Ind. Eng. Chem. Res. 2009, 48, 10827–10839
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Modeling and Control of Sequence Length Distribution for Controlled Radical (RAFT) Copolymerization Yuesheng Ye† and F. Joseph Schork* Department of Chemical and Biomolecular Engineering, UniVersity of Maryland, College Park, Maryland 20742
Controlled/living radical polymerization (CRP) allows a high degree of control over polymeric molecular architecture with narrow molecular weight distributions and desired polymer properties. Although the control of copolymer composition for RAFT copolymerization systems in a semibatch reactor has been studied, the control of sequence length receives little attention. In this work, using moment equations, a chain model and a sequence model were developed to describe both chain and sequence properties such as chain and sequence lengths and their distributions, for RAFT copolymerization in semibatch operations. It is seen that a copolymer with constant composition or linear gradient composition can be obtained through an optimized feeding policy. With the sequence model, it was possible to show the sequence length and distributions while controlling constant composition or linear gradient composition. More importantly, it was demonstrated that sequence length distribution can be designed and controlled using the sequence model. The sequence model may be used as a theoretical tool to understand the importance of sequence length and distributions in CRP. 1. Introduction Controlled/living radical polymerization (CRP) has received a great deal of attention over the past years. The advent of CRP techniques including atom-transfer radical polymerization (ATRP),1 reversible addition-fragmentation chain transfer polymerization (RAFT),2 and nitroxide-mediated radical polymerization (NMRP)3 has greatly renewed interest in radical polymerization because CRP allows a high degree of control over polymeric molecular architecture, with narrow molecular weight distributions and desired polymer properties under relatively mild conditions. In general, introducing a comonomer in CRP offers a possibility to prepare tailor-made copolymers with specific desired characteristics. Since the lifetime of a living radical chain is on the order of hours rather than seconds as in conventional free radical polymerization, it is possible to adjust monomer concentration to manipulate copolymer composition distribution and comonomer sequence distribution within a single chain. It is well-known that the copolymer composition is of great importance to the resulting copolymer properties. Thus, it is important to control copolymer composition in CRP to obtain desired properties. In a batch process, composition drift is very common due to differences in the reactivity ratios of the monomers. To control copolymer composition, a semibatch operation is commonly used. It was experimentally and theoretically demonstrated that, through optimized feeding in a semibatch reactor, copolymers with uniform composition or linear gradient composition can be successfully designed and controlled.4-8 In addition to copolymer composition distribution, monomer sequence length and distribution impact copolymer properties such as copolymer interfacial activity,9 hydrogen bonding strength,10 and crystallinity.11 Therefore, design and control of sequence length and distribution would facilitate the further optimization of copolymer properties. However, to date, there has been no modeling study published on how to design and control sequence length and distributions for a CRP system. * To whom correspondence should be addressed. Tel.: (301) 4051074. Fax: (301) 405-0523. E-mail:
[email protected]. † Current address: Chemical and Biological Engineering Department, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104.
For the modeling of sequence length and distribution for free radical copolymerization systems, most sequence models developed in the past were based on probabilistic arguments. In 1972, Ray12 developed a framework to calculate average sequence length using probabilistic functions. Arriola13 developed a model to describe sequence length for a radical copolymerization system using moment equations. However, transfer and termination reactions were not considered in the model simulation, suggesting that the simulation results may not be valid, especially at a late stage. His model was not applied to living polymerization. Tabash et al.14 borrowed the concept of a binary numeral system to describe two different monomers and introduced a new method of “digital encoding” to identify a unique sequence using a binary code for linear free radical copolymerization. However, as polymer chains grow longer, the complexity of modeling a system in this way dramatically increases, which causes significant computational cost. As a result, only a very limited chain length can be simulated by this method. Dias et al.15 developed a general kinetic method to predict sequence length distributions for nonlinear multicomponent copolymerization systems. In this model, long-chain branching was taken into account. Al-Saleh and Simon16 developed a Monte Carlo model to describe the process of olefin copolymerization. The detailed sequence distribution was obtained. However, computational cost is very high, which may result in computational difficulties if this model is employed to further study sequence control. In this work, however, a deterministic sequence model based on population balance equations was developed to calculate sequence length and sequence length distribution. For the modeling of radical copolymerization systems, a classic population balance model consists of a set of moment equations that are used to describe the reaction rate of “living” and “dead” polymeric chains. From a general point of view, moment equations are not suitable for the description of sequence length and distributions for a radical copolymerization system.17 However, a deterministic sequence model with low computational costs can be developed based on moment equations if one tracks sequence balances independent of molecular (chain)
