Modeling of α-Olefin Copolymerization with Chain-Shuttling Chemistry

Ind. Eng. Chem. Res. 2010, 49, 8135–8146

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Modeling of r-Olefin Copolymerization with Chain-Shuttling Chemistry Using Dual Catalysts in Stirred-Tank Reactors: Molecular Weight Distributions and Copolymer Composition Min Zhang* 1500 East Lake Cook Road, The Dow Chemical Company, Buffalo GroVe, Illinois 60089-6556

Thomas W. Karjala and Pradeep Jain 2301 N. Brazosport BouleVard, The Dow Chemical Company, Freeport, Texas 77541-3257

We report a kinetic model of chain-shuttling copolymerization using dual catalysts for solution R-olefin polymerization processes. This model focuses on predicting polymer properties such as the molecular weight and molecular weight distribution and the overall copolymer composition. We first validate the model through qualitative comparison between the model predictions and experimental observations reported in Arriola et al. (Science 2006, 312, 714) in both a semibatch reactor and a continuous stirred-tank reactor (CSTR). Then, examples are given to illustrate how the model can be used to examine the effects of the chain-shuttling rate constant and the chain-shuttling-agent feed rate in a CSTR. Moreover, simulations using this model demonstrate how to prepare polymers with desired properties by manipulating catalyst compositions and monomer compositions in the feed. Introduction The discovery of chain shuttling for olefin polymerization catalyzed by transition metals made it possible to prepare olefin block copolymers (OBCs) continuously on a commercial scale.1 Here, chain shuttling denotes that a growing chain can be passed between different catalyst sites so that a single polymer chain can consist of polymers originating from at least two different catalysts. When the two catalysts are able to catalyze the formation of polymer chains with distinctive properties, chain-shuttling polymerization can potentially prepare a class of block polymers that have unique block compositions, block lengths, and numbers of block distributions. The key to the success of chain-shuttling polymerization in preparing OBCs, therefore, is to identify a pair of catalysts and a chain-shuttling agent (CSA), so that the pair of catalysts has varying efficiency in incorporating comonomer into the chain and the growing chains can reversibly exchange active centers with dormant chains that are attached to the CSA.1-6 Through reversible reactions between growing and dormant chains, chain-shuttling polymerization allows for better control over chain growth and tailoring of chain architectures. Even though chain-shuttling polymerization is limited to nonpolar monomers, this chemistry has demonstrated some unique qualities over previous “living” polymerization systems such as living coordination chain polymerization,7 living free-radical polymerization,8-17 and living cationic polymerization.18 First, the previous systems have only one type of propagating active center that introduces just one polymer population at any instant, whereas chain-shuttling polymerization has two different types of propagating active centers. Thus, for the previously reported systems, the preparation of block copolymers is achieved through sequential addition of monomers in a semibatch reactor19-23 and side feed in a tubular reactor24,25 or in multistage reactors.19,26,27 These processes are able to prepare * To whom correspondence should be addressed. Tel.: 1-847-2153758. Fax: 1-847-808-3701. E-mail: [email protected].

relatively uniform block lengths for the block copolymers when semibatch or tubular reactors are utilized. Note that U.S. Patent 6,706,832 describes how to make polyethylene through a nitroxide radical by living radical polymerization, leading to the possibility of making OBCs.28 By contrast, chain-shuttling polymerization is able to avoid the process complexities by using a single continuous stirred-tank reactor (CSTR) in the presence of a dual-catalyst system to prepare OBCs. Additionally, each catalyst in a living coordination polymerization system introduces only one chain; thus, the cost ratio of polymer chain to catalyst becomes so high that a cost barrier exists for commercially viable living coordination polymerization systems. In chain-shuttling polymerization, the renewal of catalyst active centers is realized through the reversible reactions between the growing and dormant chains. Thus, each catalyst could potentially catalyze the formation of any number of polymer chains, so that the cost barrier of catalysts could be eliminated. The ability of chainshuttling polymerization to tune molecular chain architectures such as the molecular weight and molecular weight distribution and short-chain branching allows for the production of differentiated elastomeric materials that exhibit semicrystalline/high-melting-temperature blocks and amorphous/lowglass-transition-temperature blocks.29,30 Such accurate control of molecular chain architectures allows for the production of elastomeric OBCs that are easier to process, have faster setup, and exhibit better high-temperature performance and improved abrasion resistance compared to other olefin elastomers. Figure 1 provides a schematic representation of the mechanism of chain-shuttling polymerization using a dualcatalyst system, and Figure 2 illustrates formation of a new block through chain shuttling. As shown in Figure 2, in step 1, a growing chain ending with catalyst 2 is shuttled through a CSA to form a dormant chain A. Note that the portion of dormant chain A neighboring the CSA is formed through catalyst 2 and is considered block 2. Dormant chain A can

10.1021/ie100530p  2010 American Chemical Society Published on Web 07/27/2010

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Figure 1. Illustrative scheme of chain-shuttling polymerization using dual catalysts.

Figure 2. Schematic representation of block formation in chain-shuttling copolymerization using dual catalysts.

be activated through chain shuttling with a growing chain. For example, in step 2, when dormant chain A is activated by a growing chain B ending with catalyst 1 to form a growing chain C, the subsequent propagation of growing chain C leads to the formation of a new block. By contrast, dormant chain A can also be activated by a growing chain D ending with catalyst 2 to form a growing chain E as shown in step 3; the subsequent propagation of growing chain E does not form a new block but rather extends the length of block 2. Thus, not every chain-shuttling event followed by propagation leads to the formation of a new block. A new block forms only when a cross-shuttling event occurs, followed by subsequent propagation. Note that cross-shuttling refers to when a dormant chain whose portion neighboring the CSA was generated by catalyst i (i ) 1, 2) is activated through chain shuttling with a growing chain ending with catalyst j (j ) 1, 2, i * j). This new mechanism of block formation gives rise to novel block copolymers, not seen previously in traditional living polymerization. For example, running this chemistry in a CSTR leads to multiblock copolymers possessing a distribution in block length and number of blocks per chain. Furthermore, this chemistry provides a wide range of flexibility in designing new polymer materials through variation of the catalyst structure and composition, reaction conditions, or reactor configuration. For example, Hustad et al.2 reported using the chemistry termed coordinative chain-transfer polymerization (CCTP) to prepare diblock copolymers in two CSTRs in series with a single catalyst. By varying the reactor environment in each reactor, distinctly different blocks can be prepared in each reactor. Because most chains in the first reactor are in the form of dormant chains, preserving their ability to add a new block in the second reactor, flexible control of the block length can be achieved.

