Theoretical Modeling of Average Block Structure in Chain-Shuttling α

Article pubs.acs.org/Macromolecules

Theoretical Modeling of Average Block Structure in Chain-Shuttling α−Olefin Copolymerization Using Dual Catalysts Min Zhang,‡ Thomas W. Karjala,*,† Pradeep Jain,† and Carlos Villa† †

The Dow Chemical Company, 2301 N. Brazosport Boulevard, Freeport, Texas 77541-3257, United States E. I. du Pont de Nemours and Company, Experimental Station, Wilmington, Delaware 19880, United States

‡

ABSTRACT: We report the theoretical derivation of a kinetic model for the prediction of average block structures such as number-average blocks, average block length, and average number of linkage points per chain, etc., in chain shuttling polymerization in the presence of dual catalysts based on the proposed mechanism. We further investigate how the chain shuttling rate constant and virgin chain shuttling agent (CSA) feed rate affect the average block structures predicted by this theoretical model for polymers produced in a continuous stirred tank reactor (CSTR). The simulations demonstrate that the coordination of dual catalysts and CSA is the key to enabling a successful chain shuttling polymerization system.

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INTRODUCTION The discovery of chain shuttling polymerization first reported by Arriola et al.1 in 2006 has led to the rapid commercialization of olefin block copolymers (OBCs). The unique features in chain shuttling polymerization include the utilization of dual catalysts, i.e., dual active centers, each having varying ability to catalyze the formation of polymer chains with distinct molecular structures such as copolymer composition or short chain branching (SCB), long chain branching (LCB), etc., and a chain shuttling agent (CSA) that could integrate the two very different types of polymer chains into one chain, leading to the formation of OBCs. While many papers have been published to discuss the discovery of the catalyst pairs,1−4 other important aspects of the chemistry,5−9 and the unique properties of the OBC polymers,10−14 only a few papers discuss the process development issues associated with implementing the new chemistry in existing equipment15,16 or the distributions of number of blocks and lengths of those blocks.17,18 Previously, we reported a theoretical kinetic model to describe the molecular weight distribution and copolymer composition for α-olefin copolymerization with chain shuttling chemistry using dual catalysts in stirred tank reactors.15 In particular, we revealed that some unique phenomena may appear when chain shuttling reaction rates are much faster than polymerization rates. For example, the copolymer composition may not be described by the Mayo−Lewis equation.16 Also, due to the varying active center compositions, the other molecular properties such as the average molecular weights (i.e., Mn and Mw) will also vary.15 While that model provides insight into the polymerization system, it does not address a key issue regarding the formation of block structures in such chemistry and how to use the model to guide the commercialization of the chemistry to control the block structures. It is expected that the block © XXXX American Chemical Society

structures of OBCs play a significant role in influencing the morphology,18 crystallization behavior and performance for the resulting polymers.19 Thus, in this paper, we develop a theoretical model of block structures for chain shuttling polymerization in order to provide insight on how to tune the chemistry in order to achieve block polymers with varying molecular structure. Note that several researchers have modeled the sequence length distributions in free radical polymerization systems that are most closely related to the block structure modeling reported in this work.20−22 However, the sequence length distributions are considered short-range properties that can be characterized through NMR while successful characterization techniques of block structures have not been reported in the literature.

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MODEL FORMULATION A kinetic model for the purpose of polymerization process development requires us to consider the necessary structural information to be captured. When describing a polymer population containing multiblock polymer chains with two distinctive block units, we could first consider average structural properties: the average number of each type of block in a chain, the average number of linkage points in a chain, and the average length of each block. Note that it is frequently desirable to understand the detailed structural information about the blocks such as block length distribution and sequence length distribution in each block. The derivations based on statistical arguments and Monte Carlo simulations17,18,23 and their linkage to the kinetic model are discussed elsewhere. Received: March 7, 2013 Revised: May 22, 2013

