Quantitative Modeling of the Temperature Dependence of the Kinetic

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Quantitative Modeling of the Temperature Dependence of the Kinetic Parameters for Zirconium Amine Bis(Phenolate) Catalysts for 1-Hexene Polymerization Jeffrey M. Switzer, Paul D. Pletcher, David Keith Steelman, Jungsuk Kim, Grigori A Medvedev, Mahdi M. Abu-Omar, James M. Caruthers, and W. Nicholas Delgass ACS Catal., Just Accepted Manuscript • DOI: 10.1021/acscatal.8b01989 • Publication Date (Web): 27 Sep 2018 Downloaded from http://pubs.acs.org on September 28, 2018

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Quantitative Modeling of the Temperature Dependence of the Kinetic Parameters for Zirconium Amine Bis(Phenolate) Catalysts for 1Hexene Polymerization Jeffrey M. Switzer,†* Paul D. Pletcher,‡ D. Keith Steelman,‡ Jungsuk Kim,† Grigori A. Medvedev,† Mahdi M. Abu-Omar,‡#* James M. Caruthers,† and W. Nicholas Delgass† †Davidson School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, Indiana 47907; ‡Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, Indiana 47907. #Current Address: Department of Chemistry and Biochemistry, University of California, Santa Barbara, Santa Barbara, California 93106. *Corresponding Author addresses: [email protected]; [email protected]

Abstract The chemical kinetics for a series of three zirconium amine bis(phenolate) catalysts for poly 1hexene polymerization have been examined as a function of temperature. Detailed modeling of the experimental data has yielded the activation parameters for many of the reaction rate constants, including those for propagation, initiation, chain transfer, and monomer misinsertion and recovery. While existing literature is sparse, the results herein generally agree with previously published values, specifically that for the propagation rate constant the magnitude of the enthalpy of activation is low (12 kcal mol−1 and below), and the magnitude of the entropy of activation is moderate (up to −27 cal mol−1 K−1). With regard to the remaining rate constants, Arrhenius behavior is observed in most cases despite the complexity of the temperature dependence in the two step adsorption/insertion kinetics. The rate expression for these reactions approaches an Arrhenius form in certain limiting cases of the relative elementary rate constants. The reaction rate data are compared against these limiting cases, leading to the conclusion that docking, with an early transition state, is rate limiting for propagation and placing bounds on the interpretation of most of the remaining constants. The challenges encountered in assigning temperature dependent rate constants for the polymerization reactions is indicative of both the significant complexity in modeling a reaction in which thousands of species are present and the extreme care that is required in generating reproducible data. Keywords: 1-hexene; polymerization; catalysis; kinetic modeling; mechanism; temperature dependence; activation parameters; Eyring analysis

Introduction The effect of temperature on olefin polymerization by single-site catalysts has been studied by a number of research teams. The results of these studies provide the general conclusions that