10.1021/ie901032y CCC: $40.75 2009 American Chemical Society Published on Web 10/29/2009
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Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009
Table 1. Reaction Scheme of the Chain Model initiation
RAFT activation kr11
kd,f
I f 2R*
T + Pn T R* + TPn kr12
Table 2. Reaction Scheme of the Sequence Model RAFT transfer kRAFT11
Pn + TQr 798 TPn + Qr
Active Sequences initiation
RAFT activation
kd,f
kr11
L*g + T T TL*g + R*
I f 2R* ki1
R* + A f P1
kr21
T + Qn T R* + TQn kr22
ki2
kr12
kRAFT12
L*g + TO*s 798 kRAFT12
kRAFT21
TL*g + O*s
Qn + TQr 798 TQn + Qr kRAFT22
ki1
kRAFT31
Pn + TPr 798 TPn + Pr
R* + B f Q1
RAFT transfer
kRAFT12
kRAFT21
kr21
R* + A f L*1 O*g + T T TO*g + R* O*g + TO*s 798 kr22
kRAFT32
kRAFT22
TO*g + O*s propagation
termination
termination kRAFT31
ki2
kp1
Pn + A f Pn+1
kp2
Pn + B f Qn+1
kp3
Qn + A f Pn+1
ktc1
Pn + Qr f Mn+r
ktc1
Pn + Pr f Mn+r
ktc3
Qn + Qr f Mn+r
ktd1
Pn + Qr f Mn + Mr
L*g + TL*s 798
R* + A f O*1
kRAFT32
TL*g + L*s
ktd2
Pn + Pr f Mn + Mr
ktd3
Qn + Qr f Mn + Mr
kp4
Qn + B f Qn+1
propagation kp1
L*g + A f L*g+1
kp2
L*g + B f O*1
kp3
balances.13,18,19 Using this sequence model, the current work tracked the sequence properties while controlling copolymer composition (uniform composition or linear gradient composition), and also investigated direct sequence distribution control in a semibatch reactor. The model has been developed for RAFT CRP, since living polymerization allows the most opportunity for sequence distribution control. However, the model can be used directly to simulate free radical copolymerization, and can be easily extended to other CRP systems (ATRP or NMRP).
O*g + A f L*1
In this work, a chain model and a sequence model were developed to describe the chain properties and sequence properties, respectively. The chain model can be used to calculate the “classic” properties of copolymers such as numberand weight-average chain lengths, composition, and conversion. However, to account for the number- and weight-average sequence lengths and polydispersity of sequence length, a chain model is not adequate and a sequence model may be used.18 Table 1 shows a typical reaction scheme for RAFT copolymerization from which the chain model was derived.19 Note that in the RAFT transfer reaction, a RAFT species may react with an active radical to form an intermediate radical species, and then this intermediate undergoes fragmentation resulting in transfer of the RAFT agent from one polymer chain to the other.18 However, for simplicity this model assumed that the intermediate radicals undergo a fast RAFT fragmentation and the existence of intermediate radicals was not taken into account. For the purpose of this study, this treatment should be adequate. From the reaction scheme of a chain model, it is seen that a traditional chain model considers polymerization only from the chain point of view, i.e., with no consideration of internal sequence. A reaction scheme of a sequence model, as shown in Table 2,19 however, considered internal sequences during the propagation and termination reactions. Thus, the sequence
ktc1
L*g + L*s f Lg+s
ktc2
L*g + O*s f Lg + Os
ktc3
O*g + O*s f Og+s
termination ktd1
L*g + L*s f Lg + Ls
ktd2
L*g + O*s f Lg + Os
ktd3
O*g + O*s f Og + Os
kp4
O*g + B f O*g+1
Inactive Sequence propagation kp2
2. Model Development
termination
L*g + B f Lg
termination ktc1
L*g-s + L*s f Lg
ktd1
propagation kp3
O*g + A f Og
termination ktc2
L*g + O*s f Lg
ktd2
L*g + L*s f Lg + Ls
L*g + O*s f Lg
termination
termination
ktc3
O*g + O*s f Og
ktd3
O*g + O*s f Og + Os
ktc4
O*g + L*s f Og
ktd4
O*g + L*s f Og
properties can be determined by keeping track of the activeness or inactiveness of each sequence in the process of free radical copolymerization. However, a sequence model loses track of any information on a polymer chain. Therefore, the calculation of chain length and copolymer chain polydispersity has to rely upon a chain model. As a result, to fully describe the chain properties and sequence properties for a radical copolymerization system, both chain and sequence models must be employed. With the moments defined moment equations for the chain model and the sequence model can be derived.19 It should be noted that either a terminal model or a penultimate model may
Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009
be used to describe the effect of the monomer adjacent to a propagating radical on its reactivity of propagation reaction. For simplicity, a terminal model was adopted for this study. Note that the penultimate effect on propagations may be significant in some copolymerization systems. Use of a penultimate model would add significant complexity to the model, but is by no means impractical. The purpose of this work was to investigate sequence length and distribution for the cases in which copolymer properties such as copolymer composition (constant composition or linear gradient composition) were controlled, and to design copolymers with constant sequence length through controlled radical semibatch polymerization. The derivation of a semibatch model is well documented in the literature.4,5,7 The semibatch model developed in this work is briefly described as follows. In a semibatch reactor, the mass balance for species i takes the following form
