Ind. Eng. Chem. Res. 2009, 48, 4245–4253
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Design of Copolymer Molecular Architecture via Design of Continuous Reactor Systems for Controlled Radical Polymerization Amin Zargar and F. Joseph Schork* Department of Chemical and Biomolecular Engineering, Building 090, UniVersity of Maryland, College Park, Maryland 20742
In controlled radical polymerizations, the lifetime of a growing chain is approximately the residence time in the reactor train. Under these conditions, it is possible to use various continuous reactor trains (CSTRs and PFRs in series) to manipulate the molecular architecture. This work illustrates the design, by iterative simulation, of continuous reactor trains to produce highly specific molecular architectures. In addition to determining the typical parameters of polydispersity, molecular weight, and copolymer composition, the simulation produces information about the sequence structure in the copolymer. Prior work has shown the ability to use moment equations to analyze the sequence structure of batch polymerization and the potential to vary batch reactor conditions to produce copolymers with different sequences from the head to the tail of the copolymer. This work extends the technique of sequence distribution analysis through population balances to continuous reactor trains molecular architectures with specific molecular weight, copolymer composition, and sequence distribution. Introduction In free-radical polymerization, the lifetime of a growing polymer chain is 1-10 s. Given that the average residence time in a continuous reactor is on the order of magnitude of hours, the residence time distribution (RTD) has little effect on the molecular weight distribution (MWD) of polymer polymerized by the free-radical mechanism in continuous reactors. Likewise, given the short lifetime of a radical chain, it is not possible to manipulate the copolymer composition or sequence distribution within a single chain by adjusting the comonomer concentrations. However, with the advent of controlled radical polymerization (CRP) chemistries, the lifetime of a single growing chain is on the order of hours. In the case of CRP, it is worthwhile revisiting the effects of the reactor type and RTD on the architecture of polymers made by these chemistries. As with truly living polymerizations, the MWD of polymer produced by CRP mirrors the RTD. What is more interesting, however, is the potential ability of the polymerization reaction engineer to specify the copolymer composition distribution (CCD) and copolymer sequence distribution (CSD) by specifying the design of a continuous reactor system for CRP. This work explores this idea and makes use of a new method of determining copolymer sequence distribution in CRP.1 Controlled radical polymerization (CRP) has shown the potential for improvements in the manipulation of polymeric molecular architecture, with NMP (nitroxide-mediated polymerization), ATRP (atom-transfer free-radical polymerization), and RAFT (reversible addition-fragmentation chain-transfer polymerization) producing polymers with narrow molecular weightdistributionsanddesiredcopolymersequencedistributions.2-6 Although the ideas proposed in this work apply equally to NMP, ATRP, or any other form of CRP, we have chosen to simulate RAFT polymerization to demonstrate the effect of reactor design on molecular architecture. In 1995, Krstina et al. published a new controlled radical polymerization (CRP) technique involving reversible addition-fragmentation chain transfer (RAFT) that produced polymers with low polydispersity.7 Mathematical models, using * To whom correspondence should be addressed. E-mail:
[email protected].