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when reaction temperature increases, catalyst activity increases,1–6 polymer chain length decreases4–7 and stereoregularity decreases.2,3,5,7–9 The explanation for these results based on reaction kinetics is that while the rate of chain propagation increases with rising temperature, competing reactions for chain transfer, epimerization, and regioerrors increase more quickly. While these temperature studies date to the 1970s with work by Andresen, et al., on ethylene polymerization by alkylaluminum-activated titanium metallocenes,1 it wasn’t until much later that attempts were made at calculating activation parameters for the chemical reactions that govern single-site polymerization. Reliable activation parameters are dependent upon robust kinetic rate constants. However, less precise kinetic parameters are often used because they are easier to determine. Polymerization activity, defined as the average rate of polymer mass created per catalyst site, provides an approximate measurement of how fast a catalyst grows polymer and is therefore an industrially important parameter. In 2000, Alt and Köppl presented a comprehensive review on the activity of ethylene polymerization by Group IV metallocenes.10 They compared catalyst structures with the highest and lowest activities in an attempt to understand why some catalysts have higher activity than others. Ultimately, they identified steric crowding around the active site as a key feature that influences activity, although they admitted that a quantitative relationship was impossible to provide. One reason may be that activity, reported often as a single value, may not be constant throughout a reaction. In batch scale polymerization, decreasing monomer concentration will cause activity to fall; even semi-batch scale polymerization, where monomer concentration is maintained at a constant value, can have varying activity due to side reactions such as dormant site formation or catalyst deactivation. While activity can be a useful measurement in screening large numbers of catalysts for specific applications, it is not appropriate for making quantitative mechanistic comparisons between catalytic systems and is not the fundamental quantity needed for determining activation parameters. To move beyond this limitation, the kinetic rate constants for each step of the polymerization mechanism must be extracted from experimental data. Rytter and coworkers provided one of the earliest examples.11,12 In 1998 they published a study11 of ethylene and propylene polymerization by several metallocene zirconium dichloride catalysts, activated by methylaluminoxane (MAO), over large (approx. 100 °C) temperature ranges. They measured catalyst activity as a function of time, reporting an average value over an hour, but concluded that a kinetic model with rate constants provides a better description of the polymerization behavior than just the activity. Their kinetic model included reactions for “activation” (analogous to what is commonly called initiation), propagation, “latent site” formation (akin to dormant site formation), chain termination, and deactivation of the catalyst. Their “corrected activity,” i.e. the propagation rate constant, differs in some cases by over an order of magnitude from the activity measurement, again reflecting the need for an accurate kinetic analysis. This proper analysis also allowed them to determine activation enthalpy and entropy for each rate constant, providing predictive ability at additional temperatures. A number of studies followed in which data were collected and used to extract the rate constants for a polymerization mechanism, including the propagation,13–22 initiation,13–15,23 chain transfer,13–16,24 and deactivation25 rate constants. However, the temperature dependence of the rate constant was only determined in some cases.15–19,25 In 2001, Liu, et al., published a


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complete kinetic treatment for 1-hexene polymerization by the “single-site” catalyst rac-(C2H4(1indenyl)2)ZrMe2/B(C6F5)3.15 They collected data over a 60 °C temperature range and provided both rate constants and activation parameters for the initiation, propagation, and chain transfer reactions, which were assigned through the analysis of the data one response at a time rather than by modeling the data as a whole. A similar study followed from the same team with analogous results for propylene polymerization,16 eventually leading to a broader body of work in which activation parameters were collected for several catalysts activated by either MAO or by boron complexes.17,18,26,27 Ciancaleoni, et al., summarize a number of these findings,26 hypothesizing that the slow propagation rate constants seen in these catalyst systems (as compared to industrially relevant systems) are due to the relatively larger in magnitude entropy of activation (−30 cal mol–1 K–1 or more) and relatively small enthalpy of activation (5–10 kcal mol–1). The accuracy of the activation parameters depends on accurate rate constants over a large temperature range. Unfortunately, the piecemeal approach to kinetic modeling may not provide the accuracy required. For example, the Liu, et al., study15 was repeated by Novstrup, et al.,28 but the modeling approach highlighted the simultaneous fitting of all data, which included the precise shape of the molecular weight distribution (rather than just averages) as a function of time. A key finding was the correction of the propagation rate constant by a factor of approximately 2 at 0 °C. Corrections at the other temperatures would likely lead to changes to the entropy and enthalpy of activation for the reactions. This article examines 1-hexene polymerization by a series of three zirconium amine bis(phenolate) catalysts at several temperatures, with each catalyst containing a different pendant group. Results at 25 °C have been discussed in our previous work.29–31 Data have been collected as a function of time in a batch reactor, and mechanisms and rate constants are assigned by modeling the entire set of data for a given catalyst at various temperatures simultaneously. These results are then critically examined and discussed, with the understanding that there are current limitations in available experimental techniques that prevent precise analysis of all parameters. Nevertheless, a number of important features of the kinetic model have been discovered.