Y aj )
(1)
where V, Ci,f, Vf, and Ri are the volume of reaction system, the feeding concentration of species i, the volumetric feeding flow rate, and the reaction rate of species i, respectively. From eq 1, the concentration of species i in the reactor, Ci, can be expressed as VfCi,f Ci dV dCi ) + Ri dt V V dt
∑ i)1
∑
DPn )
+ +
Y b1 Y b0
1 1 VRM,imwi F F Mi P i)1
+
Z a1 Z a0
+
+ +
Z b1 Z b0
∑ n [Q ] j
n
n)0 ∞
Z aj )
∑ n [TP ] j
n
n)0 ∞
Z bj )
∑ n [TQ ] j
n
n)0
and ∞
Dj )
∑ n [M ] j
n
(j ) 0, 1, 2). From the sequence model, the number-average sequence length and weight-average sequence length of monomer i and polydispersity of monomer i can be calculated by SLn,A )
SLn,B )
)
SLw,A ) (3)
where VS,f, Ff, xi, RM,i, FMi, and FP are the volumetric feeding flow rate of the solvent, total molar feeding flow rate of monomers (no solvent), mole fraction of monomer i, reaction rate of monomer i, density of monomer i, and density of the polymer. Thus, by inserting eq 3 into eq 2 the kinetic reaction equations for the chain model and the sequence model19 can be modified accordingly to obtain the mass balance equations for the semibatch reactor. With the initial conditions given by the system, a set of ordinary differential equations that consist of the chain model and the sequence model can be solved numerically using the “ode15s” routine in MATLAB and chain properties and sequence properties can be determined. From the chain model, the number-average chain length, weight-average chain length, and polydispersity of copolymer can be calculated by Y a1 Y a0
n
∞
Y bj )
(2)
(
2
Ff ximwi FMi
j
n)0
The calculation of reaction volume is given by 2
∑ n [P ]
n)0
d(VCi) ) VfCi,f + VRi dt
dV ) VS,f + dt
+ D1 + D0
SLw,B )
PDIA ) PDIB )
Sa1 + T a1 + Sja1 Sa0 + Ta0 + Sja0 Sb1 + T b1 + Sjb1 Sb0 + Tb0 + Sjb0 Sa2 + T a2 + Sja2 Sa1 + Ta1 + Sja1 Sb2 + T b2 + Sjb2 Sb1 + Tb1 + Sjb1 SLw,A
SLw,B
∞
∑ g [L* ] j
g
g)1 ∞
Taj )
∑ g [TL* ] j
g
g)1
(4)
Sjaj )
Y a2 + Y b2 + Z a2 + Z b2 + D 2 Y a1
+
Y b1
+
PDI ) where
Z a1
+
DPw DPn
Z b1
∞
∑ g [L ] j
g
+ D1
∞
(5)
Sbj )
∑ g [O* ] j
g
g)1 ∞
Tbj )
(6)
∑ g [TO* ] j
g
g)1
and
(8)
(9)
(10)
(12)
SLn,B
where Saj )
(7)
(11)
SLn,A
g)1
DPw )
10829
∞
10830
Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009 ∞
∑ g [O ]
Sjbj )
Fi )
j
g
g)1
Rmi
)
2
DPn
(19)
DPn,target
∑R
mi
(j ) 0, 1, 2). For the monomer conversion and copolymer composition, either the chain model or the sequence model can be used. For example, in the chain model the conversion of monomer i can be expressed as xi )
∫ V dt - C V + C ∫ V dt t
Ci,0V0 + Ci,f Ci,0V0
f
0
i
(13)
t
i,f
f
0
where Ci,0 and V0 are the initially charged concentration of monomer i and the initial volume in the reactor. Here we assumed that the feed concentration of monomer i is constant during the reaction. As a comparison, the conversion of monomer i can also be calculated by the sequence model as xi )
Si1 + Ti1 + Sji1
(14)
Si1 + Ti1 + Sji1 + Ci
It is seen that, in the sequence model, the conversion can be directly calculated from the moments by the ratio of monomer i incorporated in copolymer chains to the total charged amount of monomer i, and that the integration of total feed of monomer i is not needed. Similarly, the cumulative copolymer composij i, can be calculated by the chain model as tion, F Ci,0V0 + Ci,f
ji ) F
2
V0
∑C
i,0
+
t
f
0
i
2
f
(15)
2
∫ V dt ∑ C t
0
i)1
∫ V dt - C V i,f
i)1
-V
∑C
i
i)1
or by the sequence model as ji ) F
Si1 + Ti1 + Sji1 2
∑
(Si1
+
Ti1
+
(16)
Sji1)
i)1
where Fi is the instantaneous copolymer composition and DPn,target is the target number-average chain length. Since DPn,target is a given or preselected constant value, the instantaneous copolymer composition of component i continuously and linearly increases with respect to number-average chain length. As mentioned before, the lifetime of a living radical chain is on the order of hours, which enables controlling the copolymer composition of a single chain from mostly monomer A to mostly monomer B along the chain (gradient composition). Based on the constraint equation (eq 19), the control of linear gradient composition drives the system toward the target numberaverage chain length, i.e., Fi ) 1, which may cause difficulty in the optimization of feed rate at a later stage. In this work, the optimization was stopped before the value of the objective function exceeded 1 × 10-6. To control chain properties or sequence properties through semibatch operation, the calculation scheme is briefly given as follows: (1) At a given time t, for a small time step ∆t, an optimization method was used to find an optimal feed flow rate for the semibatch model that consists of the chain model and the sequence model. (2) This feed flow rate was put back into the semibatch model, and the chain properties and sequence properties were calculated. (3) A check was done as to whether the constraint equation was satisfied again. (4) If so, the reaction was moved to reaction time t + ∆t, and steps 1-3 were repeated until the end of the reaction was reached. In the control of copolymer properties through a semibatch operation, the most important step is to find a proper feed flow rate so that the constraint equation is satisfied over the reaction time, i.e., step 1. In this work, the method of nonlinear model predictive control (NMPC) was employed to optimize the feed flow rate. In the semibatch model developed above, the only manipulated variable is the feed flow rate. A general NMPC objective function takes the following form:21
