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Ind. Eng. Chem. Res. 2005, 44, 5932-5949
Kinetic Modeling and Prediction of Polymer Properties for Ethylene Polymerization over Nickel Diimine Catalysts Dennis P. Lo† and W. Harmon Ray* Department of Chemical Engineering, University of WisconsinsMadison, 1415 Engineering Drive, Madison, Wisconsin 53706-1691
A kinetic model is developed for predicting the polymerization rate, chain-branching characteristics, and molecular weight from ethylene polymerization over homogeneous and supported nickel diimine catalysts. Seven dimensionless parameter groups are able to describe the polymer properties and kinetic behavior over a broad range of temperatures and pressures. Relative to existing models in the literature, the proposed model is simpler, requires fitting of fewer kinetic parameters, and can be more easily extrapolated over a wide range of reaction conditions. As such, the model can be more readily incorporated into reactor process models for performing dynamic simulations. Experimental data from the literature are used to fit the model kinetic parameters. The model is able to describe the kinetic behavior of both homogeneous and heterogeneous nickel diimine catalysts, and the results compare favorably with existing models in the literature. Introduction Since their discovery in 1996 by Brookhart et al. in collaboration with E. I. DuPont de Nemours and Co.,1 nickel diimine ethylene polymerization catalysts have received wide attention in both academic and industrial circles. Several research groups have published experimental2-25 and computational26-32 studies examining the catalyst chemistry and the polymerization mechanism. Nickel diimine catalysts are also prevalent in industrial patent literature,1,33-48 with claims made from a comprehensive list of chemical companies. In general, late-transition-metal catalysts are attractive compared to conventional early-transition-metal ethylene polymerization catalysts (chromium oxide, ZieglerNatta, and metallocene) because of their ability to produce branched polyethylene without a comonomer and their potential to polymerize polar-functionalized monomers.49 This new mechanism of branch formation can give rise to novel branching structures, not seen previously with early-transition-metal catalysts. By variation of the catalyst structure or reaction conditions, polyethylene varying from linear semicrystalline to highly branched amorphous polymer can be produced from ethylene homopolymerization using nickel diimine catalysts. Furthermore, nickel diimine catalysts have exhibited high activities and reasonable stability under typical polymerization conditions. We choose to focus our efforts on understanding the kinetic behavior of nickel diimine catalysts because they are capable of producing a broad range of the existing slate of polyethylene products and have the potential to produce a new generation of polyethylene products. The work presented in this paper discusses kinetic modeling of nickel diimine catalysts for ethylene polymerization. In a subsequent paper, the kinetic model presented herein will be incorporated into a reactor process model in order to perform dynamic simulations * To whom correspondence should be addressed. Tel.: (608) 263-4732. Fax: (608) 262-0832. E-mail:
[email protected]. † Current address: DuPont Surfaces, Tonawanda, NY 14150.
Figure 1. General structure of R-diimine catalysts (Ni or Pd). R1-R4 indicate various catalyst ligand groups attached to the R-diimine backbone. X indicates metal axial coordination sites occupied (e.g., by monomer and polymer chains) during polymerization.
Figure 2. Chain-growth mechanism.
of reactor-grade transitions and to develop strategies for reactor operation with these new catalysts. The general structure of the active catalyst center is shown in Figure 1. Nickel diimine catalysts employing a wide variety of R-diimine backbone ligand groups have been reported to successfully polymerize ethylene in the literature. Polyethylenes with molecular weights from 30 000 to 1 000 000 g/mol and up to ∼120 branches per 1000 carbons have been produced at activities of up to 106 (kg of polymer)/(mol of Ni)/h, depending on the polymerization temperature and pressure and catalyst design.49 Chain growth occurs through a two-step process of monomer coordination and insertion, in a fashion analogous to that of chain propagation observed in earlytransition-metal-catalyzed coordination polymerization (Figure 2). The polymerization activity is directly related to the rate of monomer consumption through the chaingrowth mechanism. Repeated trapping and insertion of monomer lead to further chain growth. Because the
10.1021/ie040197u CCC: $30.25 © 2005 American Chemical Society Published on Web 03/01/2005
Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 5933
Figure 3. Chain-walking mechanism for short-chain branch formation.