Experimental Section General procedure All manipulations were performed in a vacuum manifold or under dry inert atmosphere in a glove box using air sensitive techniques under N2 or Ar atmosphere. Solvents were distilled over activated alumina and a copper catalyst using a solvent purification system (PPT—Pure Process Technology) and degassed before being stored over activated molecular sieves. Tetrabenzylzirconium was purchased from STREM and used as received. The monomer 1hexene was purchased from Aldrich and purified by distillation over a small amount of dimethyl bis(cyclopentadienyl)zirconium and stored over activated molecular sieves. Tris(pentafluorophenyl)borane was purchased from STREM and purified by sublimation. Diphenylmethane was purchased from Aldrich, degassed, and stored over molecular sieves.


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CD3OD was purchased from Cambridge Isotopes and used as received. D8-toluene was degassed and stored over molecular sieves. 1H and 2H NMR experiments were performed on a Varian INOVA600 MHz or Bruker DRX500 MHz spectrometer. The ligands and precatalysts were prepared following literature procedures.29–31

NMR scale quenched polymerization of 1-hexene with Zr[tBuONXO]Bn2 (X = SMe, THF, NMe2) The time dependent concentrations of different species were monitored by quenching the samples using d4-methanol at times representing 30%/60%/90% conversion of the initial 1hexene amount. The procedure for NMR scale polymerization is based on literature.29,30 As an example of a typical polymerization, Zr[tBuONSMeO]Bn2 (21.0 mg, 0.02625 mmol) was dissolved into 3.5 ml toluene under continuous stirring using a stir bar. Three 1 mL aliquots of the precatalyst solution were then separated into vials containing pierceable screw-top caps. A 3 ml syringe, needle, and a vial containing the precatalyst solution were placed into a N2 bag. The vial in the bag was submerged into a constant temperature bath at the requisite temperature. At 35 °C, an oil bath was used, at 0 °C an ice bath was used, at −17 °C a dichlorobenzene/dry ice bath was used, and at −20 °C, an acetone/dry ice bath was used. Meanwhile, tris(pentafluorophenyl)borane (14.1 mg, 0.0275 mmol), 1-hexene (0.4208 grams, 5 mmol), and diphenylmethane (8.4 mg, 8.33 mmol) were added to a 5 mL volumetric flask and diluted to the mark using d8-toluene. A 1.5 mL aliquot of this stock solution was added into each of three NMR tubes containing a pierceable septum. These monomer/activator solutions were then placed into a spectrometer and allowed to equilibrate to room temperature, and the initial monomer concentration was measured relative to the diphenylmethane standard. The sample was placed in the respective temperature bath and allowed to equilibrate to temperature. The catalyst solution was then added to the monomer/activator solution by piercing the cap while the syringe remained in the N2 bag. The reaction mixture was then shaken for 30 seconds outside of the bath before being returned to the bath. The reactions were quenched at different times by CD3OD. Each sample was dried, dissolved in hexane, filtered through alumina to remove dead catalyst, dried again, and placed under vacuum for 12 hours to get a total polymer weight, which was consistently within an expected range based on initial monomer weight and monomer conversion percentage. The material was analyzed by 1H NMR to determine the conversion of monomer in accordance with literature procedure.29,30 For vinyl analysis, 1.2 mL of CDCl3 was added to the dried polymer to completely dissolve the polymer. A 1 mL aliquot was removed and placed into an NMR tube. Diphenylmethane (70.0 mg, 0.42 mmol) dissolved in CDCl3 in a 5 mL volumetric flask was used as an internal standard using the method of standard additions, where 10 microliter aliquots were added to the sample to quantify the amount of end groups through 1H NMR. The sample was then dried and reweighed to verify the total concentration of vinyl groups. The two polymer samples were recombined and dried. For 2H analysis, a similar procedure to that for vinyl analysis was followed. Following quenching, 1.2 mL of dichloromethane was added to the dried polymer sample, and the polymer was dissolved. A 1 mL aliquot was removed and placed into a NMR tube. D6-benzene (80.0 mg, 0.95 mmol) was dissolved in dichloromethane in a 5 mL volumetric flask to be used as a standard.