i)1
P
Thus, it is seen that cumulative copolymer composition can be calculated directly from the moments of the sequence model as well. To control the copolymer properties through a semibatch operation, the feed rates of the monomers should be optimized based on a constraint equation that defines the target properties of the copolymer. Although the design and control of copolymer composition and linear gradient composition through a semibatch operation has been studied in previous work,4,5,7 the sequence model developed in this work is able to investigate further the sequence properties while controlling the chain properties. Moreover, the sequence model allows the investigation of the design and control of copolymer sequence distribution properties such as constant sequence length. For the control of constant composition or constant sequence length, the constraint equations are given as
min J
u(k|k),u(k+1|k),...,u(k+M-1|k)
)
∑ w [y(k + j|k) - y (k)]
2
y
s
P-1
P-1
∑ w [u(k + j|k) - u (k)]
2
u
+
j)0
s
+
j)0
∑w
∆u[u(k
+ j|k) -
j)0
u(k + j - 1|k)]2 (20)
Fi ) constant
(17)
where us, ys, u(k + j|k), and y(k + j|k) represent the set-point trajectories for the input and output variables, input u(k + j), and output y(k + j) calculated from the information at time t, respectively. wy, wu, and w∆u are the weights, and each variable defines the contribution of the variable to the objective function. In this study, we assumed that the values of wu and w∆u are 0. The set point of the output was defined by a constraint equation. The control horizon M was set to 2 and the prediction horizon P was set to 4 in this study. An additional constraint for the manipulated variable of feed flow rate was defined as follows.
SLn,i ) constant
(18)
u1 g 0
where Fi is the instantaneous copolymer composition. For the control of linear gradient composition, we defined the constraint equation for monomer i as follows.20
(21)
At each control interval, the optimization problem generated by the NMPC was solved by the “fmincon” routine in MATLAB.
Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009 22
Table 3. Reaction Conditions for the MMA RAFT Polymerization reaction condition
value
unit
reaction temperature monomer concentration solvent concentration initiator concentration RAFT agent concentration
60 6.7418 2.6085 0.005862 0.004716
°C mol · L-1 mol · L-1 mol · L-1 mol · L-1
3. Results and Discussion Model Validation. To validate the chain model and the sequence model developed above for the semibatch reactor, the simulation results were first tested against experimental data and compared with previous simulation results in the literature.22 The experimental data were taken from Example 26 in the patent from Chiefari et al.:23 the RAFT polymerization of methyl
methacrylate (MMA) at 60 °C in the presence of 2-cyanoprop2-yl 1-pyrrolecarbodithioate with AIBN as an initiator. The reaction conditions and model parameters are listed in Tables 3 and 4, respectively. In this study, the Ross-Laurence gel effect correlation, based on free volume assumptions, was employed to describe the RAFT copolymerization (Table 4). The reaction was assumed isothermal and all the temperature-correlated parameters were calculated at the reaction temperature (T ) 60 °C). These experimental data were also used by Zhang et al. to simulate the living free radical polymerization with RAFT chemistry. This experimental example is MMA homopolymerization in a batch reactor, and the experimental data and simulation results from the work of Zhang et al. were taken to compare with simulation results of this work. In order to simulate homopolymerization with the current model, monomer
Table 4. Model Parameters for the MMA RAFT Polymerization22 model parameters
unit
f ) 0.65
kd ) 1.0533 × 1015 exp(-30704/RT)
s-1
kp ) 4.92 × 105 exp(-4353/RT)
L · mol-1 · s-1
ki1 ) kp
L · mol-1 · s-1
ktd /ktc ) 2.483 × 103 exp(-4073/RT)
L · mol-1 · s-1
ktd + ktc ) 9.8 × 107 exp(-701/RT)
gt )
{
gp )
{
0.10575 exp(17.15VF - 0.01715T) if VF > [(0.1856-2.965) × 10-4]T 2.3 × 10-6 exp(75VF)
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if VF < [(0.1856-2.965) × 10-4]T
if VF > 0.05 1.0 -5 7.1 × 10 exp(171.53VF) if VF < 0.05
VFS ) 0.025 + 0.001(T - TgS) VFM ) 0.025 + 0.001(T - TgM) VFP ) 0.025 + 4.8 × 10-4(T - TgP) FM ) 0.9654 - 0.00109T - 9.7 × 10-7T2
g · cm-3
FP ) [(1.18-1.0) × 10-3]T
g · cm-3
TgM ) -106
K
TgS ) -102
K
TgP ) 114
K
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Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009 Table 5. Kinetic Parameters for the RAFT Copolymerization kinetic parameters ki1 ) ki2 ) kp1 ) kp4
unit L · mol-1 · s-1
r1 ) kp1 /kp2 r2 ) kp4 /kp3 krij ) kr kRAFTij ) kr
Figure 1. Model validation by comparison with experimental and simulation results: (a) conversion of MMA with reaction time; (b) number-average chain length with respect to the conversion of MMA; and (c) polydispersity with respect to the conversion of MMA. Solid dot points are experimental data, dotted lines stand for the results from Zhang’s work, and solid lines are this work.