intrinsic rate of monomer trapping is much greater than that of monomer insertion except at very low monomer concentrations, the catalyst resting state exists as the alkyl-olefin complex.2 The electronic and steric properties of the catalyst ligands, as well as the temperature and monomer concentration, influence the overall polymerization rate. In addition to monomer trapping, the β-agostic alkyl-hydride catalyst complex can also undergo β-hydride abstraction, leading to chain walking, the mechanism of branch formation. If the hydride is reinserted into the polymer chain with the opposite regiochemistry, the nickel atom effectively migrates from the terminus to the first interior carbon of the live polymer chain (Figure 3). At this point, monomer trapping and insertion can occur, leading to the formation of a methyl branch and resumed chain growth. Alternatively, the chain walking can continue (in either direction) along the chain, leading to the formation of a distribution of short-chain branch lengths upon monomer addition. Thus, monomer addition at an active catalyst site located at the live chain terminus leads to linear chain growth, while monomer addition at an active catalyst site located at an interior polymer chain carbon position leads to branch formation. Factors that influence the relative chain-walking to chain-trapping rate (temperature, monomer concentration, and ligand structure) govern the resultant degree of branching. Experimental characterization of nickel diimine polyethylene has shown that adjacent branches occur at least two carbons apart on the chain backbone and that hyperbranches are not formed. These observations indicate that chain walking past tertiary carbons in the polymer chain is unfavorable, likely because of steric considerations. Termination of chains occurs principally via chain transfer to both monomer and hydrogen.26 Both associative displacement3 and direct β-hydride transfer have been proposed as possible mechanisms for chain transfer to monomer. Initial density functional theory studies have indicated direct β-hydride transfer to be the more likely mechanism.26-28 Both mechanisms proceed through a transition state involving coordination of monomer to the axial site of the nickel center. This confirms the original hypothesis of Brookhart et al., who, in discovering the R-diimine catalysts, incorporated bulky ligand groups to block the metal axial site of the metal center from approaching monomer molecules in order to retard chain transfer and enable the production of high molecular weight polymer. In the presence of hydrogen, chain transfer also occurs via hydrogenolysis. For some homogeneous nickel catalysts, large reductions in catalyst activity are observed when hydrogen is present,4 limiting its utility as a practical chain-transfer agent. Hydrogen has been more successfully employed for
supported nickel diimine catalysts treated with a lowvolatility pore-filling agent.33 The chain length is observed to decrease as the hydrogen levels are increased, while the polymerization rate and branching are largely unaffected. Chain transfer to cocatalyst can also impact the polymer chain length and varies with the cocatalyst (alkylaluminum) type.5 Alkylated silanes and carbon tetrabromide have also been successfully used as chaintransfer agents. The silanes do not adversely affect the catalyst activity, whereas decreases in activity are observed for CBr4.34 There are significant differences between the conventional early-transition-metal catalysts and nickel diimine catalysts with respect to their kinetic behavior and the influence of the reaction conditions on the polymer properties. For the case of early-transitionmetal catalysts, the branching density and molecular weight are controlled independently by varying the comonomer and hydrogen compositions, respectively. For the case of ethylene polymerization over nickel diimine catalysts, the temperature and monomer concentration must be altered to regulate the molecular weight and branching density. The control of the polymer properties is more difficult because the temperature impacts both the branching density and molecular weight and the temperature and monomer concentration also dictate the polymerization rate. Hydrogen can also be used to change the molecular weight; however, the sensitivity of the molecular weight to hydrogen for nickel diimine catalysts is low, and the presence of hydrogen can also cause significant reductions in the polymerization rate. Because of these major differences, new strategies are needed for operation of polymerization