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Molecular Weight Distribution Design with Living Polymerization Reactions Eszter Farkas,† Zsolt G. Meszena,*,† and Anthony F. Johnson‡ Department of Chemical Information Technology, Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, Hungary, and IRC in Polymer Science and Technology, School of Chemistry, University of Leeds, Leeds, West Yorkshire LS2 9JT, England
It has been previously shown that it is feasible to control the molecular weight distribution (MWD) of polymers produced from living polymerization processes in flow reactors through the control of reactant feeds. Here, attention is given to the problem of establishing an inverse process model as a step toward a fully automatic control strategy for the synthesis of polymers with predefined MWD. Particular attention is given to the prediction of instantaneous reactor feed conditions for a specified product MWD. The reactor is modeled as an ideal continuous stirred tank reactor with constant monomer concentration in the reactor. The way in which a real laboratory-scale polymerization system can be developed from this approach is outlined. 1. Introduction The principle of achieving tailored molecular weight distribution (MWD) control through designed periodic oscillation of the monomer and/or initiator flow rates to flow reactors has been previously established by simulation and experimental studies on living anionic polymerization reactions carried out in our laboratories. The problem of tailoring MWD through control of monomer and initiator flows to a tubular reactor for living anionic polymerizations has been explored by Meira and Johnson.1 Novel fast algorithms for the fast calculation of MWDs with various forcing functions have been developed by Gosden et al.2,3 and Meszena et al.,4 and the simulation results were also validated by experiment.3 Neural network methods have been applied to the design of the MWD with living anionic polymerizations by Gosden et al.5 These studies have established a basis for the development of control strategies for producing tailored MWDs. Fully automatic control strategies require a number of challenging issues to be addressed that are specific to the control objectives, reactor type, and polymerization chemistry. First, the control objective is not a setpoint tracking problem, as is the case in most control situations, but is a function representing the MWD. Second, the reactor continuously operates under forced unsteady-state conditions. As a result, controllers with parameters designed and tuned under one condition may not be able to cope with the wide range of operational modes the process may experience and therefore require adaptive capabilities. Finally, it is only possible to effect significant control of MWD when the lifetime of the growing species is long compared with the period of the feed perturbation. A further complication is that polymerization reactors often exhibit complex nonlinear dynamic behavior because of the complexity of a number of physicochemical interactions, for * To whom correspondence should be addressed. Tel.: +36-1-463-4338. Fax: +36-1-463-3953. E-mail: meszena@ chem.bme.hu. † Budapest University of Technology and Economics. ‡ University of Leeds. Tel.: +44-113-343-3914. Fax: +44113-247-0676. E-mail:
[email protected].
example, the Arrhenius-type variations in the reaction rate constants with temperature and high overall reaction rates. This suggests that nonlinear control strategies should be employed in order to deliver effective control. In this work an inverse process model has been established, based on the algorithm previously developed4 (“method of monodisperse growth”, summarized in section 1.3). All simplifications used in the method, with the additional assumption of constant monomer concentration in the reactor, have been utilized to establish a formula for the inverse calculation sequence. It is central to the concept of our MWD design process to have such a straightforward and fast means of back and forth calculation procedures in order to develop an efficient control system that is aimed at not only some moments but also the full MWD envelope. In addition to the above, in this work, the assumption of instantaneous initiation has been used in order to simplify calculations. Starting from a target MWD, the design process uses the inverse process model to derive reactor conditions and input functions. The predicted reactor parameters, consequently, can be used in more sophisticated simulations as well as in experiments, and the resulting MWDs can be compared to the target. Examples presented cover isothermal living anionic polymerizations in a continuous stirred tank reactor (CSTR). The effect of periodic input functions to a singlestage CSTR on the MWD is analyzed. 1.1. Kinetic Scheme. The kinetic scheme of an ideal living polymerization with instantaneous initiation can be described by eqs 1 and 2. ki
Initiation: I + M 98 P1 kp
ki . kp
Propagation: Pj + M 98 Pj+1
j ) 1, 2, ...