In the past decade, there has been extensive modeling research on living free-radical polymerization, which also involves reversible reactions between growing and dormant polymer chains. The vast majority of articles have discussed kinetic modeling for living free-radical polymerization, focusing on understanding the development of molecular weight and polymerization rate in batch and/or semibatch reactors.31-41 Notably, Tobita42 presented a theoretical discussion of molecular weight distribution for an ideal living free-radical polymerization in which chain termination was negligible, showing that the hypergeometric function combining both the most probable distribution and the Poisson distribution represents a fundamental distribution for living free-radical polymerization. Quite a few models have been developed to address reactor design and operating issues for living free-radical polymerization. Zhang et al.19,24 developed a comprehensive kinetic model for living free-radical copolymerization and incorporated the kinetic model into a reactor model to explore various process development issues. Lemoine-Nava et al.43 investigated the optimal operating strategies for nitroxide-mediated free-radical polymerization in a semibatch reactor, whereas Wang et al.20-23 examined the strategy for controlling copolymer composition through semibatch operations using reversible addition-fragmentation transfer (RAFT) and atom-transfer radical polymerization (ATRP) chemistry. Asteasuain et al.44,45 presented a model of nitroxide-mediated radical polymerization in a tubular reactor as a design tool to optimize tailor-made molecular weight distributions using the probability generating function transformation. These modeling efforts greatly enhanced the understanding of development issues concerning living free-radical polymerization while providing a good reference for the investigation of chain-shuttling polymerization. Even though there are similarities between living freeradical polymerization and chain-shuttling polymerization, chain-shuttling polymerization using dual catalysts represents a great challenge for modeling because of the complexity of the reactive system. Hustad et al.5 reported a kinetic model for chain-shuttling chemistry to examine the effect of the reversibility of chain transfer in olefin polymerization on the number- and weight-average molecular weights and demonstrated the utility of the model in the discovery process for this novel chemistry. However, the model did not consider copolymerization using dual catalysts in the presence of a CSA. To the best of our knowledge, this article provides, for the first time, a model for chain-shuttling polymerization using dual catalysts, allowing for the in-depth investigation of the effects of reactor conditions on molecular architectures. In this article, we present a kinetic model for the prediction of the molecular weight, the molecular weight distribution, and the copolymer composition. Note that the modeling work on block length and block length distribution shall be reported in a sequel to this article. The kinetic model then is incorporated into a well-mixed tank reactor model for the examination of batch, semibatch, and CSTR operations. Next, we validate the model through qualitative comparison between the model predictions and the reported experimental observations of Arriola et al.1 Finally, we demonstrate the use of the model through several examples including investigations of the effects of the chain-shuttling rate constant, CSA feed rate, catalyst composition, and monomer composition. We reveal that chain-shuttling polymerization operating in a single CSTR using a dual-catalyst system presents some unique

Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010 Table 1. Reaction Scheme of Chain-Shuttling Polymerization reaction instantaneous activation of catalyst without deactivation

scheme

Table 2. Definitions of Moments for Various Polymer Populations and Vector Operations46 polymer population

instantaneous

Cati. 98 Cat*i

moments ∞

growing polymer µi,j g )

instantaneous

Cat*i + Mj. 98 Pδi,jj

chain shuttling to virgin CSA

Pi,j n_

i kp,jk

+ Mk f

∑ n_ P

g i,j n_

n_)0_ ∞

dormant polymer propagation

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ξjg )

Pi,k n_+δk

∑ n_ Q g

j n_

n_)0_

i kCS0

i,k j Pi,j n_ + CSA f Qn_ + fkPδk

∞

dead polymer νg )

∑ n_ D g

n_

n_)0_

chain shuttling to polymeryl CSA chain transfer to hydrogen

thermal termination

i kCS

j' j i,j' Pi,j n_ + Qn_′ f Qn_ + Pn_′

Pi,j n_

i ktrH

+ H2 f Dn_ +

fkPδi,kk

i kth

2

bulk polymer

λg )

i,j g

i)1

vector exponentiation

_ng )

i,k Pi,j n_ f Dn_ + fkPδk

phenomena in tuning the polymer properties and optimizing polymerization behavior.

2

∑ ∑µ ∏

j)1

2 i)1

We consider a representative kinetic scheme for solution R-olefin polymerization in the presence of CSA. To illustrate the advantages of tailoring molecular chain architectures through chain shuttling, we consider dual catalysts that introduce two polymer populations with distinct polymer properties. For convenience, we denote catalyst 1 as a good comonomer incorporator (i.e., catalyst 1 readily incorporates comonomer into a polymer chain) and catalyst 2 as a poor comonomer incorporator (i.e., catalyst 2 has difficulty incorporating comonomer into a polymer chain). Thus, catalysts 1 and 2 yield polymers with significantly different copolymer compositions. For example, it is not unusual in the marketplace to use ethylene as a monomer and butene, hexene, or octene as the comonomer to prepare polyethylene. A good comonomer incorporator (catalyst 1) prepares polymer having a high comonomer content and, thus, having low crystallinity and a low glass transition temperature. The building block resulting from catalyst 1 is denoted as a soft block. By contrast, a poor comonomer incorporator (i.e., catalyst 2 in this work) leads to an ethylene-rich polymer with high crystallinity and a high melting temperature. Such a building block is denoted a hard block. To adequately describe the polymer chain architecture in this system, we introduce notations that can capture the key features of the chains as follows: growing chain, Pi,j n_ ; dormant chain attached to the CSA, Qjn_; and dead chain, Dn_. Note that n_ is a vector in which the first element records the number of monomer units in the chain and the second element records the number of comonomer units in the chain. i (i ) 1, 2) denotes the type of catalyst, whereas j (j ) 1, 2) describes the type of monomer at the chain end neighboring the catalyst active center in a growing chain or neighboring the chain-shuttling agent in a dormant chain. Using the above notation regarding polymer chains, the kinetic scheme is represented in Table 1. For chain shuttling to virgin CSA, chain transfer to hydrogen, and thermal termination, we have lumped the reactions involved for the purpose of simplifying modeling. Using chain shuttling to virgin CSA as an example, chain shuttling to virgin CSA involves two steps.