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The sample was then analyzed utilizing the method of standard additions. It was then dried and weighed to determine the percentage of polymer analyzed and total amount of active sites from deuterium labeling. The procedure listed above was used for all catalyst systems, with the measured amounts adjusted to provide the desired experimental concentrations.

Gel permeation chromatography (GPC) analysis The procedure used to analyze polymer samples using GPC methods was taken from Novstrup et al.,28 and it is summarized here. Poly(1-hexene) samples were added to THF at room temperature and allowed to dissolve for 4 h. Solutions were then passed through a 0.2 µm filter to remove any particulate matter. The GPC analysis was performed on a Viscotek TDAmax. Samples were injected through a 200 µL injection loop and passed through three Viscotek T6000M 10 µm General Mixed Org columns in series in a 35 °C oven at a flow rate of 1.0 mL min−1. The analysis made use of the differential RI detector, a viscometer, and two light scattering detectors angled at 90° and 7°. Molecular weights were assigned by way of the standard triple detection calibration method. System parameters were calibrated with a 99,000 g mol–1 polystyrene standard. The calibration was verified through the analysis of a broad standard, SRM 706a, provided by the National Institute of Standards and Technology.

Kinetic modeling analysis In order to determine the kinetic models for each data set, the methods of analyzing rich, time dependent data described in our previous work have been employed.29 Such methodology was found sufficient to produce good fits of the experimental data in all cases, shown below in Figure 1. Errors in the rate constants were assigned by using the standard errors as calculated through the weighted least squares optimization routine, also described in our previous work.29

Results Three catalyst systems were studied to gauge the effects of temperature changes on 1-hexene polymerization. The precatalysts were: Zr-tBu4[ONXO]Bn2 [X = THF (1), NMe2 (2), SMe (3)] (see Scheme 1). In all cases, B(C6F5)3 was used to activate the precatalyst. All experiments were carried out in toluene. Each catalyst system was studied at three temperatures: 1 and 2 were studied at −20 °C, 0 °C, and 25 °C; and 3 was studied at −17 °C, 22 °C, and 35 °C. The mechanisms and rate constants for 1 and 2 at 25 °C and 3 at 22 °C have been previously published,29–31 but we revisit them here.

Scheme 1. Reaction and catalyst precursors


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(b)

(a)

(c)

(d)

10 -4

dwt/d(log MW)

10 0

(i)

1 1

0.5

0 0

3000

10 -3

1.5

2

10 -1 6000

0 0

6000

Time (s)

12000

0

Time (s)

10 0

3000

6000

Time (s)

10 -4

2

10 -3

3

4 2

(ii)

1

2

10 -2

1

0 0

500 1000 1500

0 3.5

Time (s)

10 -1 0

1500

4.5

10 -4

4

2

1

2

1

0.5

4.4

0 0

4.8

1500

dwt/d(log MW)

6 4 2

-2

0 500

0

1000

0

Time (s)

3000

500

1000

Time (s) 3

10 -3

[Vinyls] (M)

2

2 1

1

0

0 4

4.5

0

5

dwt/d(log MW)

10 0

10 -1

1

2

Time (s)

log(MW)

(vi)

1500

Time (s)

10 -4

0

(v)

3000

Time (s)

log(MW)

10 -1

1000 1500

10 -3

1.5

0 4

500

Time (s)

3

0 3.6

3000

0

Time (s)

10 0

10

500 1000 1500

6

Time (s)

(iv)

0 0

log(MW)

10 0

(iii)