(1) (2)
1.2. Component Balances. Global mass balances of initiator, monomer, and live ends in an isothermal CSTR with instantaneous initiation, constant volumetric flow rate, and constant monomer concentration in the reactor can be described by eqs 3 and 6.
10.1021/ie034329f CCC: $27.50 © 2004 American Chemical Society Published on Web 10/05/2004
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I(t) ) 0
(3)
Min(t) - Iin(t) - M -rp ) 0 τ
M ) constant
dP Iin(t) - P(t) ) dt τ
(4) (5)
where τ is the average residence time. The rate of propagation, rp, is given in eq 6. ∞
rp ) kpPM
where P )
Pj ∑ j)1
(6)
1.3. Method of Monodisperse Growth. A fast, approximate MWD calculation method has been previously established (method of monodisperse growth4) for living anionic polymerization. The main assumption of the method is that dispersity caused by chain growth can be neglected. It is assumed that the dispersity caused by the reactor configuration dominates other effects in the final MWD. In this section, the method is presented in a format necessary for the development of the inverse calculation sequence of the MWD design process, i.e., with all of the simplifying assumptions embodied. In the method of monodisperse growth, it is assumed that polymerization gives rise to truly monodisperse chains. The length of all chains initiated at a certain time, t0, is taken to be uniform with an average length, µ, calculated from eq 7 for instantaneous initiation.
µ(t) ) 1 +
∫0tkpM dt
(7)
With this assumption, the MWD becomes a continuous function of the chain length. From now onward, the continuous nature of the MWD is emphasized by way of using P(j) instead of Pj. In the case of isothermal conditions and constant monomer concentration in the reactor, the average chain length is a linear function of time, as is shown in eq 8,
µ(t) ) 1 + kpMt ) 1 +
Da t τ
(8)
where Da is the Damkohler number: Da ) τkpM. The number chain length distribution (NCLD) of truly monodisperse chains of length µ and amount I0 is a Dirac δ distribution centered over µ as shown in eq 9,
P(j) ) I0δ(j-µ)
(9)
which will be the NCLD (approximately) of the polymer produced if the reaction is taking place in a batch reactor and initiation is instantaneous. For this situation, the predicted NCLD is shown and compared to the theoretical one (i.e., the Poisson distribution) in Figure 1. The cases of noninstantaneous initiation and reactor configurations other than batch are calculated on the basis of the algorithm reported by Gosden et al.1 At time t, the NCLD of a sample from a CSTR with instantaneous initiation and constant monomer concentration in the reactor is estimated with eq 10,
P(j,t) )
1 -(j-1)/Da j-1 e Iin t - τ Da Da
(
)
(10)
Figure 1. NCLD predicted by the method of monodisperse growth (Dirac δ) vs theoretical NCLD (Poisson distribution) for a batch process with instantaneous initiation.
where, as throughout the text, parentheses stand for functional arguments:
(
Iin t - τ
j-1 ) Iin(t′)|t′)t-τ[(j-1)/Da] Da
)
The overall NCLD of the product collected is defined by eq 11.
P h (j,tend) )
1 1 -(j-1)/Da e tend Da
t -τ[(j-1)/Da] Iin(t) dt ∫t)0 end
(11)
It can be seen from eqs 10 and 11 that the shape of the MWD is defined by the reversed initiator input profile (weighted with the internal age distribution). 1.4. Moments and Averages of the MWD. The most frequently used parameters of the NCLD are the moments, λk, the number-average chain length, µn, the weight-average chain length, µw, and the dispersity index, Dn, with the definitions as shown in eqs 12-15: ∞
jkPj ∑ j)1
(12)
µn ) λ1/λ0
(13)
µw ) λ2/λ1
(14)
Dn ) µw/µn ) λ0λ2/λ12
(15)
λk )
The number-average chain length, µn, is the expected value, and the dispersity index, Dn, is related to the variance of the normalized NCLD:
Dn ) 1 + σ2/µn2
(16)
The concentration of the chains is usually measured not in terms of moles but rather by weight; consequently, it is the weight MWD (WMWD), i.e., weight concentrations versus molecular weight, which can be readily compared to experimental results. Hence, it is preferred in the presentation of the results. 2. MWD Design in a CSTR The MWD design process uses an inverse of a simplified process model (eq 11) to derive approximate reactor conditions (Figure 2). The predicted reactor parameters have been used in simulations carried out both with the simple model and with a more sophisticated process
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Figure 2. Overview of the MWD design process.