∑ξ

j g

+ νg

j)1

(ni)gi

First, the growing chain transfers to a CSA to form a dormant chain attached to CSA (i.e., polymeryl CSA) and releases the active center that can propagate to form a new chain i kCS0

Kinetic Model for Chain-Shuttling Polymerization

2

+

j Pi,j n_ + CSA 98 Qn_ + Cat* i

(1)

Second, the active center immediately adds a monomer or comomomer to form a growing chain instantaneous

Cat*i + Mk. 98 Pδi,kk

(2)

Here, δk is a unit vector with its kth element equal to 1 and the rest elements of its equal to 0. For brevity, the above two steps can be combined as one step assuming that the addition of monomer k to the primary active center is proportional to the monomer composition, fk i kCS0

j i,k Pi,j n_ + CSA 98 Qn_ + fkPδk

(3)

Reported CSAs such as diethylzinc and triethylaluminum have dual or triple functionalities that are available for chain shuttling.1 For simplicity, we consider a monofunctional virgin CSA in this article. The treatment does not affect the conclusions drawn from this investigation. Additionally, we consider that chain transfer to virgin CSA is determined only by the type of catalyst and is not affected by the type of last monomer inserted. Chain-transfer reactions such as thermal termination and chain transfer to hydrogen experience two steps as well, but for simplicity, these reactions are written directly in one step (cf. Table 1), as this treatment does not affect the model formulation. Rate of Change of Species. Based on the kinetic scheme, the rate of change for each species of interest can be written. Note that we denote the rate of change of species i due to reactions as ri and the concentration of small molecule j as Cj. For the sake of the compactness of the kinetic model, we introduce the definition of moments for polymer populations of interest in Table 2. Note that some of the leading moments are used directly in the expressions of the rate of change of species, and the meaning of the leading moments is briefly addressed. The details can be found in the Nomenclature section. The rates of change of species are as follows:

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Rate of change of monomer

Table 3. Calculation of Average Bulk Polymer Properties

2

r Mj ) -

2

∑ ∑k i)1 k)1

i i,k p,kjµ0 CMj

(4)

average polymer properties number-average chain length, DPn

i)1

λ0 λ2 2

∑λ

2

∑k i)1

i i trHµ0CH2

δi

(5)

Here, µ0 ) ∑k)1µ0 denotes the total concentration of growing chain of catalyst type i. 2

∑λ

weight-average chain length, DPw

Rate of change of hydrogen

i

2

δi

Here, µi,k 0 denotes the concentration of growing chain of catalyst type i with last inserted monomer of type k.

rH2 ) -

calculation method

i,k

i)1

polydispersity, Zp

DPw DPn λδj

copolymer composition, Fpoly,j

Rate of change of catalyst (considering all forms of catalyst) rcati ) 0 (6)

2

∑λ

δi

i)1

Rate of change of CSA

average molecular weight of repeat unit, Mw,av

2

rCSA ) -

∑

i kCS0 µi0CCSA

poly,iMw,i

(7)

i)1

a

i)1

Rate of change of growing chain Pi,j n_ 2

rPn_i,j ) -

∑

2

i kp,jk Pi,j _ CMk n

+

k)1

∑

i kp,kj Pi,k n_-δjCMj

-

-

i ktrH CH2Pi,j n_

i i,j kth Pn_

+

i ktrH fjδ(n_

-

δj)CH2µi0

+

i kth fjδ(n_

-

δj)µi0

(8)

2 where δ(i) is the Kronecker function and ξ0 ) ∑i)1 ξi0 is the total concentration of dormant chains.

Rate of change of dormant chain Qjn_ 2

rQn_j )

2

∑k

i i,j CS0CCSAPn_

∑k

+

i)1

2

i i,j CSξ0Pn_

i)1

-

∑k

i i j CSµ0Qn_

(9)

i)1

Rate of change of dead chain Dn_ 2

rDn_ )

2

∑ ∑ (k i)1 j)1

i i,j trHPn_ CH2

+ kithPi,j n_ )

DPnMw,av

weight-average molecular weight, Mw

DPwMw,av

a

+

i i i j i kCS0 fjδ(n_ - δj)CCSAµi0 - kCS Pi,j n_ ξ0 + kCSQn_µ0

number-average molecular weight, Mn

-

k)1

i CCSAPi,j kCS0 _ n

2

∑F

(10)

Calculation of Average Polymer Properties. We utilize the method of moments to calculate the average polymer properties of interest.50 The moments for various polymer populations in the system have been previously defined in Table 2. By employing such definitions, we can derive equations regarding the rates of change of moments. The derivation is a standard procedure that can be found in Zhang et al.19 For simplicity, the equations regarding the rates of change for the leading moments are provided in the Appendix. Calculations of such average polymer properties as number-average chain length (DPn), weight-average chain length (DPw), and copolymer compositions are summarized in Table 3. Reactor Model. A general tank reactor model is developed for the chain-shuttling polymerization process, which is written in such a way that batch, semibatch, and CSTRs are described.19 Material balance equations are written for the species of interest and the leading polymer moments. The mixture in the reactor is assumed to be homogeneous. Only monomer, solvent, and polymer have significant volume contributions. Other components such as catalyst, hydrogen, and CSA exist only in trace amounts. Their contributions to the overall volume are negligible. In this article, we assume isothermal operation so that the reactor energy balance need not be considered.