population balances, have been used to produce simulations that determine important characteristics of RAFT polymerizations.8,9 With the kinetic rate constants and reactor conditions, the entire CRP can be simulated to determine characteristics of the copolymer with accuracy.10 However, until recently,1 the quantification of the sequence structure of copolymers had been limited to probabilistic models.11 The probabilistic functions developed by Ray provide average sequence lengths; the population balance approach used here allows for the calculation of the moments of the CSD and can thus provide more information about the distribution of sequence lengths. With the advent of CRP, copolymers with composition gradients, as well as block copolymers and uniform-composition copolymers, have been polymerized for their unique characteristics. With the advances in CRP enabling unique sequence synthesis, further study is being done to understand the effects of sequence distribution on polymer properties.12-14 The sequence model used in this work has been shown to provide information on the effect of composition drift in RAFT polymerization in batch reactors.1 Also, shot polymerizations have illustrated the potential of varying reactor conditions to produce copolymers with different compositional polymer segments. In this work, that basis is extended to continuous reactors. Continuous stirred tank reactors (CSTRs) and plug-flow reactors (PFRs) have been used to produce a variety of copolymers. An analysis of CRP with these reactors demonstrates the utility of the sequence model but also illustrates the potential usefulness of these reactors in tailoring copolymer sequence distributions. Reactor series enable a simple path toward manipulating reactor conditions and feeds to design a specified copolymer. The use of CSTRs and PFRs in series allows the reaction engineer to selectively use the properties of these reactors to specify molecular architecture. CSTRs typically have low monomer conversions with high polydispersities (in CRP), whereas the continuous input of fresh feed at constant concentration provides a uniform copolymer composition. Conversely, PFRs have high conversions with low polydispersities but are susceptible to compositional drifting. Hence, using
10.1021/ie801422w CCC: $40.75 2009 American Chemical Society Published on Web 03/31/2009
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the reactors in conjunction could produce long uniform polymer segments with high conversions and low polydispersities. Kinetic Model The mathematical model used in this work has been described previously.1,15 The model consists of two independent subdivisions: the chain model determines the MWD, conversion, and copolymer composition, whereas the sequence model determines the distribution of sequences. Together, these two models provide a description of the molecular architecture of the copolymer. Certain simplifying assumptions have been made in the model. The primary goal of this work is to illustrate the use of reactor design to specify the overall molecular architecture of the copolymer produced. These simplifications do not detract from this goal, and the approach can be extended to account for these assumptions, but this work does not require them. The penultimate effect is ignored to reduce complexity (only the terminal unit affects the kinetic rate constant).8 Similarly, the effects of the degree of polymerization on the propagation, transfer, and termination rate constants and the reactivity ratios have been ignored.2,16-18 The macro-radical intermediate (P-T-P) formed during RAFT transfer has not been considered in our model, although in some processes, it could be stable enough to retard polymerization, initiate new chains, and participate in bimolecular termination.18-24 These events will reduce the “living” character of the system but should not substantially affect the sequence distribution. Lastly, branching has also been ignored. The reactions of the model are listed in Table 1. The classic model denotes three polymer chains: propagating radical chains with terminal unit A or B (Pn and Qn, respectively), dormant