A

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Table 1. Definitions of Several Leading Moments for Various Polymer Populations polymer population growing polymer

leading moment

definition ∞

μ0i

∞

2

2

μ0i = ∑n = 0 ∑m = 0 ∑k = 1 ∑ j = 1 Pin,̲k, m, j̲ and in fact μ0i = Cat i

μ0i , j

∞

μ0i , j =

∞

2

∑ ∑ ∑ Pin,̲k,m,j̲ n=0 m=0 k=1 ∞

μ0i , k , j

μ0i , k , j =

∞

∑ ∑ Pni ,̲ ,km,̲ j n=0 m=0

dormant polymer

∞

ξ0

ξ0 =

∞

2

2

∑ ∑ ∑ ∑ Q kn̲,,jm̲ n=0 m=0 k=1 j=1 ∞

ξ0k , j

ξ0k , j =

∞

∑ ∑ Q kn̲,,jm̲ n=0 m=0

bulk polymer

∞

λ0, δl

λ0, δl =

∞

2

2

∞

2

∞

2

2

∑ ∑ ∑ ∑ ∑ m̲ δl Pin,̲k,m,j̲ + ∑ ∑ ∑ ∑ m̲ δlQ ni ,̲ ,jm̲ n=0 m=0 i=1 k=1 j=1 ∞ ∞ + m̲ δl Dn̲ , m̲ n=0 m=0

n=0 m=0 i=1 j=1

∑∑

∞

λ0

λ0 =

∞

2

2

∞

2

n=0 m=0 i=1 k=1 j=1 ∞

λ δl

λ δl =

∞

2

∞

2

∞

∑ ∑ ∑ ∑ ∑ Pin,̲k,m,j̲ + ∑ ∑ ∑ ∑ Q in,̲j,m̲ + ∑ ∑ Dn̲ ,m̲ ∞

2

2

2

n=0 m=0 i=1 j=1 ∞

∞

2

n=0 m=0 2

∑ ∑ ∑ ∑ ∑ n̲ δl Pin,̲k,m,j̲ + ∑ ∑ ∑ ∑ n̲ δlQ in,̲j,m̲ n=0 m=0 i=1 k=1 j=1 ∞ ∞ + n̲ δlDn̲ , m̲ n=0 m=0

n=0 m=0 i=1 j=1

∑∑

• Propagation

First, we introduce the notations for calculating the average number of each block in a chain.

kpi , jl

i,k,j • Growing chain, Pn,m k,j • Dormant chain, Qn,m • Dead chain, Dn,m

Pin,̲ k, m, j̲ + Ml ⎯→ ⎯ Pin,̲ i+, l δl , m̲ + (1 − δ(i − k)) δi

Here kp,jli is the propagation rate constant for a growing chain with last inserted monomer j to add monomer l at the active center originating from catalyst i. Additionally, the Kronecker delta function δ(i−k) is used to capture the formation of a new block when cross chain shuttling followed by propagation occurs. Note that

Here 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. m is a vector in which the first element records the number of blocks formed from catalyst 1, while the second element records the number of blocks formed from catalyst 2. Moreover, i denotes the type of catalyst attached to a growing chain, j the type of monomer at the chain end adjacent to the catalyst active center in a growing chain or neighboring to the chain shuttling agent in a dormant chain, and k the type of terminal block in the chain formed from catalyst k. Note that the superscript and subscript i, j, k, l in the notations crossing the entire paper can be 1 or 2, respectively. Consistent with the proposed kinetic scheme in our previous paper,15 we rewrite the kinetic scheme using the above notations as follows.

⎧ 1 when i = k δ(i − k) = ⎨ ⎩ 0 when i ≠ k

• Chain transfer to hydrogen, H2 Note that for reactions such as chain transfer to hydrogen, thermal termination, and chain shuttling to virgin CSA, we lump the reaction scheme together for the purpose of simplifying modeling. Using chain transfer to hydrogen as an example, the reaction has two steps: First, the growing chain transfers to hydrogen to form a dead chain and releases the active center that can propagate to form a new chain.