Note that Mw,i is the molecular weight of monomer i.

Model Validation. Arriola et al. showed that using dual catalysts alone in a semibatch reactor leads to polymers with broad molecular weight distributions (MWDs; Mw/Mn ) 13.6) whereas adding a CSA (diethylzinc) leads to polymers with narrow MWDs (Mw/Mn ) 1.33).1 Moreover, carrying out chainshuttling polymerization using dual catalysts in a CSTR leads to polymers with narrow distributions (Mw/Mn ) 2), whereas using dual catalysts alone results in polymers having broad MWDs (Mw/Mn ) 13.8). By using typical kinetic parameters, we show qualitatively how our model captures the experimental observations in both reactors using chain-shuttling chemistry. Considering typical catalyst efficiencies in the range of 10 000-1 000 000 g of polymer/g of metal, the propagation rate constant for ethylene homopolymerization is roughly estimated as on the order of 100 000 L/(mol s). In this investigation, we chose an ethylene/octene copolymerization system as an example. Default values for all other kinetic parameters for all simulations are zero except for those summarized in Table 4. Semibatch Operation. In a semibatch operation, solvent, octene, and hydrogen are loaded into a tank reactor and heated to the polymerization temperature. Ethylene is fed into the reactor until the mixture is saturated. Rigorously, the concentrations of reactants should be estimated using the PC-SAFT equation of state.47,48 However, for illustrative purpose, we assume that the amounts of ethylene and octene in Table 5 are completely dissolved in the solvent. We then use the volume additivity principle to calculate the concentrations of the reactants. For example, ethylene has an initial concentration of 2.63 mol/L. At time t ) 0, a single shot of catalyst is injected into the reactor, allowing for the start of polymerization. Ethylene is continuously fed into the reactor during polymerization, maintaining the pressure in the reactor constant. Table 5 lists the simulation conditions. In both cases, the targeted DPn value is 1500 (i.e., Mn ≈ 54 000) at time t ) 600 s. Figure 3 illustrates how the overall molar conversion and fractional conversions of ethylene and octene in the semibatch reactor evolve with respect to time. In the kinetic scheme, we assume that the CSA does not interfere with the nature of the

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Table 4. Kinetic Parameters Used for Simulation kinetic parameter

value

r112

5

r121

0.3

r212

100

r221

0.05

k1p,22

0.01

k1p,11 k2p,11

0.1

k1p,11 k2p,22

0.01

k1p,11 1 ktrH

0.05 Figure 3. Conversion versus time and evolution of copolymer composition in the semibatch reactor.

k1p,11 2 ktrH

0.0005

k1p,11 1 kCS0

k1p,11 2 kCS0

k1p,11

)

)

1 kCS

10

k1p,11 2 kCS0

10

k1p,11

Table 5. Simulation Conditions in a Semibatch Operation mass load (g) component solvent ethylene octene hydrogen catalyst metal catalyst 1 molar fraction CSA

case 1 (no CSA)

case 2 (CSA)

1000 200 700 0.088 1.5 × 10-4 0.6 0

0.072 0.27

active center for both catalysts and propagation. Additionally, the chain-shuttling rate constant in the simulation is chosen so that the chain-shuttling rate is smaller than the propagation rate. Thus, the active-center compositions for both catalysts should remain the same in both the presence and absence of CSA, leading to the coincidence of the conversion curves in the two cases. Moreover, the evolution of the copolymer composition in the bulk polymer in both cases is not affected by the CSA. The assumption that the CSA has no effect on the incorporation rate of the monomers into the polymer chains allows for the freedom in controlling the bulk copolymer composition through manipulation of the catalyst composition, monomer composition, and operating temperature without the need to consider the CSA effect. Note that this conclusion holds true regardless of the type of reactor when the CSA has the appropriate chain-shuttling rate constant. However, when a CSA is very efficient (i.e., has a relatively large chain-shuttling constant), the effect of the CSA on the polymerization rate can be quite dramatic, and we shall discuss this effect in a later section. Figure 4 shows the evolution of the polymer properties Mn and Mw/Mn in the semibatch reactor for both cases. When dual catalysts are employed in the absence of a CSA, the lifetime

Figure 4. Evolution of the polymer properties Mn and Mw/Mn in the semibatch reactor in both the absence and presence of CSA.

for a growing chain is very short, so the chains quickly grow to a certain length and stop through chain transfer. Thus, it is difficult to control chain growth in this case. The slight drift of Mn is due to the depletion of hydrogen and the less reactive comonomer in the reactor. The bulk polymer assumes a very broad distribution (Mw/Mn ≈ 14) because of the presence of two distinct polymer populations with number-average chain lengths of around 1200 and 45 000. By contrast, Figure 4 also shows that, in the presence of the CSA, the chain grows in a stepwise fashion with respect to conversion. The chain growth in the presence of the CSA is very different from that in conventional polymerization, in which chains instantaneously grow and become dead through chain transfer. Thus, chainshuttling polymerization allows for flexibility in controlling the chain growth. For instance, by allowing for varying reaction times, it is possible to prepare polymers with varying chain lengths in this semibatch reactor. Moreover, the final polymer products have a narrow MWD (Mw/Mn ≈ 2). Note that the polydispersity decreases first below 2 and then increases because dead polymers accumulate with respect to conversion from chain-transfer reactions. CSTR Operation. In a CSTR, all of the reactants are continuously fed into a well-mixed reactor. The reactor operates in overflow mode at the polymerization temperature. Table 6

