value can be assigned weighting values (s and t), as given in Equation (39), as well as the importance of each variable (w),
as given in Equation (42). All input variables in Table 8 were assigned weighing values of 1 (linear weights) and importance values of 3. Likewise, each response variable can be targeted to a specific value or range, and values for weight and importance. As is the input values, response variables were assigned weight and importance values of 1 and 3, respectively. Lastly, for each set of targeted properties, desirability values are calculated using Equation (42) to convey how well each run condition (solution) meets the desired targets. Desirability values help discern between multiple solutions that often result in numerical optimization searches. Although desirability can range from 0 to 1, in the two examples given in Table 8 we used desirability values >0.999 as a cut off value. In Set 1, run conditions that produced a product with blend Mw = 150 000 ± 7500 g mol−1 and density = 0.950 ± 0.002 g cm−3
Figure 8. Surface response plots for a) log Mw of polymer blends as a function of ethylene concentration and catalyst ratio. b) Polymer blend density as a function of ethylene concentration and catalyst ratio. For both plots, the 1-hexene to ethylene ratio, temperature, and time were set to 0.4, 130 °C, and 4 min, respectively.
Macromol. React. Eng. 2018, 1700066
1700066 (11 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
14
PSP2 Values
12 10 8 6 4 2 0 0.93
0.94
0.95 0.96 0.97 3 Density (g/cm )
0.98
Figure 9. Comparison of directional trends in blend responses.
Figure 10. Blend PDI and weight fraction polymer made on catalyst 1 for the 48 solutions in Table 8. All resins meet the criteria Mw = 150 000 ± 7500 g mol−1 and density = 0.950 ± 0.002 g cm−3.
were searched using numerical optimization methods. As show in Table 8, 48 various combinations of reactor conditions can be used to make a polymer with these targeted properties. Of course, the other polymer properties such as weight fraction of catalyst 1, PDI, SCB content, etc., may be very different for each of the 48 samples. This point is illustrated in Figure 10 where the blend PDI and the weight fraction of polymer made on catalyst 1 are plotted for the resins that meet the targeted Mw and density values and have desirability values >0.999. Conversely, if our search criteria focused on blend targets that were not in the range of the DoE, then no solutions would be found. For example, from Figure 10, a blend with Mw = 150 000 g mol−1 and density of 0.950 g cm−3 having a weight fractions of catalyst 1 = 0.25 ± 0.004 and PDI = 12.0 ± 0.1 cannot be made using the run conditions making up the DoE.
To reduce the number of solutions obtained from this approach the desired resin targets need further defining. Keeping the blend Mw and density targets the same, if we target a resin with the C1 weight fraction equal to 0.3 ± 0.004 and blend PDI equal to 10.8 ± 0.1, only one solution (Set 2) is found with a desirability value >0.999. The run conditions for this solution are given in Table 9. As previously noted, further options in these optimization programs include setting weighted values for the input or responses and well as importance values, both of which helps reduce multiple solutions. Alternately, constraints on either reactor or response variable can be imposed. Lastly, another interesting aspect of the RSM reverse response models is the resulting nontargeted response values that result in a prediction. In the numerical optimization process used to derive a set of reactor conditions that produces
Table 8. Optimization settings for targeted product. Constraints name
Lower limit
Upper limit
Weight
Weight
Importance
Set 1 goal
Set 2 goal
Input
Variable
A:CE
0.2
0.75
1
1
3
is in range
is in range
Input
B:C6/CE
0
0.8
1
1
3
is in range
is in range
Input
C:CAT/CAT2
0.6
4
1
1
3
is in range
is in range
Input
D:Temp
120
140
1
1
3
is in range
is in range
Input
E:Time
2
10
1
1
3