• Instantaneous activation of catalyst without deactivation instantaneous

Cat i ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Cat*i

i ktrH

Pin,̲ k, m, j̲ + H 2 ⎯→ ⎯ Dn̲ , m̲ + Cat*i

instantaneous

Cat*i + Mj ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Piδ, ij,,jδi

(2)

(3)

Here ktrHi is the chain transfer to hydrogen rate constant for a growing chain with the active center originating from catalyst i. Second, the active center immediately adds a monomer to form a new chain with a block identified by catalyst i.

(1)

Here Cati and Cat*i are catalyst i and activated catalyst i respectively. Mj is monomer j. δj and δI represent the unit vector with the jth and ith element equal to 1 respectively. The instantaneous activation of catalyst i starts the formation of a new chain with a block identity from catalyst i.

instantaneous

Cat*i + Ml ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Piδ, il,,l δi B

(4)

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Thus, for brevity, the two steps of the reaction can be lumped into one step by assuming that the addition of monomer l to the primary active center is proportional to the monomer composition, f l, i.e. Pin,̲ k, m, j̲

i ktrH

+ H 2 ⎯→ ⎯ Dn̲ , m̲ +

fl Piδ, il,,l δi

+

2

+Q

(7)

2 1

⎯→ ⎯ Q + (8) ′ ′ ′ ′ i Here kCS is the chain shuttling rate constant between a dormant chain and a growing chain with the active center originated from catalyst i. For the sake of simplicity in formulating the theoretical model, the mass balances of small molecules such as monomer, comonomer, hydrogen, and CSA are not described. They can be calculated through the model reported earlier.15 Thus, in the following, we focus on the rates of change of polymer chains only using the current notations of polymer chains. 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, some of the leading moments of various polymer populations are used directly in the expressions of the rate of change of species, and the definitions of the leading moments are briefly addressed in Table 1. • Growing chain 2 l=1

j

2

j

i

+ α 2μ02 (14)

2 i = −∑ kpi , jlμ0i , k , j C Ml − α iμ0i , k , j − kCS ξ0μ0i , k , j

j

l=1

i + δ( n̲ − δj)δ(m̲ − δi)f j α iμ0i − kCS ξ0 Pin,̲ i,,mj ̲ i kCS μ0i Q in,̲ j, m̲

i + kCS μ0i ξ0k , j

(9)

2

r ξ0k ,j =

2

2 i=1

(16)

Additionally, even though we can directly calculate the rate of change for the moment μi,i,j 0 through the general moment i,2,j j 1,j 2,j equations, since μi,j0 = μi,1,j 0 + μ0 and ξ0 = ξ0 + ξ0 , we can greatly reduce the complexity of the model. The incorporation of monomers into block i is directly due to the propagation event occurring through catalyst i. If we define the concentration of monomer j in block i as ζij, the rate of change of ζij can be written as

l=1 i kCS μ0i Q kn̲,,jm̲

(15)

∑ (kCSi0CCSA + kCSiξ0)μ0i ,k ,j − ∑ kCSiμ0i ξ0k ,j i=1

i r Pin,̲k,m,j̲ (i ≠ k) = −∑ kpi , jl Pin,̲ k, m, j̲ C Ml − α i Pin,̲ k, m, j̲ − kCS ξ0 Pin,̲ k, m, j̲

+

2

∑ ∑ kp2,jiμ02,1,j CM

r μ0i ,k ,j(i ≠ k)

l=1

+

(13)