reactors using nickel diimine catalysts, underscoring the importance of having a reliable predictive model for polymer properties. Existing Models Despite the widespread interest in nickel diimine and late-transition-metal catalysts, there have been only a few published studies of kinetic models for these catalysts to date. The earliest models for predicting polyethlyene branching distributions arising from nickel diimine catalysts are from the group of Soares et al. and use Monte Carlo simulations.50 The model uses five parameters, which are event probabilities, to describe the relative rates of the various reaction steps involved in chain walking, chain growth, and chain termination: β-hydride elimination, isomerization (reinsertion of the hydride into the polymer chain with the opposite regiochemistry), forward chain walking, methyl branch formation bias, and chain transfer. The Monte Carlo model is able to match experimental data from solution
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polymerization experiments quite well, providing the instantaneous branching distribution for a given set of conditions. To be able to predict branching distributions over a wide range of polymerization conditions, Simon et al.51 correlate the Monte Carlo model parameters to the monomer concentration and temperature by fitting the probability parameters to branching distribution data for a particular nickel diimine catalyst over a range of polymerization conditions. Empirical functions for the probability of forward chain walking are used to describe the functionality with the temperature and monomer concentration for all four of the probability parameters. Although these functions are able to closely fit the experimental data, they introduce an additional six coefficients per Monte Carlo probability parameter, a total of 24 parameters, that must be fit to the experimental data. Soares et al.52 also use population balances to model the chain branching for ethylene polymerization over nickel diimine catalysts. The population balances rigorously account for the catalyst chain-walking behavior by tracking all possible locations of active sites. To restrict chain walking past tertiary carbons, the model also keeps record of the last branch point locations. The resulting model must keep track of a large number of catalyst active site types (no. of active site positions × no. of branch point locations); however, the fit to branching distribution data is quite good. Because a large number of kinetic rate constants are regrouped into a smaller number of lumped parameters to facilitate solution of the model and parameter estimation, it is difficult to extrapolate the model over a range of operating temperatures. Thus, the model is not ideal for process design and control. Simon et al.6 also propose a negative rate order model with respect to the monomer concentration in order to fit an observed rate decrease in homogeneous ethylene polymerizations conducted in a chlorobenzene solvent at 30, 40, and 50 °C, with varying ethylene pressures. At 30 and 40 °C, the catalyst activity is observed to decrease at high ethylene concentrations. At 50 °C, the catalyst activity increases over the entire monomer concentration range. Simon et al. attribute the observed decrease in the catalyst activity at high ethylene concentrations to the reversible formation of a “latent” catalyst site type that is inactive to polymerization. The proposed latent site results from the coordination of two monomer molecules and is thus favored at higher monomer concentrations. Similar observations, where the apparent polymerization rate decreases as the monomer concentration is increased, are reported by Schleis et al.7 and Peruch et al.8 for polymerization of 1-hexene. However, the observed negative rate order behavior is inconsistent with ethylene polymerizations conducted by other research groups, most notably the research group of Brookhart et al.2,9 The observed rate behavior for supported nickel diimine catalysts10,35 shows no decrease at high monomer concentrations. Results from density functional theory computations also dispute the hypothesis of multiple monomer coordination to the metal diimine center to form latent sites. Ziegler et al. investigated the possibility of a resting state complex possessing two coordinated monomer molecules. They found that there is a negligible driving force for coordination of a second monomer molecule to the catalyst active site.28 Finally,