B related terms and the feed rate were set to zero, and the semibatch model became a batch homopolymerization model. Note that one major difference between this work and the work of Zhang et al. is that the current work did not consider the intermediate radical species while Zhang et al. did. Thus, the reaction kinetic parameters of RAFT chemistry should be different from theirs. In this work, it was assumed that the RAFT initiation has the same rate coefficient as the RAFT transfer rate, and that the forward and backward RAFT reaction rate constants are same, i.e., krij ) kRAFTij ) kr (i, j ) 1, 2). Thus, the number of fitting parameters was reduced to one. Figure 1 shows the comparison of simulation results with experimental data and Zhang’s work on conversion, numberaverage chain length, and polydispersity. It was found that kr ) 7 × 103 L · mol-1 · s-1 gives a good fit with experimental data, i.e., monomer conversion (Figure 1a). Figure 1b,c shows that the simulation captures the characteristic of a living polymerization, i.e., linear increase of the number-average chain length with conversion, and narrow polydispersity. It is also seen that the simulation results in this work are in good agreement with experimental data. Although the comparison with Zhang’s results shows some difference, both simulation results still agree well with each other considering the difference in RAFT chemistry between the two models. In particular, note that the sequence properties should be the same as chain properties
(i ) 1, 2; j ) 1, 2)
(i ) 1, 2; j ) 1, 2)
ktcn ) ktc
(n ) 1, 2, 3)
ktdn ) ktd
(n ) 1, 2, 3)
L · mol-1 · s-1 L · mol-1 · s-1 L · mol-1 · s-1 L · mol-1 · s-1
if comonomer B is not added for copolymerization. In the simulation results shown in Figure 1b,c, it is seen that the number-average sequence length (eq 7) is the same as the number-average chain length (eq 4) and that the polydispersity of chains (eq 6) is the same as the polydispersity of sequences (eq 11), meaning that the sequence model can be recovered to the chain model as the comonomer-related terms were set to zero. The good agreement of simulation results with experimental data indicates that this semibatch model can be utilized for further simulation study. In the following study, it was assumed that monomer A is MMA and the physical and reaction kinetic parameters are listed in Table 4. To illustrate the sequence properties in controlled living copolymerization through semibatch operation, we also assumed that monomer B has the same physical and reaction kinetic parameters as monomer A except for a larger propagation rate constant (Table 5). In other words, only the reactivity ratio was treated as a variable and other parameters were assumed to be the same as those of MMA. To show the effect of the reactivity ratio on simulation results, two sets of reactivity ratios were selected. One set is r1 ) 0.5 (r1 ) kp1/kp2), r2 ) 2 (r2 ) kp4/kp3), which represents the case that two monomers have similar reactivities. The other set is r1 ) 0.1, r2 ) 10, which represents the case that two monomers have very different reactivities causing substantial “composition drift” if two monomers are polymerized in a batch reactor. Sequence Properties for Constant Composition. For a copolymer with controlled chain properties, a simple case is to prepare a copolymer with constant (or uniform) composition. In a batch process, the composition drift is very common due to the different reactivity ratios of the monomers. As an alternative operation, a semibatch reaction offers a way to obtain constant copolymer composition by adjusting the composition drift via optimal monomer addition. In order to control the copolymer composition, a simple policy is to feed with the monomer that has the faster reaction rate using a semibatch operation mode. In the semibatch operation, we assumed that the semibatch reactor is charged with the desired amount of monomer A and monomer B such that the initial composition matched the target composition. Then monomer B (higher propagation rate) is continuously added in a semibatch reactor mode at an optimal rate to keep the composition as a constant value. To design a copolymer with a certain initial composition,
Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009
10833
a
Table 6. Operating Conditions for the Semibatch Reactor
initial concentration (mol/L) objective property constant composition (FA ) 0.5) linear gradient composition (FB ) DPn/1000) constant number-average sequence length (SLn,A ) 3) constant number-average sequence length (SLn,B ) 3) a
reactivity ratio r1 r1 r1 r1 r1 r1 r1 r1
) ) ) ) ) ) ) )
0.5, 0.1, 0.5, 0.1, 0.5, 0.1, 0.5, 0.1,
r2 r2 r2 r2 r2 r2 r2 r2
) ) ) ) ) ) ) )
2 10 2 10 2 10 2 10
feeding concentration (mol/L)
A
B
S
A
B
S
4.4945 6.1289 6.2818 6.6818 5.3934 6.4208 3.3709 5.5160
2.2473 0.6129 0.4600 0.0600 1.3484 0.3210 3.3709 1.2258
2.6085 2.6085 2.6085 2.6085 2.6085 2.6085 2.6085 2.6085
0 0 0 0 0 0 0 0
6.7418 6.7418 6.7418 6.7418 6.7418 6.7418 6.7418 6.7418
2.6085 2.6085 2.6085 2.6085 2.6085 2.6085 2.6085 2.6085
The initial reaction volume is assumed as 20 L.