4

[Active Sites] (M)

10

-1

[M]/[M] 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3

10 4

10 -3

1.5 1 0.5 0

0

300

600

0

Time (s)

300

600

Time (s)

Figure 1. Data and model fits for 1-hexene polymerization by 1–3/B(C6F5)3 in toluene. [1–3]0 = 3 mM; [B(C6F5)3]0 = 3.3 mM; [1-hexene]0 = 600 mM. Black: data; color: model fits. The rate constants for the models are in Tables 1–3. Row (i): 1 at −20 °C; Row (ii): 1 at 0 °C; Row (iii): 2 at −20 °C; Row (iv): 2 at 0 °C; Row (v): 3 at −17 °C; Row (vi): 3 at 35 °C. Column (a): monomer consumption; Column (b): molecular weight distributions at times corresponding to data in column (a); Column (c): vinylidene (solid) and vinylene (open) concentrations; Column (d): primary (solid) and secondary (open) active site concentrations.


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In all cases, the data sets consisted of the following: monomer concentration, vinylidene and vinylene concentration, primary and secondary deuterium incorporation following catalyst quenching, and molecular weight distribution of the polymer product, all as a function of reaction time in the batch reactor. Kinetic modeling methods discussed in previous work29 were used here to obtain kinetic mechanisms and rate constants that provide good fits of the data sets collected at each reaction temperature. The mechanisms were not assumed a priori to be the same, but the results of the fitting showed many similarities. In Figures 1 and 2, the model fits are color-coded by the corresponding catalyst as follows: THF pendant (1): Red; NMe2 pendant (2): Blue; SMe pendant (3): Yellow. Black represents experimental data, regardless of the catalyst. The data and model fits for 1–3 that have not been previously published are shown in Figure 1. The following observations were gathered from the data and used to help select a model: (i)

(ii)

(iii)

(iv)

In all cases, both primary and secondary deuterium labels were discovered. The labels, which originate upon the use of the quenching agent CD3OD, are assumed to affix to the growing end of the polymer, and their concentrations therefore represent the active site concentration before quenching. The presence of both primary and secondary labels is evidence that monomer misinsertion (that is, 2,1-insertion) occurs at some rate in all cases. In all cases, both vinylidene and vinylene groups were discovered. Vinylidene groups are assumed to originate from a chain transfer pathway (either monomer dependent or independent) where the reactant is a primary active site (a primary carbon is bonded to the metal), whereas vinylene groups originate from secondary active sites (a secondary carbon is bonded to the metal). Assuming that active site concentrations are constant (which they all roughly are, as seen in Figure 1d), vinyl groups will either form linearly with time (independent of monomer concentration) or will have a decreasing growth rate (dependent on monomer concentration). As seen in Figure 1, the vinyl formation rate always decreases late in the reaction at these temperatures. The vinyl formation pathways are therefore taken to be monomer dependent. This is assumed to occur through a β-H transfer to monomer pathway. All experiments shown were carried out with a 200:1 ratio of 1-hexene:precatalyst. If polymerization were “living,” one would expect a maximum chain length of approximately 16,800 g mol–1 (about 4.2 on the logarithmic scale). However, despite chain transfer reactions decreasing the chain length, in each case the maximum polymer molecular weight exceeds this value. The mechanism must account for this large molecular weight in some manner. Three possible mechanistic features that will achieve higher-than-living molecular weight are: (i) initiation is slow compared to propagation, (ii) the secondary active sites are slow to insert additional monomers, and (iii) not all of the precatalyst participates during polymerization (due to incomplete activation). In each of these three cases, the amount of working catalyst is smaller than the initial amount of precatalyst added, effectively increasing the 1-hexene:catalyst ratio. Each of these features has a different effect on the polymerization data and can therefore be distinguished from the