chains with terminal unit A or B (TPn and TQn, respectively), and dead chains (Mn). The sequence model tracks active species with terminal unit A or B (Lg* and Og*, respectively) and inactive species with terminal unit A or B (Lg and Og, respectively). Because branching has been ignored, the active species is geometrically signified as the last sequence in the copolymer, or the ending sequence. Definition of Chain and Sequence Moments and Development of Moment Equations. The mathematical modeling in this work is based on the method of moments, as mentioned earlier. Table 2 lists the definition of each moment species, and the moment equations of the overall model are produced through population balances around each species. The moment equations of the chain and sequence model can be found in ref 1. Characteristics of Chain and Sequence Models. The chain model tracks the classic parameters of the number-average chain length (NACL), weight-average chain length (WACL), polydispersity (D), conversion, and copolymer compositions of monomers A and B (CA and CB).The parameters that were tracked by the sequence model provide information of the sequence structure of the copolymer. NAA is the number-average sequence length for active sequences of monomer A, and NAB is the number-average sequence length for active sequences of monomer B. Because branching has been ignored, NAA and NAB essentially reflect the sequence at the end of the copolymer, providing the first glimpse at the molecular architecture. NIA is the number-average sequence length for inactive sequences of monomer A, and NIB is the number-average sequence length for inactive sequences of monomer B. An important note is that the last sequence of dead chains is counted as inactive; therefore, geometric inferences about the end sequences will be slightly skewed. With RAFT polymerizations, the ratio of living to dead
chains is high, thereby reducing the impact of this simplifying assumption. However, if the end sequence is much longer than all of the other sequences, the dead chains could significantly alter the tracking of the inactive sequences. The polydispersity parameters measure the uniformity of the sequence lengths: DAS is the polydispersity of the sequence length of monomer A, and DBS is the polydispersity of the sequence length of monomer B. Table 3 presents the important characteristics of the copolymer that were determined by the simulation. Results and Discussion The kinetic parameters used in this work were adapted from the literature.25 The use of “typical” kinetic parameters for CRP is an impossible task, as these parameters can be very different from system to system.24,26 However, the goal of this work is to illustrate that copolymers with different compositional polymer segments can be produced by varying reactor conditions, and these parameters were taken as applicable rate constants. The kinetic parameters used here are listed in Table 4. Stand-Alone CSTR Polymerization. Previous work has developed the sequence model to produce copolymers with batch reactors. This work shows how it can be expanded to include continuous reactors.1 In simulation one, a single CSTR polymerization was simulated to show the applicability of using continuous reactors and their effects on the sequence structures of the copolymers they produced. For this entire work, reactivity ratios were set at r1 ) 2.0 and r2 ) 0.5 to highlight the effects of composition drift in PFRs as opposed to CSTRs. In simulation one, equimolar concentrations (2 M) of A and B are fed to the reactor. The recipe consists of initiator totaling 0.3 mol % of the total of the monomers and a 9:1 molar ratio of RAFT agent to initiator. For this simulation and all others, at time zero, the CSTR is filled with reactants at their feed concentrations. Figure 1 shows the characteristics of the CSTR as the polymerization progresses. As Figure 1 shows, steady state is reached at approximately 4 times the residence time (20000 s). The degree of polymerization totals 23.3, with a 59.3% monomer conversion. As has been shown by Schork and Smulders,27 the polydispersity of a purely living polymerization in a CSTR