Figure 4. Predicted initiator input profile before (O) and after (s) feasibility checks.
Figure 3. Target and predicted MWDs.
model of the commercial package, Predici.6 Additionally, the predicted parameters can be used in experiments (not reported here), and the resulting MWDs can be compared to the target. 2.1. Inverse Process Model for a Single-Stage CSTR. The inverse calculation starts from a target MWD (Figure 3, solid line), which is the MWD of the product collected over the whole processing period. Parameters to be specified are the Damkohler number (Da ) 200) and the maximum chain length considered (jmax ) 1001; jmin ) 1). Additional parameters needed to relate the results to a certain time scale and feasible concentrations are the average residence time (τ ) 100 s) and the highest possible initiator input concentration (Iin,max ) 0.02 mol/L). For the sake of simplicity, the molecular weight of a monomer unit equals 1. The processing time derived is shown in eq 17.
jmax - 1 ) 500 Da
tend ) τ
(17)
If the MWD of a sample at tend were known, the necessary input initiator profile could be simply derived by rearranging eq 10, as is shown in eq 18.
j-1 ) P(j,tend)Dae(j-1)/Da Da
(
)
Iin tend - τ
(18)
In general, the initiator input profile can be derived from eq 11, as is shown in eqs 19 and 20 and in Figure 4 (circles):
(
∂F(j,tend) j-1 ) -(jmax - 1)Da (19) Da ∂j
Iin tend - τ
)
where
h (j,t) F(j,t) ) e(j-1)/DaP
(20)
i.e., function F is proportional to the final MWD deconvolved with the residence time distribution of the reactor.
Figure 5. Integral of the reversed initiator input profile: a monotonic function with a bounded slope.
The feasibility of the predicted initiator input profile has to be checked. If a processing window shown in eq 21 is applied, a feasible initiator input profile is obtained (Figure 4, solid line).
0 e Iin,feasible e Iin,max
(21)
It is worth noting that function F, defined by eq 20, is proportional to the integral of the reversed initiator input profile. As a consequence, (i) function F, if feasible, is a monotonically decreasing function (due to the lower bound on Iin) and (ii) the slope of F is bounded (due to the upper bound on Iin). Function F is shown in Figure 5 for the case studied. It can be seen in Figure 4 that, for example, the lower bound on Iin is applied in time intervals [300, 330] and [420, 470] approximately. On reversed time scale (Figure 5), the same can be observed in intervals [500-330, 500-300] and [500-470, 500-420], i.e., [170, 200] and [30, 80]. It is the function F that is the best functional form to analyze the constraints that are applicable to the MWD. Having been established, the feasible initiator input profile, eq 11, can be used to predict the MWD. As is shown in Figure 3, for the case analyzed, cutting of negative or too high initiator concentration values has no profound effect on the MWD. The predicted initiator input profile has been used in an additional simulation with a more sophisticated MWD calculation method. Calculations have been carried out with the commercial simulation package Predici.6 It can be seen in Figure 3 that, for the case studied, the results of the MWD design process using the assumption of monodisperse growth do not significantly deviate from those of the more precise (and much slower) calculations. Having been calculated, the initiator input profile, the monomer input profile necessary to maintain a constant monomer concentration level in the reactor, is expressed from eq 4, as is shown in eq 22,
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Min(t) ) M + Iin(t) + Da × P(t)
(22)
where the total polymer concentration, P(t), can be derived from eq 5 (with an initial condition of P(0) ) 0), as is shown in eq 23.