Note that λ0,δ1 and λ0,δ2 denote the concentrations of blocks 1 and 2, respectively. Careful examination of the above equations shows that a new block is formed under two circumstances: first, a new chain and hence a new block forms through a chain transfer reaction such as chain transfer to hydrogen, thermal termination, or chain transfer to virgin CSA and second, a dormant chain with a potential catalyst site i shuttles with a growing chain with catalyst site j (j ≠ i) and subsequently experiences propagation. k,j We require knowledge about the moments μi,k,j 0 and ξ0 for growing and dormant chains. On the basis of the polymer population equations, we have derived the general moment equations. For brevity, we do not discuss the derivation of the general equations but rather the rates of change of moments k,j μi,k,j 0 and ξ0 , written as

∑ kpi ,ljPni ,̲ −k ,lδ ,m̲ − δ C M (k ≠ i) − α iPin,̲i,,mj̲ i

+ α1μ01

i

i=1 j=1

l=1

j

2

2

rλ0, δ =

2

+

(12)

i=1 j=1

2

∑ kpi ,ljPin,̲i−,l δ ,m̲ C M

+ kthi)Pin,̲ k, m, j̲

2

∑ ∑ k1p,jiμ01,2,j CM

rλ0, δ =

Pin, k,′m, j ′

r Pin,̲i,,mj̲ = −∑ kpi , jl Pin,̲ i,,mj ̲ C Ml +

2

The equations of moments under the above notations can also be derived following standard procedures, thus average polymer properties such as DPn, DPw, and overall copolymer composition can be calculated. Even though the model formulation seems to be overly complicated, we found that since we are able to calculate average polymer properties through the model reported earlier,15 the calculation of block structures only requires the calculation of the following first moments of bulk polymers, i.e. λ0,δI. We have

Here kiCS0 is the chain shuttling to virgin CSA rate constant for a growing chain with the active center originated from catalyst i. • Chain shuttling to polymeryl CSA k ,j n̲ , m̲

(11)

i=1 j=1 k=1

i kCS 0

k′,j′ n ,m

2

i CH ∑ ∑ ∑ (ktrH

rDn̲ ,m̲ =

(6)

Pin,̲ k, m, j̲ + CSA ⎯⎯⎯→ Q kn̲,,jm̲ + fl Piδ, il,,l δi

Pin,̲ k, m, j̲

∑ kCSiξ0Pin,̲k,m,j̲

• Dead chain

Here kith is the thermal termination rate constant for a growing chain with the active center originated from catalyst i. • Chain shuttling to virgin CSA

i kCS

i=1

i=1

(5)

fl Piδ, il,,l δi

Dn̲ , m̲ +

i=1

2

• Thermal termination kthi Pin,̲ k, m, j̲ →

2

∑ kCSi0Pin,̲k,m,j̲ CCSA − ∑ kCSiμ0i Q kn̲,,jm̲

r Q kn,̲ ,jm̲ =

(10)

where i i α i = ktrH C H2 + kthi + kCS 0CCSA

• Dormant chain C

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Table 2. Calculation of Average Properties of Block i

2

r ζji =

∑ kpi ,kjμ0i ,k CM

j

average properties of block i (i = 1, 2)

(17)

k=1

number-average of block i per chain

Finally, it is also of interest to examine the average number of linkage points in a polymer chain. In the operation of CSTR to prepare OBCs using dual catalysts, the resulting polymers have a range of distributed number of blocks per chain due to both the residence time distribution of the reactor and the chemical mechanisms. Thus, predicting the average number of linkage points in a polymer chain from chemical mechanism of chain shuttling polymerization is also of practical importance. Close examination of the chemistry reveals that the formation of a linkage point only involves two reactions: a cross shuttling event followed by propagation. If we define the concentration of linkage points in a chain as λ0,linkage, the rate of change of λ0,linkage is written as 2

rλ0,linkage =

2

∑ ∑ k1p,jiμ01,2,j CM i=1 j=1

2 i

+

λ0, δi λ0

molar fraction of block i

λ0, δi 2 ∑i = 1 λ0, δi

weight fraction of block i

2

∑ j = 1 ζjiMwj 2

∑k = 1 λ0, δk Mwk average length of block i

2

∑ j = 1 ζji λ0, δi

average number of linkage points per chain

λ0, linkage λ0

2

∑ ∑ kp2,jiμ02,1,j CM

calculation method

positive number could essentially reflect the formation of block structures in the polymerization system.