approaches the residence time distribution of a CSTR (2.0). The simulation determined a polydispersity value of 1.95, which is lower than that of a purely living polymerization because of the small amount of dead polymer in the reactor.27 A highly favorable aspect of CSTRs is the high ratio of living to dead polymers, which was determined in this simulation to be 55.6. The copolymer composition fraction of monomer B begins at 63.3% at the beginning of the polymerization and quickly lowers to its steady-state value of 56.7%. (Note that all copolymer compostions are given in mole percent.) It is wellknown that the copolymer composition remains constant in a CSTR at steady state. Inactive sequences of A and B total 1.71 and 2.19, respectively. The values of the inactive sequences approximately equal the active sequences, which is the result of the lack of composition drift. The composition is uniform throughout the copolymer, revealing a great deal about the sequence structure of the copolymer. As an approximate average, two of every three sequence lengths of A total two monomers, with the third sequence length totaling one monomer. Correspondingly, four of every five sequence lengths of B total two monomers, with the fifth sequence totaling three monomers. Because the ratio of A-monomer to B-monomer end sequences strongly favors B monomer, the last sequence of the copolymer
Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4247 Table 1. Reactions of the Chain and Sequence Models Reactions of the Chain Model Initiation kd,f
I 98 2R*
ki1
R* + AfP1
RAFT Activation
RAFT Transfer
kr11
kraft11
T + PnT R* + TPn
Pn + TQr T TPn + Qr
kr12
kraft12
kr21
kraft21
T + QnT R* + TQn
Qn + TQr T TQn + Qr
kr22
kraft22
ki2
kraft31
R* + BfQ1
Pn + TPr T TPn + Pr kraft32
Propagation kp1
Pn + A 98 Pn+1
kp2
Pn + BfQn+1 kp3
Qn + AfPn+1
Termination ktc1
ktd1
Pn + QrfMn+r
Pn + QrfMn + Mr
ktc2
ktd2
Pn + PrfMn+r
Pn + PrfMn + Mr
ktc3
ktd3
Qn + QrfMn+r
Qn + QrfMn + Mr
kp4
Qn + BfQn+1 Reactions of the Sequence Model Active Sequences Initiation kd,f
RAFT Activation
RAFT Transfer
kr11
kraft31
Lg* + TT TLg* + R*
If2R*
Lg* + TOs* T TLg* + Os*
kr12
ki1
R* + AfL1*
kraft32
kr21
kraft41
Og* + TT TOg* + R*
Og* + TOs* T TOg* + Os*
kr22
kraft42
ki2
kraft52
R* + BfO1*
Lg* + TLs* T TLg* + Ls* kraft51
Propagation kp1
Lg* + AfLg+1* kp2
Lg* + BfOg* kp3
Og* + AfLg*
Termination ktc1
Lg* + Ls*fLg+s ktc2
Lg* + Os*fLg+s ktc3
Og* + Os*fOg+s
ktd1
Lg* + Ls*fLg + Ls ktd2
Lg* + Os*fLg + Os ktd3
Og* + Os*fOg + Os
kp4
Og* + BfOg+1* Propagation kp2
Lg* + BfLg
Formation of Inactive Sequences of A Termination ktc1
Lg-s* + Ls*fLg ktc2
Lg* + Os*fLg Propagation kp3
Og* + AfOg
ktd1
Lg* + Ls*fLg ktd2
Lg* + Os*fLg
Formation of Inactive Sequences of B Termination ktc3
Og-s* + Os*fOg ktc4
Og* + Ls*fOg
ktd3
Og* + Os*fOg ktd4
Og* + Ls*fOg
4248 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 Table 2. Moment Definitions for the Chain and Sequence Models Chain Model Propagating Chains
Dormant Chains
∞
Yia )
∑
∑
Zia )
ni[Pn]
n)0
∑
∞
Dia )
ni[TPn]
n)0
∞
Yib )
Dead Chains
∞
∑ n [M ] i
n
n)0
∞
∑ n [TQ ]
Zib )
ni[Qn]
i
n
n)0
n)0
Sequence Model Inactive Chains
Sjia )
Active Chains
∞
∑
∞
Sia )
gi[Lg]
g)1
Sjib )
g)1
∞
∑
∑
∞
Tia )
gi[Lg*]
g)1
∑
i
g
g)1
∞
Tia )
gi[Og]
∑ g [O *] ∞
Tib )
gi[TLg*]
g)1
∑ g [TO *] i
g
g)1
Table 3. Parameters for the Characterization of Molecular Architecture
NACL )
Y1a + Y1b + Z1a + Z1b + D1a
WACL )
Y0a + Y0b + Z0a + Z0b + D0a
Y2a + Y2b + Z2a + Z2b + D2a Y1a + Y1b + Z1a + Z1b + D1a
D ) WACL/NACL
conversion )
CA )
Y1a + Y1b + Z1a + Z1b + D1a Y1a + Y1b + Z1a + Z1b + D1a + A + B
Ainitial - Afinal Ainitial - Afinal + Binitial - Bfinal
NAA )
DAS )
S1a + T1a
NAB )
S0a + T0a
CB )
S1b + T1b S0b + T0b
Sj2a + S2a + T2a Sj1a + S1a + T1a / Sj1a + S1a + T1a Sj0a + S0a + T0a
Table 4. Model Parameters and Kinetic Rate Constantsa f ) 0.6 kd ) 10-5 s-1 ki,i ) 3 × 103 L/mol · s kr,ij ) 105 L/mol · s kraft,ij ) 105 L/mol · s
kp,ii ) 10 L/mol · s kp,ij ) depends on reactivity ratios (L/mol · s) ktc,i ) 107 L/mol · s ktd,i ) 107 L/mol · s 3
a Concentration of initiator is 0.3% with respect to the total amount of monomer; concentration of RAFT agent is 10 times the concentration of initiator.