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Table 6. Simulation Conditions in a CSTR Operation

DPn )

mass feed rate (kg/h) component

case 1 (no CSA)

case 2 (CSA)

solvent ethylene octene catalyst metal catalyst 1 molar fraction hydrogen CSA

9.57 0.91 2.27 9.07 × 10-6 0.6 1.04 × 10-4 0

1.02 × 10-4 3.63 × 10-5

(2) R(1) t θ + Rt θ + CCSA,consumed

Note that, under the same polymerization conditions, we can also calculate the intrinsic number-average chain length of the resulting polymer using each individual catalyst in the absence of the CSA. If we denote the intrinsic number-average chain length of the resulting polymer for catalyst i as DP(i) n , then we have DP(i) n )

Table 7. Summary of Results in a CSTR Operation variable overall mass conversion ethylene conversion octene conversion DPn DPw polydispersity ethylene molar fraction in bulk polymer ethylene molar fraction in block 1 ethylene molar fraction in block 2 weight fraction of block 2

case 1 (no CSA) 79.41% 95.31% 73.05% 1000 31613 31.6 0.9181 N/A N/A N/A

case 2 (CSA)

shows the simulation conditions. In both cases, the targeted DPn is 1000 at the steady state. Note that, in the presence of the CSA, the feed rate of hydrogen is reduced so that the targeted DPn value can be reached. The volume of the reactor is chosen as 3 L, and the residence time at the steady state is about 10 min. Table 7 summarizes the results at the steady state in both cases. Note that the overall mass conversions and fractional conversions of ethylene and octene are the same for the two cases. Bulk copolymer in both operations has the same copolymer composition. However, in the presence of the CSA, the molecular weight distribution becomes much narrower, with Mw/Mn close to 2. Using dual catalysts in the absence of a CSA, the bulk polymer is simply a polymer blend. Results and Discussion The production of OBCs for commercial applications is achieved in continuous processes such as those conducted in CSTRs. Thus, a CSTR model can be used to investigate how the tuning of chemistry through the choice of catalysts, CSA, and operating conditions such as catalyst split, monomer composition, and temperature affect the resulting polymer properties. In the following sections, we first develop an analytical model for Mn of the bulk polymer in a CSTR at its steady state. The analytical solution for Mn allows for an examination of the effect of the chain-shuttling rate constant in conjunction with model simulation. We then conduct a detailed analysis of the role of the CSA in determining the MWD and even the polymerization rate, as well as the effects of other important factors. Molecular Weight Model in a CSTR at Its Steady State. For a single CSTR, if there is no polymer in the feed stream and the catalyst concentrations are much smaller than that of the CSA, the number-average chain length of bulk polymers in the presence of the CSA can be calculated from the ratio of the total amount of monomer incorporated into the polymer chain to the total number of polymer chains in the reactor

R(i) p R(i) t

(12)

The total concentrations of monomer and comonomer incorporated into the polymer because of catalyst i (i ) 1, 2) in the CSTR, λ(i) 1 , are given by

1002 2049 2.03 53.5% 96.1% 18.8%

(11)

(i) λ(i) 1 ) Rp θ

(13)

By introducing the assumption that the presence of the CSA does not alter the copolymerization behavior of each catalyst (i.e., does not alter the active-center composition distribution), so that R(i) p remains the same in both the absence and presence of the CSA if other operating conditions are the same, we have 1 ) DPn

F(i) block

2

∑ DP i)1

(i) n

+

CCSA,consumed (2) λ(1) 1 + λ1

(14)

where F(i) block is the molar fraction of total monomer and comonomer incorporated into polymer because of catalyst i and is calculated as λ(i) 1

F(i) block )

(2) λ(1) 1 + λ1

(15)

Note that the assumption that behavior of each catalyst is independent of the presence of the CSA could fail under certain conditions, as we discuss later. Following a similar procedure, we can also calculate the number-average molecular weight of the bulk polymer, Mn, as 1 ) Mn

2

wt(i) block

i)1

M(i) n

∑

+

CCSA,consumed λm

(16)

(i) Here, wt(i) block (i ) 1, 2) is the weight fraction of block i, Mn is the intrinsic number-average molecular weight in the absence of CSA under the same polymerization conditions, and λm is the polymer mass concentration (in grams per liter) in the reactor. If the conversion of virgin CSA is defined as xCSA, then

1 ) a + bCCSA,fxCSA Mn

(17)

where a)

2

wt(i) block

i)1

M(i) n

∑

(18)

and b)

1 λm

(19)

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is very efficient, that is, when kCS0 is several orders of magnitude greater than the propagation rate constant.49 Nevertheless, eq 17 was able to describe the development of Mn in the case when the presence of the CSA does not affect the polymerization rate. For the purposes of clarity and illustration, we use one catalyst, say, catalyst of type k, to demonstrate that the activecenter composition distribution could be influenced by the presence of the CSA. The reaction rate of an active center with a last inserted monomer of type i is written as k k,j k k,i k k j rµ0k,i ) -kkp,ijµk,i 0 CMj + kp,jiµ0 CMi - kCSµ0 ξ0 + kCSµ0ξ0 (20)

Figure 5. Comparisons of Mn and Mw/Mn for varying chain-shuttling rate constants.