Figure 2. Comparison of conversion and copolymer composition simulation results calculated by the chain model and the sequence model.
the Mayo-Lewis equation24 can be used. As an example, we assumed that the target composition of A component in the copolymer is 0.5 (eq 17). Since the reactivity ratios are known values, to match the target copolymer composition, the initial concentration ratio of monomer A to monomer B can be calculated from the Mayo-Lewis equation. The assumption is made that the total concentration of monomer A and monomer B is the same as the MMA concentration in the batch operation. Thus, the concentrations of monomer A and monomer B can be determined. It is assumed that only monomer B is added into the semibatch reactor. The reaction conditions for the semibatch operation are summarized in Table 6. First, we compared the simulation results for conversion and copolymer composition from the chain model and the sequence model while controlling to a constant copolymer composition. Figure 2 shows that, for two different sets of reactivity ratios, the conversions calculated by eq 13 (chain model) and by eq 14 (sequence model), and the copolymer compositions of component B calculated by eq 15 (chain model) and by eq 16 (sequence model), are in excellent agreement. This indicates that system variables of the semibatch reactor such as conversion and copolymer composition can be calculated either by a chain model or by a sequence model. However, in a sequence model, moment equations can be utilized and a sequence model does not need to use monomer feed rate and monomer feed concentrations while a chain model does. Figure 3 shows simulation results for the case of controlling constant copolymer composition at two different sets of reactiv-
ity ratios. It is seen that, for both sets of reactivity ratios, component B in the copolymer composition is well controlled at F2 ) 0.5 as shown in Figure 3b,b′. Figure 3a,a′ shows that as a reaction proceeds the feeding rate of monomer B decreases with reaction time and then increases probably due to the acceleration of reaction rate caused by the gel effect. As a comparison, it is seen that the feed rate for the second set of reactivity ratios (r1 ) 0.1, r2 ) 10) is much smaller than that for the first set of reactivity ratios (r1 ) 0.5, r2 ) 2), indicating that, for a comonomer B that has a higher polymerization rate, a slower rate of addition is needed to maintain constant composition. Figure 3c,c′ shows the number-average chain lengths versus reaction time. With the sequence model developed above, we are able to show the number-average sequence lengths while controlling a constant composition at F2 ) 0.5. Figure 3d,d′ shows that number-average sequence length A and number-average sequence length B increase gradually and then level off at 2. It is seen that both number-average sequence lengths of monomer A and monomer B are about the same probably because the feed policy controlled at the same compositions for monomer A and monomer B. Polydispersity of polymer chains decreases from 2, which is typical for a RAFT system. As the reaction goes further, the polydispersity of chains reaches a minimum. For the case with a wider spread of reactivity ratios, after it passes the minimum value, it starts to increase again as shown in Figure 3e′, which is probably due to the gel effect. The polydispersities of sequence A and sequence B, different from the polydispersity of polymer chains,
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Figure 3. Simulation results for the case of constant composition (FA ) 0.5): (a, a′) feed rate; (b, b′) cumulative copolymer composition; (c, c′) numberaverage chain length; (d, d′) number-average sequence length; (e, e′) polydispersity of chains and polydispersities of sequence A and sequence B.