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alternative mechanism. It can be seen from the rate constants, shown in Tables 1–3, that all three of these mechanistic features exist for all catalysts to some degree. Table 1. Model rate constants for 1-hexene polymerization by 1/B(C6F5)3 in toluene. The rate constants at 25 °C have been revised from the original publication.30

−1

0 °C

25 °C

0.173 (±0.008)

1.56 (±0.05)

8.0 (±0.04)

0.0046 (±0.0001)

0.008 (±0.001)

0.06 (±0.01)

0.017 (±0.003)

0.019 (±0.003)

0.06 (±0.01)

s )

0.082 (±0.001)

0.344 (±0.008)

1 order

−1

0.090 (±0.001)

0.62 (±0.05)

1 order

−1

kp (M

s )

−1

−1

s )

−1

s )

kmis (M

−1

krec (M

−3

kvinylidene (10

−3

kvinylene (10 −1

–20 °C

−1

M

−1

M

−1

s )

−1

ki (M

s )

st st

0.00255 (±0.00005)

0.0397 (±0.0004)

0.39 (±0.31)

−1

s )

0.125 (±0.001)

0

0

Catalyst participation

0.83 (±0.02)

0.70 (±0.01)

0.42 (±0.01)

−3

kd (10

Table 2. Model rate constants for 1-hexene polymerization by 2/B(C6F5)3 in toluene. The rate constants at 25 °C have been revised from the original publication.29

−1

−1

kp (M

s )

–20 °C

0 °C

25 °C

0.73 (±0.04)

3.5 (±0.4)

10.5 (+1.4 / −0.6)

−1

s )

−1

0.018 (±0.007)

0.07 (±0.04)

0.09 (+0.15 / −0.04)

−1

−1

0.036 (±0.006)

0.12 (±0.05)

0.09 (+0.23 / −0.05)

s )

0.31 (±0.03)

4.2 (±0.9)

1 order

−1

0.29 (±0.04)

1.97 (±0.07)

1 order

ki > 0.09

ki > 0.2

0.22 (+0.01 / −0.06)

s )

−1

0

0

0


0.4975 (±0.0005)

0.485 (±0.006)

0.427 (+0.025 / −0.002)

kmis (M krec (M

s ) −3

kvinylidene (10

−3

kvinylene (10 −1

ki (M

−1

M

−1

s )

−1

s )

−3

kd (10

−1

M

st st


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Table 3. Model rate constants for 1-hexene polymerization by 3/B(C6F5)3 in toluene. The rate constants at 22 °C have been revised from the original publication.30 −1

−1

kp (M

s )

−1

s )

−1

s )

kmis (M krec (M

22 °C

35 °C

0.6 (+0.2 / −0.3)

10 (+4 / −3)

16 (±1)

0.16 (+0.07 / −0.09)

0.29 (±0.04)

0.037 (+0 / −0.008)

0.064 (±0.006)

0.0118 (+0.0002 / −0.0093) 0.00438 (+0.00002 / −0.00150)

−1

−1

−3

kvinylidene (10

–17 °C

−1

−1

s )

0.16 (+0.04 / −0.08)

2.1 (+0 / −0.9)

19 (±0.002)

−1

0.023 (+0 / −0.004)

1.0 (+0.1 / −0)

2.7 (±0.2)

0.00098 (+0.00066/ −0.00002)

0.028 (+0 / −0.02)

ki > 9

s )

−1

0.14 (±0.03)

5 (+3 / −2)

0


0.54 (+0 / −0.07)

0.7 (+0.3 / −0)

0.44 (±0.02)

−3

kvinylene (10 −1

ki (M

−1

M

s )

−1

s )