if the negative rate order behavior reflects the true behavior or the catalyst, the molecular weight of the polymer should also decrease at high monomer concentrations. Simon et al. do not report molecular weight data, except to mention that molecular weights steadily increase with increasing monomer concentration. Schleis et al. report molecular weights that remain relatively constant over the entire range of monomer concentrations. Peruch et al. do not report molecular weight data. Because of the disagreement in the rate behavior observed by other groups and the small theoretical likelihood of the multiple monomer coordination mechanism, it is believed that the observed decrease in the polymerization rate is due to the formation of insoluble polymer at high monomer concentrations. In obtaining kinetic rate data from solution polymerization experiments, it is imperative that the polymer remains soluble. If the polymer becomes insoluble, the observed rate will fall due to encapsulation of active sites, thereby masking the intrinsic catalyst activity. Polymer solubility is dependent on the polymer properties (molecular weight and crystallinity), solvent, and temperature. Because the properties of polyethylene produced using nickel diimine catalysts can change drastically with the temperature and monomer concentration, maintaining complete polymer solubility is more complex than it is with other catalyst systems. Polymer solubility will be highest when polymerizations are conducted at high temperatures and low concentrations because the polymer chains will be highly branched. Conversely, at high concentrations and low temperatures, the polymer will be much less soluble, having a linear and highly crystalline structure. The experimental data presented by Simon et al. are consistent with these trends. For this reason, we do not consider negative order reaction rates in our modeling work. Kinetic Model Development The excellent work of Soares’ group provides a crucial understanding of the branching properties of these catalysts, but none of their models is capable of predicting the polymer properties and production rates over a wide range of conditions using only a few fundamental kinetic parameters. Our objective is to derive a simple kinetic model that can be used in dynamic reactor process simulations for predicting the molecular weight distribution, polymer density (branching characteristics), and kinetic rate behavior arising from ethylene polymerization over nickel diimine catalysts. Because commercial olefin polymerizations consist of both homogeneous and heterogeneous processes, the model should be able to describe the kinetic behavior exhibited by both homogeneous and supported catalysts. To be useful in reactor process modeling, extrapolation of the polymer properties over a wide range of reaction conditions (temperature and monomer concentration) is required. To avoid the introduction of a large number of empirical parameters, the proposed models are based on population balances derived directly from rate expressions from the polymerization kinetic mechanism. As a result, the model parameters consist of the individual reaction rate constants arising from the kinetic mechanism. Temperature effects can thus be modeled by assuming Arrhenius functionality in the rate constants. The rate constants for monomer trapping and monomer insertion that are fitted to the polymerization rate model are also used in the branching models, thus ensuring fundamental consistency.
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Polymerization Rate Model. The kinetic rates of monomer trapping and monomer insertion govern the overall rate of polymerization. The model assumes that the intrinsic rates of monomer trapping and insertion are independent of active site location on the polymer chain. Experimental evidence justifies this assumption for monomer trapping.11 Conversely, both computational and experimental results indicate that insertion of a monomer is more difficult when the active site is located at a secondary carbon than at a primary carbon.11,27 The insertion rate constant should accordingly be higher when the active site is at the chain end. However, the difference in insertion rates is postulated to be modest so that using a single rate constant for monomer insertion regardless of the active site position should be adequate for modeling the polymerization rate because it represents a composite average of the monomer insertion rates over all active site positions. The kinetic scheme for chain growth and termination is represented by eq 1. Pk represents an active catalyst site (β-agostic complex) attached to the kth carbon of a growing chain (k ) 1 at the terminal carbon), Rk represents an active catalyst site in the resting state, and D represents a dead polymer chain. The mechanism of chain transfer to monomer is represented as direct β-hydride transfer to monomer. The resulting expression for the rate of polymerization is shown in eq 2. ktrap