are close to 1.5 probably due to equal constant composition throughout the course of copolymerization. It is interesting to note that the polydispersities of sequence A and sequence B are about the same and reactivity ratios have a negligible effect on them. Sequence Properties for Linear Gradient Composition. In addition to constant composition, preparation of a linear gradient copolymer with linear gradient composition varying
continuously along a polymer chain is also of great interest, not only from a basic scientific point of view, e.g., regarding the impacts of the composition gradient on repulsive interactions,25 interfacial activity,26 and phase segregation,27 but also because of their potential practical applications. For example, gradient copolymers have been suggested as alternatives to block copolymers for use as compatibilizers for immiscible polymer blends (interfacial activity) since it was found that the critical
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Figure 4. Simulation results for the case of linear gradient composition (FB ) DPn/1000): (a, a′) feed rate; (b, b′) instantaneous and cumulative copolymer compositions B vs scaled number-average chain length; (c, c′) number-average chain length; (d, d′) number-average sequence length; (e, e′) polydispersity of chains and polydispersities of sequence A and sequence B.
micelle concentration of gradient copolymers in homopolymers or blends is orders of magnitude higher than that of block copolymers made from the same comonomers.26 To prepare a linear gradient copolymer via semibatch operation, monomer B (with the higher propagation rate) has to be added into the reactor at a rate that will satisfy the constraint equation (eq 19). From the batch operation, it is seen that the number-average chain length builds up at the
very beginning of polymerization. Thus, to satisfy the constraint equation at t ) 0, we assumed that a certain amount of monomer B is initially charged with monomer A before the reaction and the target number-average chain length used in eq 19 is set at 1000. The concentrations of monomer A and monomer B charged in the reactor before semibatch operation and concentrations of monomer A and monomer B in the feed are listed in Table 6.
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Figure 5. Simulation results for the case of constant number-average sequence length of sequence A (SLn,A ) 3): (a, a′) flow rate; (b, b′) number-average sequence length; (c, c′) number-average chain length; (d, d′) cumulative copolymer composition; (e, e′) polydispersity of chains and polydispersities of sequence A and sequence B.
Figure 4 shows simulation results for the case of linear gradient composition. It is seen that the copolymer composition of component B increases linearly with the scaled number-average chain length as shown in Figure 4b,b′, indicating that linear gradient copolymers can be prepared by using a semibatch operation. Figure 4a,a′ shows that the feed flow rate increases gradually in the beginning and increases more and more significantly as the polymerization
proceeds. This is because, in the preparation of a linear gradient copolymer, the ratio of number-average chain length to the target number-average chain length becomes larger and larger with reaction time, which requires more and more monomer B to be added into the system. The shape of the feed flow rate curve may be dependent on the value of the target number-average chain length. Figure 4c,c′ shows the increase in number-average chain length with reaction
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Figure 6. Simulation results for the case of constant number-average sequence length of sequence B (SLn,B ) 3): (a, a′) flow rate; (b, b′) number-average sequence length; (c, c′) number-average chain length; (d, d′) cumulative copolymer composition; (e, e′) polydispersity of chains and polydispersities of sequence A and sequence B.
time. Figure 4d,d′ shows the profiles of number-average sequence length for monomer A and monomer B. Because there is only a small amount of monomer B charged into the reactor at the beginning of reaction, the number-average sequence length of monomer B is close to 1 and the numberaverage sequence length of monomer A has a high value. As more and more monomer B is added into the reactor, the number-average sequence length of monomer B goes up gradually and the number-average sequence length of mono-
mer A decreases dramatically. For the chain length distribution and sequence length distribution, Figure 4e,e′ shows that polydispersity of chains decreases with reaction time, but the polydispersities of both sequence length distributions increase with reaction time, meaning that the distribution of sequence length becomes broader, while the distribution of chain length becomes narrower. Constant Sequence Length. In the above modeling study, the sequence properties have been simulated using the
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sequence model while controlling copolymer composition via a semibatch operation. It remains to be seen if it is possible to control copolymer sequence distribution to produce a constant sequence length. As an example, we assumed that monomer B is added into the reactor to control the numberaverage sequence length. If monomer B is absent or the amount of monomer B is small in the reactor, the numberaverage sequence length of sequence A would be a large value (as shown in Figure 4), which may cause the difficulty in control of sequence length in the beginning. Thus, a certain amount of monomer B charged together with monomer A in the reactor gives the initial value of sequence length A that is close to the target. In this work, we assumed that the target value of number-average sequence length is 3 (eq 18). The operating conditions are listed in Table 6. Figure 5 shows the simulation results for the case of constant number-average sequence length of sequence A (SLn,A ) 3). Figure 5b,b′ shows that the number-average sequence length of sequence A is controllable with the optimal feed rates shown in Figure 5a,a′. It is seen that the shapes of the feed profiles are similar to the case for the