Effect of Chain-Shuttling Rate Constant in a CSTR. An ideal chain-shuttling polymerization process contains a pair of catalysts that have different capabilities to incorporate monomer and comonomer, so that hard and soft blocks can be formed as building blocks for the chains. Moreover, such a process should allow good control over polymer chain growth and block formation. Achieving this control requires a good choice of chemical components, operating conditions, and reactor type. Screening for such a good chain-shuttling polymerization process is a challenging task, best carried out by employing high-throughput screening techniques. For instance, Arriola et al.1 reported that 1600 individual polymerizations were carried out in a high-throughput laboratory to find a chain-shuttling system that showed good chain-shuttling polymerization performance. Successful chain shuttling requires that a CSA have the ability to shuttle between growing chains attached to different active centers. In the remainder of this section, we discuss the effect of the chain-shuttling rate constant as predicted by the model. Moreover, it has been observed and we have confirmed that chain-shuttling polymerization in a CSTR has the benefit of producing polymers that have broader molecular weight distributions and narrower composition homogeneity than those produced in semibatch reactors because of the difference in reactor residence time distributions for the two types of reactors. Thus, it is desirable to investigate the effect of the chain-shuttling rate constant in a CSTR. The operating conditions are the same as in case 2 of Table 6, but the chain-shuttling rate constant was allowed to vary. The chain-shuttling rate constant of case 2 in Table 4 was chosen as the baseline chain-shuttling rate constant. A relative chain-shuttling rate constant in the simulations is defined as the ratio between the chain-shuttling rate constant and the baseline chain-shuttling rate constant. Figure 5 shows how Mn and Mw/Mn are affected by the chainshuttling rate constant. As the chain-shuttling rate constant increases, Mw/Mn approaches 2. Mn initially drops, then reaches a constant, and increases again as the relative chain-shuttling rate constant increases. It seems that the prediction of Mn from the model simulation is different from the prediction from eq 17, which shows that Mn should drop initially and reach a plateau once all of the CSA has been consumed if the other operating conditions remain unchanged. Note that, in deriving eq 17, we assumed that the presence of the CSA has no effect on copolymerization kinetics, that is, that it has no impact on the active-center composition distribution for either catalyst. We have shown that this might not be the case when chain shuttling

For conventional addition polymerization, the active-center composition distribution is thought to be determined by the propagation step (i.e., the first two terms in eq 20), as the propagation rates are many orders of magnitude larger than the rates of chain-transfer reactions involving µk,i 0 . Note that we have neglected the terms involving irreversible chain-transfer reaction rates in eq 20. Using the quasi-steady-state assumption (QSSA) for µk,i 0 , we obtain the active-center composition distribution as φk,i )

kkp,jifi k fj kkp,jifi + kp,ij

(21)

where i, j ) 1, 2, and i * j. It can be argued that, if the chain-shuttling terms (i.e., the last two terms in eq 20) are much smaller than the propagation terms, then the active-center composition distribution is still described by eq 21. In this case, the polymerization rate will not be impacted by the CSA. However, in the case that the CSA is very efficient, so that the chain-shuttling terms are comparable to or much larger than the propagation terms, the active-center composition distribution will be determined by the chaink,i i shuttling reactions. Using the QSSA for µk,i 0 , we have φ ) ψ , i where ψ is the molar fraction of dormant chains with last inserted monomer i. Note that, in the case of dual catalysts, φk,i could be different from the value calculated according to eq 21, thus leading to a polymerization rate in the presence of the CSA that is different from that in the absence of the CSA. It is striking that both catalyst active centers will have the same composition. A very efficient CSA could homogenize the activecenter composition distribution. Even though one can still use eq 11 to calculate DPn, R(i) p becomes dependent on the choice of CSA, so that eq 16 does not hold any more as illustrated in Figure 5 when the relative chain-shuttling rate constant approaches 100. Effect of Virgin CSA Feed Rate in a CSTR. Once an appropriate CSA has been identified, the CSA feed rate can be manipulated to control the degree of chain shuttling and, thus, the polymer chain architecture in a CSTR. Here, we examine the effect of the CSA feed rate when other operating conditions (cf. case 2 in Table 6) are maintained the same. We chose the CSA feed rate of case 2 in Table 6 as the baseline CSA feed rate and define the relative CSA feed rate as the ratio between the actual CSA feed rate and the baseline CSA feed rate. Figure 6 shows the variations of the copolymer compositions of the bulk polymer, block 1, and block 2 with respect to the relative CSA feed rate. When the relative CSA feed rate increases from 10-4 to 1, there is not much change in the copolymer compositions. However, when the relative CSA feed rate increases beyond 1, the copolymer compositions of the bulk and block 2 decrease, whereas that of block 1 increases slightly. Additionally, Figure 6 shows the variations in the mass

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Figure 6. Variations of bulk copolymer composition, block copolymer composition, overall conversion, and fractional conversions of ethylene and octene for varying virgin CSA feed rates.

Figure 7. Comparisons of Mn and Mw/Mn for varying CSA feed rates.

conversions of ethylene, octene, and overall monomer with respect to the relative CSA feed rate. Again, when the relative CSA feed rate increases from 10-4 to 1, the mass conversions of ethylene, octene, and overall monomer remain unchanged. By contrast, when the relative CSA feed rate increases beyond 1, the conversions of ethylene, propylene, and overall monomer increase. Such observations can be explained by the fact that the presence of the CSA could potentially influence the activecenter composition when the chain-shuttling rate is much larger than the propagation rate, as shown in the case of relative CSA feed rates greater than 1. Figure 7 shows how Mn evolves when the virgin CSA feed rate changes. An increase in the feed rate of the CSA leads to decreases of both Mw and Mn. Moreover, the Mw/Mn values of the resulting polymers becomes smaller, approaching 2 at the limit. When a high level of CSA exists in the system, extension of the chain length becomes difficult. In the extreme case, when the feed rate of CSA is very high, the average chain length is

Figure 8. Variations of bulk copolymer composition, block copolymer composition, weight fraction of block 1, overall conversion, and fractional conversions of ethylene and octene for varying catalyst compositions.