−3

kd (10

M

These observations were used to assist in model selection and optimization. The overall chemical mechanism, appearing in Scheme 2, includes the following reactions: (i) propagation, (ii) initiation (which is sometimes slow compared to propagation), (iii) misinsertion and recovery, (iv) monomer dependent vinylidene and vinylene formation, and in some cases (v) monomer independent catalyst deactivation. Catalyst participation was also less than 100% in all cases. The similarity of all these models allows us to compare rate constants and activation parameters among all reaction temperatures and catalysts. The bounds on the rate constants in Tables 1–3 were determined by optimizing the values of the rate constants over the entire data set available for each catalyst at each temperature. In an ideal case the optimization procedure would converge to a unique global minimum—the set of rate constants that provides best fit to the data. Then using the known uncertainty in the experimental data the error bounds on the converged rate constants can be obtained via standard statistical methods. Unfortunately, the ideal case does not occur for any of the investigated systems. Instead it was observed that rather than converging to a single point, the optimization procedure results in a region in the multi-dimensional space, where all sets of values of the rate constants in that region result in a comparable quality fit to the experimental data. This is illustrated in greater detail in the SI, where examples of different sets of rate constants and the corresponding fits to the experimental data are shown. Obviously, no onedimensional drawing can fully represent what in reality is a complex region in multi-dimensional space, so Figure 2 contains a compromise. When an interval is shown for a given rate constant, it means that if this particular rate constant is assigned a value outside the interval then the fit to experimental data is not possible no matter what values are used for the rest of the rate constants. However, it should be remembered that the rate constants are highly correlated, which means that a set where each rate constant is randomly chosen from within its interval is unlikely to result in a fit to data. Also it should not be assumed that the “best” value of a particular rate constant is located in the middle of the interval; rather, all values within the interval should be considered as equally probable. For the sake of consistency, this method was performed for our previously published data for 1 and 2 at 25 °C and 3 at 22 °C,29–30 and Tables


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1–3 also show these revised rate constant values and error bounds. Note that the previously published values for the rate constants fall within the intervals shown. The rate constants in Tables 1–3 are shown graphically in Figure 2 as Arrhenius plots. Although, of course, more data would have been desirable, we chose to measure rates at three temperatures in order to test the linearity of the plots. Figure 2a shows that kp obeys typical linear Arrhenius behavior for all catalysts over the temperature range studied. Figure 2 also shows that, other than possibly ki, the remaining rate constants (kmis, krec, kvinylidene, and kvinylene) for all catalysts obey Arrhenius behavior as well, or very nearly so, with the largest deviations occurring for kmis and krec of 1 and kvinylidene of 3. With regard to chain transfer, while the mechanism for 3 is the same at all temperatures, there appears to be a change in the chain transfer mechanism for 1 and 2 when the temperature changes. As previously noted, the rate of vinyl group formation slows as monomer is consumed in the lower temperature experiments (−20 and 0 °C). Our previously published results29,30 show that at the higher reaction temperature Scheme 2. Kinetic mechanism for 1–3. The (25 °C) vinyl groups form at a constant scheme includes both monomer dependent and rate regardless of the monomer monomer independent chain transfer with concentration. Thus the low temperature reinitiation, both of which are observed depending chain transfer rate constants for 1 and 2 on the precatalyst and reaction temperature. cannot be compared to the high temperature rate constants. We may still estimate activation parameters for monomer dependent chain transfer from the two available data points. Figures 2e and 2f show the line between the two low temperature data points to provide a rough comparison with the results from 3, which undergoes monomer dependent chain transfer at all temperatures.


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ACS Catalysis

kp

-2 -4 -6

kmis

-6 -8 -10

(a)

-8 3.2

3.6 -1

1/T (K )

4

10

(c) 3.6 -1

1/T (K ) 10

4 -3

kvinylidene

-9

-6 -10

-12 3.2

-3

ki

-2

(b)

-11 -13

(d)

-14 3.2

3.6 -1

1/T (K ) 10

4 -3

(e)

-15 3.2

(f) 3.6 -1

1/T (K ) 10

4 -3

Figure 2. Arrhenius plots for kinetic rate constants describing kinetic behavior of 1–3. Red: 1, Blue: 2, Yellow: 3. Solid lines show satisfactory linear fits. Dashed lines represent either poor linear fits or fits based on only two data points. Bars extending above upper vertical axis represent rate constants with uncertain upper bounds.