Pk + M 98 Rk kinsert
Rk 98 P1 kctM
Rk 98 D + P1 kctH
Pk + H2 98 P1 + D Rp ) kinsert
∑k [Rk]
(1) (2)
The steps of catalyst activation and deactivation are assumed to be independent of the site position. Thus, these mechanisms only influence the total number of active sites. Because high molecular weight polymer is obtained, chain-transfer rates are much slower than those of monomer insertion (kctM , kinsert). Consequently, chain transfer does not significantly affect the populations of resting state complexes, Rk, or alkylhydride metal complexes Pk. In addition, although their intrinsic rates are different, the observed rates of trapping and insertion are equal because they are sequential reaction steps and are the only mechanisms by which Pk and Rk species are generated and consumed. Thus, we can write
Rp ) ktrap[M]
∑k [Pk] ) kinsert∑k [Rk]
(3)
We can also make use of the total catalyst site balance
C0* )
∑k [P ] + ∑k [R ] k
k
(4)
Thus, combining eqs 3 and 4, one obtains expressions for the total concentration of alkyl-hydride metal complexes and resting state complexes as
C0*
K+1
PT )
∑
[Pk] )
k)1
1 + kr[M]
and K+1
RT )
∑
[Rk] )
k)1
Rp )
ktrap[M]C0*
kr[M]C0* 1 + kr[M]
where kr ) ktrap/kinsert
1 + kr[M]
(5)
(6)
At low monomer concentrations, the rate of polymerization is first order in monomer concentration. At high monomer concentrations, the rate of polymerization is independent of the monomer concentration. This behavior is consistent with the observed rate data reported in the literature. The rate of polymerization can be written in dimensionless form as the ratio of the overall polymerization rate to the rate of monomer trapping:
R p* )
Rp ktrap[M]C0*
)
1 1 + kr[M]
(7)
Thus, Rp* is a function of the dimensionless quantity kr[M], the ratio of the monomer trapping to monomer insertion rates. The dimensionless polymerization rate has a maximum value of 1 when the rate of monomer insertion dominates the rate of monomer trapping (e.g., conditions of low monomer concentration). In this case, the overall polymerization rate is thus equal to the rate of monomer trapping; in effect, monomer is rapidly inserted into the polymer chain following monomer trapping. At higher monomer concentrations, the dimensionless rate decreases asymptotically to zero as monomer insertion becomes the rate-limiting step and the catalyst exists more and more predominantly in the alkyl-olefin resting state (Figure 4). The temperature can also have a large influence on kr[M] if the difference in the activation energies for monomer trapping and insertion is significant. Knowing the value of kr[M] allows one to determine, using eq 7, the fraction of the maximum activity attainable for a particular catalyst, temperature, and monomer concentration. Chain Transfer and Molecular Weight Model. A simple analytical expression for the chain-transfer rate to monomer is derived in a fashion analogous to that of the polymerization rate. The rate of chain transfer to monomer from eq 1 is
RctM ) kctM
∑k [Rk]
(8)
As explained above, the chain-transfer mechanism here is assumed to follow direct β-hydride transfer from the growing polymer chain to trapped monomer. Using eqs 3 and 4, we can derive an expression for the rate of chain transfer.
RctM )
kctMkr[M]C0* 1 + kr[M]
(9)
Similar to Rp, the rate of chain transfer to monomer is first order with respect to the monomer concentration at low monomer concentrations and zero order with respect to the monomer concentration at high monomer concentrations.
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Molecular Weight Distribution Model. The instantaneous molecular weight distribution is described by the most probable (Schultz-Flory) distribution because at any given time and active site position there are only two possible events that can occur that influence the chain length. Live polymer chains can either grow by the addition of monomer through trapping and insertion or terminate through monomer trapping and subsequent chain transfer. The most probable distribution requires the calculation of one parameter, R, the probability of propagation, which yields a numberaverage chain-length distribution, fn, given by
fn ) (1 - R)Rn-1
Figure 4. Overall dimensionless reaction rate as a function of the (trapping rate)/(insertion rate) ratio.