is in range
is in range
7914
10 434
1
1
3
is in range
is in range
mol−1)
Primary response
C1 Mw (g
Primary response
C2 Mw (g mol−1)
60 090
275 138
1
1
3
is in range
is in range
Primary response
SCB C1 (per 1000 TC)
0
5.2
1
1
3
is in range
is in range
Primary response
SCB C2 (per 1000 TC)
0
29.3
1
1
3
is in range
is in range
Wt frac C1
0.185
0.751
1
1
3
is in range
target = 0.3 (± 0004)
Derivative
Blend Mw (g mol−1)
142 500
157 500
1
1
3
target = 150000
target = 150000
Derivative
mol−1)
Primary response
5 355
19 065
1
1
3
is in range
is in range
Derivative
Blend PDI
3.8
14.2
1
1
3
is in range
target = 10.8 (±0.1)
Derivative
Blend density (g cm−3)
0.948
0.952
1
1
3
target = 0.950
target = 0.950
Derivative
Blend PSP2
0.7
19.2
1
1
3
is in range
is in range
Solutions with desirability >0.999
48
1
Blend Mn (g
Macromol. React. Eng. 2018, 1700066
1700066 (12 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
Table 9. Run conditions and predicted responses for Set 2 given in Table 8. Run conditions CE
C6/CE
Cat 1/Cat 2
Temp
Time
0.61
0.097
2.3
122.7
3.03
Prediction
95% Cl low
95% Cl high
95% PI low
95% PI high
9 672
9 546
9 799
9 357
9 997
Response C1 Mw (g mol−1) −1
210 224
202 095
218 680
192119
230 035
SCB C1 (/1000 TC)
0.6
0.5
0.6
0.5
0.7
SCB C2 (/1000 TC)
3.9
3.8
4.0
3.7
4.1
C2 Mw (g mol )
Wt frac C1
0.30
0.2944
0.3058
0.2885
0.3122
Blend Mw (g mol−1)
149 986
141 530
158 946
134166
167 671
Blend Mn (g mol−1)
14 286
13 707
14 889
13145
15 525
Blend PDI
10.8
10.4
11.2
9.9
11.7
Blend density (g cm−3)
0.950
0.948
0.952
0.945
0.955
9.9
8.9
11
7.7
12.2
Blend PSP2
the targeted responses, the response surfaces for the other nontargeted structural properties can also be accessed and the respective values calculated. For example, in Table 8, blend Mw, density, weight fraction of polymer made on catalyst 1, and blend PDI were used in the optimization criteria for Set 2. Yet the predicted responses not used in the optimization criteria (e.g., Mw values for C1 and C2) are all in good agreement with the predicted blend Mw, as well as the predicted blend Mn (also not used in the optimization criteria). This is evident by comparing the calculated derivative responses from the predicted component molecular weights using Equations (22) and (23). As shown in Table 10, the predicted results all four molecular weight responses are relatively self-consistent even though they were estimated using separate, independent models. Even the PDI values calculated from the derivative blend data (twice removed from the primary input) fell within the 95% confidence limits for the independent blend PDI prediction. The connectivity of these various response surfaces makes the reverse RSM model a powerful tool for catalyst and resin design.
of runs needed to adequately reproduce and predict results obtained through complex reaction and property models. Furthermore, by using RSM reverse response models and optimization software we showed that the reactor design space can be searched to provide a set of reactor conditions that produces a bimodal resin with a specific set of resin properties. Multiple solutions (reactor conditions) that provide similar targeted properties can be prioritized on how well they meet the targeted values using the desirability objective function provided in the software package or by simply applying constants to response targets, reactor variables or both. The methods described in this study will aid researchers in the design of catalyst and bimodal resins for specific applications as well lead to a better understanding of these systems. To that end, future efforts will be directed to refining the initial kinetic model described in this work. The feasibility of applying these methods was successfully demonstrated digitally using fundamental kinetic and property models. In Part 2 of this paper these methods will be applied to experimental catalyst systems. The application of the RSM in the digital studies should prove a helpful tool in guiding subsequent laboratory work.