small. In this case, the system does not even have highmolecular-weight polymer. Note that the effects of the chainshuttling rate constant and the virgin CSA feed rate in a CSTR on the MWDs and block structures of the resulting polymers share many common features. Increasing either the chainshuttling rate constant or the CSA feed rate leads to narrower MWDs. However, the chain-shuttling rate constant, when it is large enough to convert all of the virgin CSA into dormant chains, has less of an effect on Mn whereas increasing the virgin CSA feed rate leads to a decreased Mn value. Thus, the combination of the feed rates of virgin CSA and hydrogen can be used to control the molecular weight in chain-shuttling polymerization. Effect of Catalyst Composition in the Presence of Chain Shuttling. Catalyst composition in the feed is another important variable that can be manipulated to tune molecular chain architectures. Figure 8 shows the effect of the catalyst composition on the bulk copolymer and block copolymer compositions. Because catalyst 1 easily incorporates octene into the polymer, it is intuitive to assume that increasing the molar fraction of catalyst 1 in the feed will lead to an increasing octene molar fraction in the soft block. However, even though increasing the molar fraction of catalyst 1 in the feed leads to an increase in the overall octene content in the bulk polymer, the molar fraction of octene in the soft block actually decreases. Close examination shows that increasing feed rates molar fraction of catalyst 1 in the catalyst feed leads to an increase in the octene conversion and a slight drop in the ethylene conversion (cf. Figure 8). Thus, the molar fraction of octene in the reactor drops. Increasing the catalyst 1 composition leads to an increase in the weight fraction of block 1 generated by catalyst 1. Figure 9 shows that Mn increases significantly as the molar fraction of catalyst 1 increases whereas the polydispersities of polymers remains around 2 for a wide range of catalyst compositions. Catalyst composition is one of the variables that can be manipulated for the development of olefin block copolymers with desired properties.

Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010

Figure 9. Variations of Mn and Mw/Mn for varying catalyst compositions.

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Figure 11. Variations of Mn and Mw/Mn for varying ethylene weight fractions in the feed.

ethylene weight fraction in the feed is changed. Even though the average chain length changes, the polydispersities remain around 2 because of chain shuttling. The delicate balance between the average length and the copolymer composition of each block is the key to obtaining desired properties. Model predictions show that the monomer feed composition is also a variable that can be used to tune block properties. Conclusions

Figure 10. Variations of bulk copolymer composition, block copolymer composition, weight fraction of block 1, overall conversion, and fractional conversions of ethylene and octene for varying ethylene weight fractions in the feed.

Effect of Monomer Composition in the Feed in the Presence of Chain Shuttling. If the catalyst composition and feed rate, CSA feed rate, and other operating conditions are unchanged but is variedmonomer weight composition in the feed is varied, certain effects on product properties are predicted by the model. Note that, for this analysis, we chose a CSA feed rate so that chain shuttling exists, leading to desired block properties. Additionally, the total monomer mass feed rate was assumed to remain constant. Figure 10 shows that, as the ethylene weight fraction in the feed increases, the molar fractions of ethylene in both the bulk polymer and the two blocks increase whereas the weight fraction of hard blocks in the bulk polymer remains nearly flat. Moreover, the overall mass conversion and the fractional conversions of ethylene and octene increase when the ethylene weight fraction in the feed increases because the less reactive octene tends to decelerate polymerization. Figure 11 demonstrates how polymer properties such as Mn vary when the

A model for chain-shuttling copolymerization using dual catalysts was developed in this study. Validation of the model was carried out qualitatively against experimental observations. Simulations demonstrate how this model can be employed for improved process development and operation. Model predictions show that operating a chain-shuttling copolymerization process in a CSTR leads to the production of unique olefin block copolymers with control of such chain properties as the molecular weight and molecular weight distribution, copolymer composition, and block composition. Selection of a CSA with appropriate chain-shuttling rate constants is also important in controlling the molecular weight distribution. A chain-shuttling rate constant that is too large could alter the active-center composition in the CSTR, leading to different polymerization behavior from that in the absence of the CSA. It is also possible to observe a varying polymerization behavior when a large amount of CSA is used for a more moderate chain-shuttling agent. Moreover, the model was used to analyze the effects of catalyst composition in the feed and feed rate of the CSA on the properties of the resulting polymers. Acknowledgment This work is dedicated to Professor W. Harmon Ray on the occasion of his 70th birthday. The authors thank Dr. Ted Carnahan for helpful discussions on this topic. Nomenclature a ) defined in eq 18 b ) defined in eq 19 Cat*i ) activated catalyst of type i Ci ) concentration of species i, where i can be Mj, H2, Cati, and CSA CCSA,consumed ) concentration of total consumed virgin CSA in a CSTR

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Dn_ ) dead chain with n_ monomer units DPn ) number-average chain length of bulk polymer DP(i) n ) number-average chain length of bulk polymer using catalyst of type i alone in the absence of CSA DPw ) weight-average chain length of bulk polymer (i) Fblock ) molar fraction of total monomer and comonomer incorporated into polymer due to catalyst i (i ) 1, 2) Fpoly,j ) molar fraction of monomer j in bulk polymer; i.e., copolymer composition with respect to monomer j i fcat ) molar fraction of catalyst of type i fi, fj, fk, fl ) molar fractions of monomers i, j, k, and l, respectively, in the reaction mixture kiCS0 ) chain-shuttling rate constant for virgin CSA to transfer with growing chain with catalyst of type i i kCS ) chain-shuttling rate constant for dormant chain to transfer with growing chain with catalyst of type i i kp,kj ) chain propagation rate constant for catalyst of type i with last inserted monomer k to add monomer j i kth ) chain thermal termination rate constant for catalyst of type i i ktrH ) chain transfer to hydrogen rate constant for catalyst of type i Mi ) monomer of type i Mn ) number-average molecular weight Mw ) weight-average molecular weight Mw,av ) average molecular weight of repeat unit in bulk polymer Mw,i ) molecular weight of monomer i n_ ) vector in which the kth element is used to record the number of monomer k that has been incorporated into the chain Pi,j _ monomer units and n_ ) growing chain of catalyst type i with n last inserted monomer j Qjn_ ) dormant chain with n_ monomer units and last inserted monomer j ri ) rate of change of species i due to reactions, where i can be j Mj, H2, Cati, CSA, Pi,j n_ , Qn_, Dn_, and various leading moments k k rijk ) reactivity ratio, defined as rijk ) kp,ii /kp,ij R(i) (i ) 1, 2) ) polymerization rate through catalyst i p Rt(i)(i ) 1, 2) ) termination rate to form dead polymer chains through catalyst i (i) wtblock ) weight fraction of block i that is created through catalyst of type i xCSA ) conversion of virgin CSA in a CSTR Zp ) polydispersity of bulk polymer