is, on average, B monomer. With that geometric position as a starting point, the copolymer can be approximated from the tail to head with the information gleaned from the inactive sequences. The dispersity of A sequences is 1.41, which is
Binitial - Bfinal Ainitial - Afinal + Binitial - Bfinal
NIA )
DBS )
Sj1a Sj0a
NIB )
Sj1b Sj0b
Sj2b + S2b + T2b Sj1b + S1b + T1b / Sj1b + S1b + T1b Sj0b + S0b + T0b
slightly lower than that for B sequences, which is 1.54. This might be due to the higher concentration of B monomer in the copolymer. A flow diagram of the process is presented in Figure 2, with a cartoon “lifesaver” model of the rough characterization of the copolymer from the sequence determinations made from the sequence model. Series of Continuous Reactors. In controlled radical polymerization, series of reactors can be combined to produce copolymers with varied sequence structures. As has been mentioned, PFRs and CSTRs can be combined to produce polymer blocks of substantially different sequence distributions. In the following two simulations, the same series of three reactors with different reactor conditions produces copolymers
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Figure 1. Simulation one. (a) NACL, copolymer composition, conversion, and dispersity plotted versus time. (b) Monomer length of the ending sequence, monomer length of the inactive sequences, ratio of living to dead polymer chains, and ratio of end sequences plotted versus time.
Figure 2. Simulation one. Flow diagram of simulation with cartoon lifesaver model of the copolymer. (A monomer shown as gray; B monomer shown as white.)
Figure 3. Simulation two, first CSTR block. (a) NACL, copolymer composition, conversion, and dispersity plotted versus time. (b) Monomer length of the ending sequence, monomer length of the inactive sequences, ratio of living to dead polymer chains, and ratio of end sequences plotted versus time
with varying compositional characteristics. Simulation two consists of a CSTR-PFR-CSTR train to produce a block of constant copolymer composition (from the first CSTR) rich in one monomer at the head portion, and a copolymer of constant copolymer composition (from the second CSTR) rich in the other monomer at the tail portion. The PFR between the two CSTRs provides a gradient block connecting the two end blocks. Pure A monomer (2 M) is fed to the first CSTR with the same recipe of initiator and RAFT agent as in previous simulations. The results of the simulation of the first CSTR are shown in Figure 3. The polymer that is produced has an NACL of 21.6, with a conversion of 53.5%. This low conversion is coupled with a high polydispersity of 1.96. With a feed of pure A monomer, the copolymer is 100% composed of A monomer. The dispersity of sequence A is 1.93. Once the CSTR has come to steady state, its output is fed as an input to a PFR reactor (see Figure 4). In addition, an equimolar mixture (2 M) of A and B monomers is fed to the PFR reactor. The PFR has the same residence time as the CSTR (20000 s), and no initiator or RAFT agent is added, so that no new chains are initiated and the copolymer is only elongated.
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Figure 4. Simulation two, PFR block. (a) NACL, copolymer composition, conversion, and dispersity plotted versus time. (b) Monomer length of the ending sequence, monomer length of the inactive sequences, ratio of living to dead polymer chains, and ratio of end sequences plotted versus time.
The sequence model is reset to allow a fresh analysis of the compositional structure of the next polymerization segment. The copolymer at the end of the PFR has a degree of polymerization of 72.7, meaning that the (average) chain was extended by 51.1 monomer units. The overall conversion of the two reactors was raised to 67.8%, and the overall polydispersity was lowered to 1.22. The reduction in polydispersity in the PFR is expected, because a single PFR carrying out RAFT polymerization has a theoretical minimum polydispersity of 1.0, although the theoretical minimum will never be reached However, there was evidence of composition drift as the copolymer composition of B started at 57.5% and then drifted downward to 50.5%. This is highlighted in the drift exhibited in the inactive sequences of A and B. The inactive sequence of B began at 2.32 and steadily decreased to a value of 2.03, and in a mirror image, the inactive sequence of A began at 1.72 and steadily increased to a value of 1.99. Because the copolymer composition at the end of the segment was approximately 50%, and the final inactive sequence values were approximately equal; the head (CSTR) and tail (PFR) portions were approximate mirror images. Using the initial inactive sequence values of A and B (1.72 and 2.32) to geometrically determine the head portion and then reversing these values for the tail portion
Figure 5. Simulation two, second CSTR block. (a) NACL, copolymer composition, conversion, and dispersity plotted versus time. (b) Monomer length of the ending sequence, monomer length of the inactive sequences, ratio of living to dead polymer chains, and ratio of end sequences plotted versus time.