Chain transfer to monomer via associative displacement can easily be modeled by replacing the chaintransfer mechanism shown in eq 1 by ktrap
k
The number-average molecular weight, weight-average molecular weight, and polydispersity as defined for the most probable distribution are given by
Mn )
k
P + M 98 R
kctM
RctM )
∑k [Pk]
kctMC0* 1 + kr[M]
(11)
∑k [Pk]
(13)
Again, using relationships given by eqs 3 and 4, we can write
RctH )
kctH[H2]C0* 1 + kr[M]
(14)
Here, it is assumed that the presence of hydrogen or other chain-transfer agent does not affect the rate of polymerization. The same analysis can be performed for any chaintransfer agent, X (e.g., cocatalyst, silanes, solvent, etc.), where the chain-transfer mechanism is of the form kctX
Pk + X 98 P10 + D
(15)
Provided the presence of chain-transfer agent X does not affect the rate of polymerization, the rate of chain transfer to X is then given by
RctX )
kctX[X]C0* 1 + kr[M]
(16)
(18)
The probability of propagation can be determined from the rate expressions for propagation (monomer addition) and chain transfer derived above. When chain transfer to monomer is the only mechanism for chain termination, the probability of propagation can be derived using eqs 6 and 9:
R)
(12)
Equations 9 and 12 only differ by a factor of kr[M] in the numerator. The rate of chain transfer to hydrogen from eq 1 is
RctH ) kctH[H2]
Zp ) 1 + R
(10)
The chain-transfer rate expression (11) would then replace eq 8, resulting in the rate expression for chain transfer shown in eq 12.
RctM ) kctM
M0 1-R
1+R Mw ) M0 1-R
kinsert
Rk 98 P1 Pk + M 98 D + Rk
(17)
kinsert Rp ) Rp + Rct kinsert + kct
(19)
Here, the molecular weight depends only on the temperature. When hydrogen and/or other chaintransfer agents are present, the total rate of chain transfer is
Rct )
kctMkr[M]C0* + kctH[H2]C0* + kctX[X]C0* 1 + kr[M]
(20)
The probability of propagation then becomes
R)
ktrap[M] ktrap[M] + kctMkr[M] + kctH[H2] + kctX[X]
(21)
The resulting expression for the number-average chain length, DPn, is
DPn ) 1 +
Rp )1+ Rct 1 (22) kctH kctX kctM [H ] + [X] + kinsert ktrap[M] 2 ktrap[M]
Here, the molecular weight distribution is dependent on the temperature as well as the relative concentrations of hydrogen and chain-transfer agent to monomer. It is assumed that the presence of hydrogen or chaintransfer agent does not affect the rate of polymerization, which is behavior consistent with certain supported nickel diimine catalysts.
Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 5937 Table 1. Definition of State Variables state
no. of states
initial conditions
description
[Pk]
K+1
0 for all k
K+1 [Rk] ∑k)1 Bk
1 K 1
Rtot 0 for all k 0
live chain with the active site (alkyl-hydride complex) located at position n (the kth interior carbon) live chain with the active site in the resting state (alkyl-olefin complex) number of branches of length k total number of carbons in polymer chains
TC
Figure 5. DPn profiles (bottom) with increasing monomer concentration resulting from chain transfer to monomer, from chain transfer to cocatalyst (or other chain-transfer agent), and from both chain transfer to cocatalyst and monomer. The chain-transfer rate and polymerization rate profiles are plotted qualitatively at the top and middle, respectively.
In the absence of hydrogen, chain transfer to cocatalyst can explain the increase in molecular weight with the monomer concentration that is sometimes observed. At low monomer concentrations, rates of chain transfer to monomer are low, and chain transfer to cocatalyst (or other chain-transfer agent) can dominate the control of molecular weight. As the monomer concentration is increased, chain transfer to monomer increases, becoming the dominant chain-transfer mechanism. The chainlength profiles with increasing monomer concentration that arise from these chain-transfer mechanisms are depicted in Figure 5. Chain-Walking and -Branching Density Model. Existing models for chain branching in the literature are complex and require a large number of kinetic or empirical parameters to capture all of the details of the chain-walking mechanism and resulting chain-branching distribution. In this work, a simpler approach to modeling the chain branching is taken in order to minimize the computation time and number of required parameters. Emphasis is placed on the ability of the model to capture the features of the branching distribution that govern the final material properties of the polymer. In this respect, it is important for the model to be able to accurately predict the number frequency of short-chain branches. Similarly, in terms of modeling the branch-length distribution, the model focuses on capturing the number of methyl branches. Because the branching distribution is predominantly methyl branches, the exact distribution of longer short-chain branches does not have a major impact on the final polymer properties.