4. Conclusions In these studies, we have demonstrated the utility of using properly designed RSM models in mapping the structure–property space of bimodal polyethylene. This was accomplished by using a five factor IV-optimal design, validation and augmentation methods which significantly reduces the typical number
Supporting Information Supporting Information is available from the Wiley Online Library or from the author.
Acknowledgments Table 10. Connectivity between primary and derivative responses. Derivative response
Calculated from C1 + C2
DoE predicted
Blend Mw (g mol−1)
150 058
149 986
Blend Mn (g mol−1)
14 557
14 286
10.3
10.8
Blend PDI
Macromol. React. Eng. 2018, 1700066
Calculated from DoE predicted
The authors thank Dr. Martin Bezener from Stat-Ease, Inc for his helpful discussions concerning RSM calculations and Dr. Youlu Yu for SEC-IR data. The authors also thank Chevron Phillips Chemical Company LP for support of this work.
Conflict of Interest 10.5
The authors declare no conflict of interest.
1700066 (13 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.advancedsciencenews.com
www.mre-journal.de
Keywords bimodal polyolefins, desirability functions, polymerization kinetics, response surface methods, structure–property relationships Received: November 27, 2017 Revised: February 13, 2018 Published online:
[1] J. B. P. Soares, T. McKenna, Polyolefin Reaction Engineering, Wiley-VCH, Weinheim, Germany 2012. [2] G. V. Schultz, Z. Phys. Chem. 1935, B30, 379. [3] G. V. Schultz, Z. Phys. Chem. 1938, A182, 127. [4] G. V. Schultz, Z. Phys. Chem. 1939, B43, 25. [5] G. V. Schultz, Z. Phys. Chem. 1939, B43, 47. [6] P. J. Flory, J. Am Chem. Soc. 1936, 58, 1877. [7] P. J. Flory, J. Am Chem. Soc. 1937, 59, 241. [8] F. R. Mayo, F. M. Lewis, J. Am. Chem. Soc. 1944, 66,1594. [9] J. B. P. Soares, Macromol. React. Eng. 2014, 8, 235.
Macromol. React. Eng. 2018, 1700066
[10] SEC-IR5 data obtained on a solution polymerized, dual metallocene catalyzed resin supplied by University of Alberta. [11] P. J. DesLauriers, D. C. Rohlfing, E. T. Hsieh, Polymer 2002, 43, 149. [12] P. J. DesLauriers, ACS Symp. Ser. 2005, 893210. [13] P. J. DesLauriers, D. C. Rohlfing, Macromol. Symp. 2009, 282, 136. [14] M. P. McDaniel, P. J. DesLauriers, Kirk-Othmer Encyclopedia of Chemical Technology, John Wiley & Sons, Inc., New York 2015, pp. 1–40. [15] J. Janzen, D. Register, ANTEC 1996, 54, 2190. [16] Y. L. Huang, N. Brown, J. Polym. Sci., Part B: Polym. Phys. 1990, 28, 2007. [17] Y. L. Huang, N. Brown, J. Polym. Sci., Part B: Polym. Phys. 1991, 29, 129. [18] R. Seguela, J. Polym. Sci., Part B: Polym. Phys. 2005, 43, 1729. [19] R. Seguela, Macromol. Mater. Eng. 2007, 292, 235. [20] B. Xiong, O. Lame, J. Chenal, C. Rochas, R. Seguela, G. Vigier, Macromolecules 2015, 48, 2149. [21] R. H. Myers, D. C. Montgomery, C. M. Anderson-Cook, Response Surface Methodology: Process and Product Optimization Using Design Experiments, 3rd ed., John Wiley & Sons, Hoboken, NJ 2009. [22] L. D’Agnillo, J. B. P. Soares, A. Penlidis, J. Polym. Sci., Part B: Polym. Phys. 2002, 40, 905. [23] G. Derringer, R. Suich, J. Qual. Technol. 1980, 12, 214.
1700066 (14 of 14)
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