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i

i=1 j=1

RESULTS AND DISCUSSIONS Commercial production of OBCs can be achieved in continuous processes such as CSTRs. Thus, it is of interest to examine the effects of the kinetic variables and operating variables on the average block structures as predicted by our theoretical model. While many factors play a role in chain shuttling polymerization and affect the molecular structure of the resulting polymer, there are two variables that appear important for the formation of block structures in this process: chain shuttling rate constant and virgin CSA feed rate. In the following sections we will examine how these two factors affect the block structures in a CSTR environment, where the steady state mass balance for generic moment s, takes the following form in terms of the average reactor residence time τ: 0 = s f − s − τrs (20)

(18)

We use this model to describe as an illustration the average block structures for chain shuttling polymerization carried out in a single CSTR using dual catalysts, but the rate expressions listed so far are applicable to other reactor configurations, under steady state or dynamic conditions, at constant or variable temperature, with perfect or imperfect mixing. Characterization techniques for block structures have been rapidly developed since the emergence of OBCs prepared through chain shuttling chemistry.24 Li Pi Shan et al. developed a method to quantify the block structure via temperature rising elution fractionation (TREF) fractionation, but the approach does not describe the details of the block structures such as number-average block and average block length, etc.24 While the exploration of more advanced characterization methods is still ongoing, we would like to pursue the theoretical modeling of average block structures based on the proposed mechanism. A technique was proposed by Li et al.18 to estimate the average number of blocks per chain, generating values between 2.2 and 3.5 for this quantity when applied to samples synthesized using the techniques described by Arriola et al.1 In combination with this technique or characterization techniques to appear in the future, our work thus provides a theoretical understanding and the basis for estimating the kinetic properties of systems used to generate multiblock structures. Additionally, we are also interested in learning how the intrinsic chemistry and operating conditions can affect the block structure development. Such understanding can facilitate the commercialization of the new chemistry by determining the key variables for manipulation. The average block properties are calculated as shown in Table 2. Note that the number-average block per chain is a misnomer. For a polymer population without block structures in the presence of a single catalyst, the average number of blocks per chain is one. In the absence of chain shuttling while using multicatalysts, the number-average block per chain is equivalent to the molar fraction of polymers resulting from each catalyst. When the number-average blocks per chain is smaller than one, it does not tell exactly whether the polymer population contains block polymers or not. Thus, we also need to use the numberaverage linkage point per chain, the ratio of the total linkage points to the total number of chains, to better illustrate the formation of block structures of the polymer population as a

To be consistent with our earlier paper,15 we use the same operating conditions to show the effects of the two factors. For clarity, the baseline simulation conditions are summarized in Table 3. The hypothetical kinetic parameters for the Table 3. Simulation Conditions in a CSTR Operation component

mass feed rate (kg/h)

solvent ethylene octene catalyst metal catalyst 1 M fraction hydrogen CSA

9.57 0.91 2.27 9.07 × 10−6 0.6 1.02 × 10−4 3.63 × 10−5

simulations are listed in Table 4 in which the propagation rate constant for ethylene homopolymerization is roughly estimated as on the order of 100 000 L/(mol s). These kinetic parameters do not necessarily represent the actual chemistry practiced commercially but instead are chosen to illustrate the qualitative behavior of the theoretical model derived in this paper. Estimates of actual values of some of these kinetic parameters can be obtained through other sources.3 Effect of Chain Shuttling Rate Constant on Block Structures in a CSTR. Careful selection of CSA and catalyst D