It is also important to recognize that for HDPE there are ∼5 branches per 1000 carbons and even the most highly branched specialty polyethylene on average produced from nickel diimine catalysts contains at most 100-120 branches per 1000 carbons in the chain backbone. Thus, on average, from 10 to 200 methylene units separate each branch point, and for any given polymer chain, the branch points are sufficiently sparse such that the opportunities for an active site to chain walk beyond the last branch point are rare and can be neglected. This assumption is an important simplification because it allows for chain walking to be modeled without keeping track of the location of the most recent branch point. A system of linear ordinary differential equations is used to model the chain-walking behavior. The state variables are defined in Table 1. As previously defined in the polymerization rate model, Pk and Rk denote an active catalyst site (alkyl-hydrogen β-agostic complex and resting state complex, respectively) attached to the kth carbon of a growing chain (k ) 1 at the terminal carbon). Initially, all catalyst sites are assumed to be active and in their resting states. Site deactivation is neglected for these initial model feasibility studies, making the total number of active sites constant. Because the rates of catalyst deactivation are slow relative to chain walking and chain growth, catalyst deactivation affects only the total polymer yield and does not have a large impact on the polymer properties. The maximum branch length tracked, K, is 20. It is found that the model fit is not sensitive to increasing K beyond 20. Thus, the number of branches longer than 20 carbons is assumed to be negligible, and the total number of state equations is 43. At each position on the chain, the active site can undergo chain walking forward, chain walking backward, or monomer trapping and insertion (eq 23), and these events are represented by their kinetic rates (eq 24). As discussed previously in the development of the polymerization rate model, the overall observed rates of trapping and insertion are equal and the kinetic rate constants for trapping and insertion are assumed not to depend on the active site location. The rate of chain transfer is also negligible relative to the rates of trapping, insertion, and chain walking and so is not considered in the resulting state equations. kfk, kbk
Pk 798 Pk+1 ktrap
Pk + M 98 Rk kinsert
Rk 98 P1
(23) k
Rcwf,k ) kfk[P ] Rcwb,k ) kbk[Pk] Rtrap ) ktrap[Pk][M] Rinsert ) kinsert[Rk]
(24)
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A system of ordinary differential equations resulting from population balances are used to describe the distribution of active site positions on live chains as well as the evolution of the branching distribution as a function of the monomer concentration, [M], and time, t.
d[P1] dt
Table 2. Model Cases
) -kf1[P1] + kb2[P2] - ktrap[P1][M] + K+1
kinsert
[Rk] ∑ k)1
(25)
d[P2] ) kf1[P1] - (kf2 + kb2)[P2] + kb3[P3] dt ktrap[P2][M] (26) d[Pk] ) kf(k-1)[Pk-1] - (kfk + kbk)[Pk] + kb(k+1)[Pk+1] dt 2 < k < K + 1 (27) ktrap[Pk][M] d[PK+1] ) kfK[PK] - kb(K+1)[PK+1] - ktrap[PK+1][M] dt (28) d[Rk] ) ktrap[Pk][M] - kinsert[Rk] dt dBk ) RBk ) kinsert[Rk+1] dt d(TC) dt
(29) total no. of branches )
∑ [R ] ) k)1 K+1
2ktrap[M]C0*
k)1
1 + kr[M]
∑ [Pk] )
K+1
) 0 ) ktrap[M]
∑ k)1
(35) (31)
K+1
[Pk] - kinsert
[Rk] ∑ k)1
(32)
When the quasi-steady-state assumption is applied to eq 29, eq 30 can be expressed in terms of the rates of monomer trapping at each active site location:
dBk ) RBk ) ktrap[Pk+1][M] dt
∑ kBk [mol/L]
k)1
K+1
dt
(34)
K
The monomer concentration is held constant in the polymerization runs from which the experimental data are generated. Because the polymer chains grow very fast, high molecular weights are achieved after several seconds. Assuming that the long-chain hypothesis is valid, all of the chains will have the same short-chain branching distribution and total branches per 1000 carbons, BPTC. Thus, the quasi-steady-state assumption can be made for the system of ordinary differential equations (25)-(29), giving rise to a system of algebraic equations. The distribution of catalyst positions in the resting state, [Rk], is not required for the calculation of the branching distribution. Equation 29 can thus be rewritten simply in terms of the total active sites in the resting state:
[Rk] ∑ k)1
∑ Bk [mol/L]
k)1
total no. of carbons in branches )
k
2ktrap[M]
d
K
0 < k < K + 1 (30)
K+1
) RTC ) 2kinsert
The total number of branch points and the number of carbons in branches is then given by the zeroth and first moments of the branching distribution, Bk.