Discussion In order to model polymerization behavior, it is desirable to have a model that includes as complete a description of the detailed kinetic pathway as possible. In the case of polymerization catalysts studied here, this would mean, for example, that chain growth should be modeled by the two-step docking–insertion Cossee–Arlman reaction,32 which assumes that the monomer must first dock (or adsorb) reversibly to the catalyst site before it reacts irreversibly to add to the growing polymer chain, as shown in Scheme 3 for 1,2-insertion of 1-hexene.

Scheme 3. Cossee–Arlman mechanism with two elementary reaction steps: (1) reversible 1hexene monomer docking and (2) irreversible 1,2-insertion of the docked monomer into a hydride or alkyl group. Assuming a forward docking rate of k1, a reverse desorption rate of k−1, and an insertion rate of k2, and making the steady-state approximation, the full reaction rate is given by:


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r=

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k1k 2 Ctot ⋅ M k1M + k −1 + k 2

(1)

Note that in Equation 1 Ctot represents catalyst both with and without docked monomer but excludes the non-participating catalyst. While Equation 1 is first order in active catalyst, the reaction order in monomer will not always be first order, and the apparent rate constant,

k1k2 ( k1M + k −1 + k 2 ) , is not that of a single elementary step and therefore does not have the Arrhenius form. Quantifying the rate constant of each of these elementary steps separately requires experimental data that measures the species formed in each elementary reaction, requiring a significantly more elaborate experimental procedure than was available here. There are, however, three limiting cases described below for which the apparent rate constant does take the Arrhenius form, allowing extraction of further fundamental information. 1. k2 >> k1M + k −1 . This case represents a reaction that is rate limited by the docking process. Once a monomer docks it will very quickly insert into the polymer, leading to a catalyst coverage near zero. The reaction rate simplifies to

r = k1 ⋅ Ctot ⋅ M

(2)

The reaction appears first order in both monomer and catalyst, and the single parameter rate constant k1 represents the docking step of the reaction. The apparent entropy and enthalpy of activation will be those of k1. Since adsorption essentially always involves a loss of degrees of freedom of motion for the monomer, we expect ∆S‡ to be negative and relatively large in magnitude. We note that full loss of translational freedom in the adsorption transition state would yield the highest magnitude, but that value could be smaller in magnitude for an early transition state. On the other hand, ∆H‡ can take a wide range of values, but is not likely to be unusually high. 2. This case and case 3 occur when insertion is the rate limiting step.

This can be

represented as ( k1M + k −1 ) >> k2 . This leads, in general, to

r=

k1 ⋅ k2 ⋅ Ctot ⋅ M K1 ⋅ k2 ⋅ Ctot ⋅ M = k1 ⋅ M + k −1 K1 ⋅ M + 1

(3)

where K1 ≡ k1 k −1 is the equilibrium constant for monomer docking/adsorption to the catalyst site. This expression will not generally show Arrhenius behavior, but does have two limiting cases that do. We note that K1M

( K1M + 1)

is equal to the fractional

coverage of monomer on the catalyst. When K1M >> 1 , the coverage is equal to 1, and the rate reduces to

r = k 2 ⋅ Ctot

(4)

Because all the catalyst sites are covered with monomer, the rate becomes independent of monomer, i.e. zero order. The insertion of the docked monomer into the growing


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ACS Catalysis

polymer involves relatively constrained species all along the reaction coordinate and leads to the expectation that the activation entropy, ∆S‡, should be small in magnitude, but can be either positive or negative. Again, the activation enthalpy can take a wide range of values. 3. If, contrary to case 2 above, K1M

Quantitative Modeling of the Temperature Dependence of the Kinetic

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