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the absence of chain shuttling, there should be no formation of block copolymers. Thus, the dual catalyst system can lead to a polymer blend with two polymer populations originating from each catalyst. By contrast, when choosing a high chain shuttling rate constant, the average block lengths can theoretically drop to a point where the synergy of the two distinct blocks disappears. Figure 1 shows the effect of the relative kCS on the average block structures in a CSTR, where the relative kCS is defined as the ratio between the chain shuttling rate constant and the baseline propagation rate constant in Table 4. When relative kCS is small, i.e. less than 10−1, it appears that there are very few block polymers formed in the system as shown in near zero number-average linkage points per chain. As relative kCS increases, both the number-average blocks per chain for block 1 and 2, and the number-average linkage points per chain increase. Even when relative kCS increases to 1, the numberaverage linkage point per chain is still below 1, suggesting that there are still polymers that have not incorporated both blocks into the chains. When relative kCS increases beyond 1, the number-average block per chain for block 1 and 2 becomes large and the number-average linkage point per chain is more than 1. Thus, the system starts to have significant amounts of multiblock copolymers. Additionally, the average length of block 1 and 2 decreases as relative kCS increases. When relative kCS is in the range of 1 to 10, the average length of block 1 is in the range of about 500 to 100 while the average length of block 2 is in the range of 700 to 70. It is our conjecture that the block length of each block is long enough for the resultant polymer to show synergistic physical properties from both blocks. Effect of Virgin Chain Shuttling Agent Feed Rate on Block Structures in a CSTR. Once an appropriate CSA has been identified, the CSA feed rate can be manipulated to

Table 4. Hypothetical Kinetic Parameters Used for Simulation kinetic parameter

value

1 r12

1

1 r21

0.1 100

2 r12 2 r21

0.01

k1p ,22

0.01

k1p ,11 0.1

kp2,11 k1p ,11

0.01

kp2,22 k1p ,11

0.01

1 ktrH

k1p ,11 0.0001

2 ktrH

k1p ,11 1 kCS 0

k1p ,11 2 kCS 0

k1p ,11

=

=

1 kCS

10

k1p ,11 2 kCS

10

k1p ,11

pairs is a key to the success of chain shuttling polymerization.. Thus, the choice of CSA with appropriate chain shuttling rate constant can determine whether the chain shuttling polymerization system can deliver multiblock copolymers as desired. In

Figure 1. Effect of chain shuttling rate constant on average block structures in a CSTR. E

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Figure 2. Effect of virgin CSA feed rate on average block structures in a CSTR.

structures is the average length of each block. As the virgin CSA feed rate increases, the average length of both blocks decreases. When the relative virgin CSA feed rate approaches 100, the average lengths of both blocks drop into the single digits, thus indicating homogenization of the resulting polymers. Consistent with model predictions, Park et al. experimentally found that increasing CSA level pushes the polymer in the direction of a random copolymer and thus suppress the occurrence of mesophase separation transition in the melt.13 Note that in the simulations, we have maintained the rate of hydrogen feed rate into the systems unchanged. In practice, when the virgin CSA feed rate increases to a certain level, it is desirable to shut off hydrogen feed as CSA alone can be used to control both molecular weight and block structures of the resulting polymers.

achieve control over both molecular weight and block structures of the resulting polymers in a CSTR. Here, we examine the effect of the CSA feed rate when other operating conditions are maintained the same. We chose the CSA feed rate in Table 1 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 2 illustrates the effect of virgin CSA feed rate on average block structures in a CSTR. Note that the role of virgin CSA is 2-fold: one is to regulate molecular weight and the other is to regulate block structures of the resulting polymers. Therefore, when virgin CSA feed rate increases, the total number of polymer chains increases, leading to the decrease of average molecular weight of the resulting polymers. Concurrently, the total number of each block and the total number of linkage point also increase as the reaction rate of cross chain shuttling increases. Since the average number of block and linkage points per chain is defined as the ratio of the total number of blocks and linkage points to the total number of chains, the variations of the average number of blocks and linkage points per chain is reasonably reflected in Figure 2. As virgin CSA feed rate increases, the number-average block per chain increases initially. However, when virgin CSA feed rate increases to a certain point, the number-average block per chain actually decreases. A similar trend is also observed for the average number of linkage points per chain. Note that the rate increases of the total number of blocks is the sum of rate terms in eqs 14 and 15) while that of the total number of linkage points is reflected in eq 18. By contrast, the rate increase of the total number of chains is directly proportional to the relative CSA feed rate. The model predictions could be attributed to the difference between the increase of the total number of blocks and linkage points to that of the total number of chains as relative CSA feed rate increases. Nevertheless, for the purpose of forming block polymers, a higher virgin CSA feed rate is preferred. However, another aspect of the block