provides a rough geometric approximation. With these values for the head of the copolymer segment, two of every three sequences of B totals two monomers, and the third sequence totals three monomers. Conversely, two of every three sequences of A totals two monomers while the third sequence totals one monomer. The exact reverse analysis holds true for the tail section. Therefore, the head section of the copolymer has a copolymer composition of A monomer of approximately 44%, and the tail has the copolymer composition of A monomer of approximately 56%. The ratio of end sequences favored B at the beginning and end of the polymerization. Combined with the values of the ending sequence at these points, the first sequence of the head of the copolymer and the last sequence of the tail of the copolymer can be approximated. The dispersities of A and B monomers were 1.53 and 1.52, respectively, which reflect the equality of the copolymer composition. The output of the PFR is fed to a CSTR with a fresh feed of pure B monomer (2 M; see Figure 5). The final reactor in the train, a CSTR, extended the (average) chain by 30.8-mers to a final NACL of 103.5, with a conversion of 73.5%. The total
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Figure 6. Simulation two. Flow diagram of simulation with cartoon lifesaver model of the copolymer. (A monomer shown as gray; B monomer shown as white.)
Figure 8. Simulation three, CSTR block. (a) NACL, copolymer composition, conversion, and dispersity plotted versus time. b) Monomer length of the ending sequence, monomer length of the inactive sequences, ratio of living to dead polymer chains, and ratio of end sequences plotted versus time.
Figure 7. Simulation three, first PFR block. (a) NACL, copolymer composition, conversion, and dispersity plotted versus time. (b) Monomer length of the ending sequence, monomer length of the inactive sequences, ratio of living to dead polymer chains, and ratio of end sequences plotted versus time.
polydispersity rose slightly to the still-low value of 1.24. A high living-to-dead ratio of 23.2 was maintained. Inactive sequences of 1.40 and 3.53 for A and B, respectively, were obtained. Because there is no composition drift in a CSTR,
the average sequence structure can easily be approximated. Roughly, three of every five sequences of A total one monomer, and two of every five total two monomers. Using the same logic, roughly every other sequence of B is three monomers or four monomers. The same analysis was performed as in the previous scenario to produce a lifesaver model of the copolymer (Figure 6). This cartoon snapshot highlights the greater degree of composition drift with this wider difference in reactivity ratios. The flow diagram shows that, with a simple reactor train, a copolymer can be produced that has a variable sequence structure. A homogeneous portion of A monomer is polymerized at the head block. A middle (gradient) block is polymerized that slightly favors B monomer toward the beginning of the block and then almost equivalently favors A monomer toward the end of the block. Last, a tail block is formed that has a constant copolymer composition and sequence structure and is dominated by B polymer. The final scenario (simulation three) involves a train consisting of PFR-CSTR-PFR. The goal of this simulation is to produce a relatively constant compositional gradient along the copolymer. In the first reactor in the series, A and B monomers
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Figure 10. Simulation three. Flow diagram of simulation with cartoon lifesaver model of the copolymer. (A monomer shown as gray; B monomer shown as white.)
Figure 9. Simulation three, second PFR block. (a) NACL, copolymer composition, conversion, and dispersity plotted versus time. b) Monomer length of the ending sequence, monomer length of the inactive sequences, ratio of living to dead polymer chains, and ratio of end sequences plotted versus time.