control of constant copolymer composition. Interestingly, we can see from Figure 5d,d′ that the copolymer compositions are also close to constant values. This suggests that controlling to a constant sequence A length may result in constant composition, even though the sequence length for B is not controlled. It is also interesting that the polydispersities of sequence A and sequence B are near constant values over the reaction time shown in Figure 5e,e′. As a comparison, Figure 6 shows the simulation results for the case of constant number-average sequence length of sequence B (SLn,B ) 3). Again, Figure 6b,b′ shows that the number-average sequence length is controllable with the optimal feed rates shown in Figure 6a,a′. It is interesting to note that the profiles of cumulative copolymer composition and polydispersities of sequence A and sequence B are very similar to those shown in Figure 5, but the labels are switched. Comparing Figure 5d,e with 5d′,e′ and Figure 6d,e with 6d′,e′, we can see that, for the different sets of reactivity ratios, the copolymer composition and the polydispersities of sequence A and sequence B are the same when the number-average sequence length is controlled as a constant value. Again, this may be not true in general, but this holds for the sets of parameters just investigated. 4. Conclusion In this work, a chain model and a sequence model were developed to calculate the sequence length and distributions for controlled radical copolymerization via RAFT chemistry in a semibatch reactor. Two sets of reactivity ratios that cover a wide range of monomer reactivities were investigated. The sequence length and distributions produced while controlling constant copolymer composition and linear gradient copolymer composition were calculated by using the sequence model. It is seen that number-average sequence lengths are also almost constant values during the control of constant composition, while one sequence length decreases and the other increases gradually during the control of linear gradient composition. The polydispersity of polymer chains decreases for both cases, but the polydispersities of sequences are very different from those of chains. It is also possible to control the number-average sequence length at a certain value using an optimized feeding policy. Interestingly, the copolymer compositions are also almost constant values during the control of sequence length. Simulation results suggest that the constant copolymer composition may
result in constant sequence length, or vice versa. It may be possible to control both sequence lengths by manipulating feeds of both monomers. In this case, the copolymer composition would be fixed as well. Alternately, one could control copolymer composition and one sequence length, again, by manipulating feed of both monomers. The ability to track sequence length with the present model makes these types of open-loop optimizations possible. With a sequence model, it is seen that it is possible to control an additional level of molecular architecture: monomer sequence, which provides additional insights into understanding the effect of sequence and distribution on the copolymer properties. More importantly, this model allows one to predict sequence distribution and relate this to polymer properties. Nomenclature A ) monomer of A B ) monomer of B Ci,0 ) initial charged concentration of monomer i, mol · L-1 Ci ) concentration of species i in the reactor, mol · L-1 Ci,f ) feeding concentration of species i, mol · L-1 Dji ) jth moment of dead chain with terminal unit i (i ) a, b; j ) 0, 1, 2) DPn ) number-average chain length DPw ) weight-average chain length f ) efficiency of initiator Ff ) molar feeding flow rate of monomers, mol · s-1 Fi ) instantaneous copolymer composition j i ) cumulative copolymer composition F gp ) propagation gel effect factor gt ) termination gel effect factor I ) initiator kd ) initiator decomposition rate constant, s-1 ki ) kinetic rate constant for the first propagation step, L · mol-1 · s-1 kr ) RAFT activation reaction rate constant, L · mol-1 · s-1 kRAFT ) RAFT transfer reaction rate constant, L · mol-1 · s-1 kp ) propagation rate constant, L · mol-1 · s-1 ktc ) termination by combination rate constant, L · mol-1 · s-1 ktd ) termination by disproportionation rate constant, L · mol-1 · s-1 L ) inactive sequence of A (internal sequence of A) L* ) active sequence of A (end sequence of A) M ) dead polymer chain O ) inactive sequence of B (internal sequence of B) O* ) active sequence of B (end sequence of B) P ) living polymer radical chain with terminal unit A PDI ) polydispersity of polymer chains PDIi ) polydispersity of sequence i Q ) living polymer radical chain with terminal unit B R* ) radical from initiator or leaving agent Ri ) reaction rate of species i Rm,i ) reaction rate of monomer i Sjji ) jth moment of inactive sequence i (i ) a, b; j ) 0, 1, 2) Sji ) jth moment of active sequence i (i ) a, b; j ) 0, 1, 2) SLn,i ) number-average sequence length of sequence SLw,i ) weight-average sequence length of sequence i T ) RAFT agent Tg ) glass transition temperature, K TP ) polymer chain with terminal unit A bound to RAFT agent TQ ) polymer chain with terminal unit B bound to RAFT agent Tji ) jth moment of active sequence i bound to RAFT agent (i ) a, b; j ) 0, 1, 2) u ) input variable V ) volume of reaction system V0 ) initial volume in the reactor
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Vf ) volumetric feeding flow rate, L · s VF ) free volume fraction VS,f ) volumetric feeding flow rate of solvent, L · s-1 w ) weight factor defined in eq 18 xi ) mole fraction of monomer i y ) output variable Yji ) jth moment of living radical chain with terminal unit i (i ) a, b; j ) 0, 1, 2) Zji ) jth moment of dormant chain with terminal unit i (i ) a, b; j ) 0, 1, 2) Greek Symbols R ) thermal expansion coefficient F ) density, g · cm-3 Subscripts i ) monomer i (1 ) MMA, 2 ) comonomer) M ) monomer P ) polymer s ) set point defined in eq 18 S ) solvent
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ReceiVed for reView June 25, 2009 ReVised manuscript receiVed September 14, 2009 Accepted September 30, 2009 IE901032Y