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CONCLUSIONS A theoretical model for chain shuttling polymerization in the presence of dual catalysts and CSA was developed to allow the description of average block structure. The model was incorporated into a CSTR model to examine the effect of chain shuttling rate constant and virgin CSA feed rate on the average block structures. Selection of a CSA agent with appropriate chain shuttling rate constants is important to enable block polymers with balanced number of blocks and average block length that result in commercially useful polymer properties. Additionally, it is possible to have a chain shuttling rate constant that allows for the polymerization to demonstrate different polymerization behavior in the presence of chain shuttling agent while preparing a block polymer with balanced properties. The increase of virgin CSA feed rate leads to the increase of blocks in the polymer but there is a limit when a minimum length of the average block is also required. In combination with current and future characterization techniF

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and Copolymer Composition. Ind. Eng. Chem. Res. 2010, 49, 8135− 8146. (16) Zhang, M.; Carnahan, E. M.; Karjala, T. W.; Jain, P. Theoretical Analysis of the Copolymer Composition Equation in Chain Shuttling Copolymerization. Macromolecules 2009, 42, 8013−8016. (17) Anantawaraskul, S.; Somnukguande, P.; Soares, J. Monte Carlo Simulation of the Microstructure of Linear Olefin Block Copolymers. Macromol. Symp. 2012, 312, 167−173. (18) Li, S.; Register, R.; Weinhold, J.; Landes, B. Melt and Solid-State Structures of Polydisperse Polyolefin Multiblock Copolymers. Macromolecules 2012, 45, 5773−5781. (19) Zuo, F.; Mao, Y. M.; Li, X. W.; Burger, C.; Hsiao, B. S.; Chen, H. Y.; Marchand, G. R. Effects of Block Architecture on Structure and Mechanical Properties of Olefin Block Copolymers under Uniaxial Deformation. Macromolecules 2011, 44, 3670−3673. (20) Arriola, D. J. Modeling of Addition Polymerization Systems. Ph.D. Dissertation, University of Wisconsin: Madison, WI, 1989. (21) Zargar, A.; Schork, F. J. Design of Copolymer Molecular Architecture via Design of Continuous Reactor Systems for Controlled Radical Polymerization. Ind. Eng. Chem. Res. 2009, 48, 4245−4253. (22) Ye, Y. S.; Schork, F. J. Modeling and Control of Sequence Length Distribution for Controlled Radical (RAFT) Copolymerization. Ind. Eng. Chem. Res. 2009, 48, 10827−10839. (23) Wang, L.; Broadbelt, L. J. Explicit Sequence of Styrene/Methyl Methacrylate Gradient Copolymers Synthesized by Forced Gradient Copolymerization with Nitroxide-Mediated Controlled Radical Polymerization. Macromolecules 2009, 42, 7961−7968. (24) Li Pi Shan, C.; Hazlitt, L. G. Block Index for Characterizing Olefin Block Copolymers. Macromol. Symp. 2007, 257, 80−93.

ques, our work provides a theoretical understanding and the basis for estimating the kinetic properties of systems used to generate multiblock structures, and becomes a tool for proper selection of reactor conditions leading to multiblock structures with desired properties.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (T.W.K.) [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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dx.doi.org/10.1021/ma4004902 | Macromolecules XXXX, XXX, XXX−XXX