ξ0 ) total concentration or zeroth moment of dormant chains; ξ0 2 ) ∑i)1 ξi0 ξi0 ) concentration or zeroth moment of dormant chains with monomer i directly attached to CSA ξδj l ) first moment with respect to monomer l for dormant chains with last inserted monomer j ξδl ) first moment with respect to monomer l for dormant chains; 2 note that ξδl ) ∑j)1 ξδj l i,j φ ) molar fraction of growing chains having an active center from catalyst of type i and ending with monomer j ψj ) molar fraction of dormant chains with monomer j directly attached to CSA Acronyms ATRP ) atom-transfer radical polymerization CCTP ) coordinative chain-transfer polymerization CSA ) chain-shuttling agent CSTR ) continuous stirred-tank reactor MWD ) molecular weight distribution OBC ) olefin block copolymer RAFT ) reversible addition-fragmentation transfer

Appendix In this section, the reaction rates of the leading moments are described. Concentration or zeroth moment of growing polymer 2

rµ0i,j ) -

∑k

2

i i,j p,jkµ0 CMk

k)1 i kCS µi,j 0 ξ0

∑k k)1

i i,k p,kjµ0 CMj

i i - Riµi,j 0 + fjR µ0 -

i + kCS ξj0µi0

(22)

where i i Ri ) ktrH CH2 + kith + kCS0 CCSA

Note that, for convenience, we define φi,j as the molar fraction of growing chains ending with monomer j with active center generated by catalyst i. Additionally, for each catalyst separately and the total catalysts, we have

Greek Symbols i i i Ri ) parameter defined as ktrH CH2 + kth + kCS0 CCSA δ(i) ) Kronecker delta function, where δ(i) ) 1 when i ) 0 and δ(i) ) 0 when i * 0 θ ) residence time of the reactor λ0 ) concentration or zeroth moment of bulk polymer λ(i) 1 ) total concentration of monomer and comonomer incorporated into polymer due to catalyst i (i ) 1, 2) in the CSTR λ2 ) second moment of bulk polymer λδk ) first moment with respect to monomer k of bulk polymer λm ) polymer mass concentration in the CSTR µ0 ) total concentration or zeroth moment of growing chains; µ0 2 2 ) ∑i)1 ∑j)1 µi,j 0 i µ0 ) concentration or zeroth moment of growing chains with active 2 i i center generated by catalyst i; µi0 ) ∑j)1 µi,j 0 or µ0 ) µ0fcat i,j µ0 ) concentration or zeroth moment of growing chains of catalyst i i,j type i and last inserted monomer j; note that µi,j 0 ) µ0fcatφ µδi l ) first moment with respect to monomer l for growing chains with 2 active center generated by catalyst i; note that µδi l ) ∑j)1 µδi,jl µδi,jl ) first moment with respect to monomer l for growing chains with active center generated by catalyst i and last inserted monomer j

+

2

rµ0i )

∑r j)1

2

µ0i,j

) 0,

rµ0 )

∑r i)1

µ0i

)0

(23)

Concentration or zeroth moment of dormant polymer 2

2

∑k

rξ0j )

i i,j CS0µ0 CCSA

i)1 2

j)1

2

∑k

i i,j CSµ0 ξ0

i)1

-

∑k

i i j CSµ0ξ0

i)1

2

∑r

rξ0 )

+

ξ0j

)

∑

(24)

i kCS0 µi0CCSA

i)1

In our model, we do not count virgin CSA as dormant chains; thus, the net increase in dormant chains is due to the consumption of virgin CSA to form polymeryl CSA. After complete consumption of the virgin CSA, the net generation rate of dormant chains is zero. First moment of growing polymer 2

rµδi,j ) l

∑k

i p,kjδ(j

k)1

i i,j i i - l)µi,k 0 CMj - R µδl + fjδ(j - l)R µ0 i i kCS µδi,jlξ0 + kCS ξδj lµi0 (25)

Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010 2

rµδi )

∑r

)

∑k

l

j)1 2

k)1

µδi,j

l

i i - Riµδi l + flRiµi0 - kCS µδi lξ0 + kCS ξδlµi0

i i,k p,klµ0 CMl

(26) Note that, as an approximation, we use µδi,jl ) µδi lφi,j. First moment of dormant polymer 2

2

rξδj )

∑

rξδ )

∑k

l

i kCS0 µδi,jlCCSA +

i)1

i)1

2

l

∑

2

i kCS µδi,jlξ0 -

+

i)1

i i j CSµ0ξδl

(27)

i i CSµ0ξδl

(28)

i)1

2

i i CS0µδlCCSA

∑k 2

∑k

i i CSµδlξ0

-

i)1

∑k i)1

Concentration or zeroth moment of bulk polymer 2

rλ0 )

∑Rµ

i i 0

(29)

i)1

First moment of bulk polymer with respect to monomer k 2

rλδ ) k

2

∑ ∑k

i i,j p,jkµ0 CMk

i)1 j)1

(30)

Second moment of bulk polymer 2

rλ2 ) )

2

∑ ∑r

m)1 l)1 2 Riµi0 i)1 2 2

∑ 2

λδ

m+δl

2

+

2

2

∑ ∑ ∑k

i i,k p,kjµ0 CMj

i)1 j)1 k)1 2 2 kip,kjµδi,kl CMj i)1 j)1 k)1 l)1

∑∑∑∑

+

(31)

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ReceiVed for reView March 15, 2010 ReVised manuscript receiVed June 22, 2010 Accepted July 1, 2010 IE100530P