are continuously fed to a PFR reactor in concentrations of 2 and 0.5 M, respectively. The same recipe is followed as earlier for the initiator and RAFT agent concentrations. The results are shown in Figure 7. The NACL is 25.5, with a conversion of 70.1%. The composition drift is evident, as the copolymer composition begins at 69.6% and ends at 74.9%. The polydispersity is 1.22, and the dispersities of A and B sequences are 1.25 and 1.73 respectively. The analysis of the sequence distribution is as the same as before. The end sequence of A begins at 2.18 and ends at 3.85. The end sequence of B begins at 1.30 and ends at 1.25. The inactive sequence of A begins at 2.18 and ends at 3.40. Therefore, the sequences of A are shorter at the beginning of the block than at the end. The inactive sequence of B begins at 1.30 and ends at 1.33. This means that one of every three sequences consists of two monomers and the other two consist of a single B monomer. The end sequence favors A at the beginning and end of the block, meaning that the first and last sequences of this polymer segment are A monomer. The output of this PFR reactor is fed to a CSTR along with a continuous fresh feed of A (2 M) and B (1.85 M). The
composition of the feed was iteratively selected to produce a polymer segment that was equivalent in composition. The results are shown in Figure 8. At the output of the CSTR, the NACL is 58.0, meaning that it was extended by 32.5-mers. The overall conversion was slightly lowered to 63.6%, and the polydispersity rose slightly to a value of 1.45. The copolymer composition of A and B sequences totalled 50.1% and 49.9%, respectively. The nearequivalent composition is underscored by the equality of the values of the dispersities of A and B sequences, which totaled 1.50 and 1.49, respectively. At steady state, the inactive sequences of A and B total 2.00 and 1.99, respectively. Therefore, every two sequences of A are followed by two sequences of B, and because the ratio of end sequences favored B at the end of the polymerization, the sequence structure of this polymerization segment can be very simply approximated. Once the CSTR reached steady state, its output was fed to a PFR reactor with a feed of pure B monomer (2 M). No additional A monomer was added, so that the composition of the tail segment of the copolymer would be approximately the reverse of the composition of the head segment of the copolymer. The results are shown in Figure 9. At the end of the second PFR, the NACL was 95.2, meaning that it was extended by 37.2-mers. The overall conversion was a high 80.2%, with a low polydispersity of 1.22. The copolymer composition of B begins at 78.1 and ends at 73.4 at the end of the polymerization, which is approximately the composition of A monomer at the head segment of the copolymer. Clearly, most of the unreacted monomer leaving the CSTR is A monomer, as the composition of A still reached 26.7% in the tail segment of the copolymer. Overall, a copolymer with a low polydispersity, high conversion, and high degree of polymerization with a relatively uniform compositional gradient was obtained by manipulating the reactor types and operating conditions. A summary of simulation three is shown in Figure 10. Conclusions This work has shown that the method of moments can be used to produce a mathematical model that provides insight into the sequence structure of copolymers. The molecular architecture of copolymers can be approximated with the model developed. CSTRs produce polymer segments with no composition drift, which is evident from various parameters, notably from the equivalence of active and inactive sequences from the sequence
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model. However, shorter chains are produced, with less conversion, and higher polydispersity. PFRs produce longer chains, with higher conversion and lower polydispersity, but there is composition drift. If one is concerned with controlled radical polymerizations, where the lifetime of a growing chain is approximately the residence time in the reactor train, the combination of these various configurations allows tight control of molecular architecture. The combination of these two reactors allows their strengths to complement each other, to iteratively form copolymers with lower polydispersity, constant copolymer composition, and a sequence structure more closely approaching the desired. Nomenclature A ) monomer of A B ) monomer of B f ) efficiency of initiator g ) specific number of monomers in sequence chain I ) initiator L ) internal sequence of A L* ) end sequence of A M ) dead chain n ) specific number of monomers in polymer chain O ) internal sequence of B O* ) end sequence of B P ) polymer chain with terminal unit A Q ) polymer chain with terminal unit B r ) total number of monomers in polymer chain R* ) radical from initiator or leaving agent s ) total number of monomers in sequence chain T ) RAFT agent TP ) polymer chain with terminal unit A bound to RAFT agent TQ ) polymer chain with terminal unit B bound to RAFT agent
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ReceiVed for reView September 22, 2008 ReVised manuscript receiVed January 22, 2009 Accepted January 23, 2009 IE801422W