0 < k < K + 1 (33)
The number of branches per 1000 carbons in the backbone can thus be calculated: K
1000
∑ Bk
k)1
BPTC )
K
TC -
(36)
∑ kBk
k)1
The representation of the monomer trapping reaction as irreversible is a simplification used by our model. However, the inclusion of reversibility in the monomer trapping step does not affect the overall branching distribution predicted by the model because it is probabilistic and given the validity of the long-chain hypothesis. In other words, while chain walking, the probability of trapping followed by monomer insertion versus the case of trapping followed by the reverse “untrapping” step and continued walking is the same at all active site positions. Five different model cases with varying numbers of chain-walking parameters are considered in order to investigate chain-end effects (Table 2). The number of kinetic parameters depends on the level of detail used to model the influence of the chain-end effects on chain walking. Obviously, an infinite number of chain-walking rate constants could be used to describe the kinetics of chain walking at each active site location on a live polymer chain. However, this would not be a practically useful or tractable model. In our studies, we have
Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 5939
limited the number of required model parameters by considering models incorporating two to five kinetic parameters. Thus, in addition to the kinetic parameters fit to the polymerization rate model (kinsert and ktrap), these model cases require fitting an additional two to five chain-walking rate constants. Case 1 is a twoparameter model that distinguishes chain walking forward from the chain end only and assumes that the rate of chain walking is independent from the direction at other active site positions. Case 2 is identical with case 1 but has an additional kinetic rate constant to distinguish chain walking forward from chain walking backward at nonterminal active site positions. Case 4 is a three-parameter version of case 1, where chain walking backward to the chain end is also treated distinctly. Case 5 is the most detailed, requiring estimation of five chain-walking parameters. Chain-end effects are modeled by distinguishing all of the chain-walking rate constants involving the terminal or first interior carbon from chain walking at locations further from the chain end. Case 3 is a simplified version of case 5. Chain walks originating from either position 1 or 2 are defined distinctly from chain walks at other positions. However, chain walking forward is not differentiated from chain walking backward. The quasi-steady-state equations for the active site position distribution can be solved recursively (similar to the approach of Soares et al.52) to yield an explicit solution for the active site distribution as follows. The concentration of each active site position can be written explicitly as a function of the concentration of active sites at adjacent positions. For simplicity, the recursive solution method is shown for the three-parameter model case 3 (see Table 2). Here we rename the chain-walking rate constants: kw1 ) kf1, kw2 ) kf2 ) kb2, and kw ) kf ) kb. When the quasi-steady-state assumption is assumed and the equations are rearranged, eqs 25-28 can be rewritten:
kw2[P2] + kinsert [P1] )
[P2] ) [P3] ) [Pk] )
∑k [Rk]
kw1 + ktrap[M] kw1[P2] + kw[P3] 2kw2 + ktrap[M] kw2[P3] + kw[P4] 2kw + ktrap[M]
kw[Pk-1] + kw[Pk+1] 2kw + ktrap[M] [PK+1] )
(37)
(38)
kw[PK] kw + ktrap[M]
θk )
θ1 )
1 kw1 + ktrap[M] kw2 kw1 + ktrap[M]
kw
3