P(t) ) e-t/τ
∫0t1τ Iin(Θ) eΘ/τ dΘ
(23)
2.2. Periodic Input Functions to a CSTR. In general, our MWD design process does not pose restrictions other than boundedness (see eq 21) on the input functions to the reactor. However, to minimize waste, periodic input functions would probably be used in a continuous production process. It has to be pointed out that, in the example studied in section 2.1, the product collected from the CSTR has not been mixed with the material that remains in the reactor. If a periodic input profile is applied to the reactor, the final MWD can be defined as the limit shown in eq 24.
h j(t) ) lim P tf∞
1 e-(j-1)/Da lim tf∞ t Da
(
Figure 6. Single period of a periodic initiator input.
∫0t-τ[(j-1)/Da] Iin(Θ) dΘ) (24)
Because Iin(t) is periodic, it can be seen that the final MWD, independent of the functional form of the initiator input, tends to a distribution shown in eq 25, where
h j(t) ) lim P
e-[(j-1)/Da] Iin Da
(25)
Figure 7. Calculated product MWDs after 2, 4, 8, 16, 32, and 64 periods.
Iin is the integral average of the initiator input in a period. The above formula is equivalent to the SchulzFlory distribution shown in eq 26 for instantaneous initiation
reactor configurations other than CSTR are being investigated and will be published elsewhere.
tf∞
Iin j-1 Pj ) q Da
(26)
with the approximation shown in eq 27, where q is
ln q ) -ln(1 + 1/Da) ≈ -1/Da
(27)
defined by eq 28.
q)
1 1/Da + 1
(28)
Approximation (27) is reasonable if Da . 1, which is the case in most of the systems. To illustrate the development of the final MWD toward the Schulz-Flory distribution, calculations have been carried out with the simple process model (eq 11) for a number of periods of an initiator input function shown in Figure 6, namely, for periods 2, 4, 8, 16, and 32. The corresponding MWD envelopes are shown in Figure 7. It is obvious from eq 25 and Figure 7 that the range of possible MWD envelopes from a CSTR is limited. However, examples like those in section 2.1 demonstrate that there is considerable scope for the MWD design process even with a single-stage CSTR, especially if the processing time can be set to several times higher than the average residence time. In addition to the above,
3. Other Reaction Schemes The design process described in section 2 has been developed for living anionic polymerization. It is not yet clear whether other reaction schemes could be treated in a similar manner. The basis of the process, the method of monodisperse growth (section 1.3), can be used to model systems with some termination and transfer reactions (spontaneous, impurity, and monomer termination, instantaneous termination by impurities of a continuous feed, spontaneous chain transfer, chain transfer to monomer, and impurity transfer). These reactions or any others that are first order with respect to active centers can be treated. The distribution of the active centers is not affected by the source of the active centers, i.e., originally present, initiated, or generated in a transfer reaction. The key feature utilized is that the scope is restricted to transfer reactions producing active centers of unit length. Consequently, the method is readily applicable to cases including these reaction steps. In contrast, the treatment of the above-mentioned termination reactions needs additional integrations. The above-mentioned termination reactions do not affect the distribution of the active centers, and they are first-order reactions with respect to the concentration of active centers. It is not yet clear how disproportionation, combination, or condensation reactions could be treated. Treatment of reaction schemes other than ideal living anionic polymerization is being investigated and will be published elsewhere.
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However restricted the scope of the method is, there are commercial processes that can be approximated with the model of ideal living anionic polymerization. Solution polymerization of styrene or methyl methacrylate in polar solvents with an alkyllithium initiator is an example. The attempts that are being made to utilize the derived feeds in real reactor experiments will be described at a later date. 4. Conclusions It has been demonstrated that both design and simulation calculations can be carried out with the method of monodisperse growth. These calculations are essential parts of a fully automated control algorithm for living polymerization reactors. The formulas necessary to derive the reactor conditions and input functions from a target MWD have been established for a single-stage CSTR. It has been proven that periodic input functions applied to a CSTR restrict the possible shapes of the MWD because the product MWD tends to a SchulzFlory distribution with increasing process time. However, it has been demonstrated that there is considerable scope for the MWD design process even with a single-stage CSTR, especially if the processing time can be set to several times higher than the average residence time. Acknowledgment We acknowledge the Engineering and Physical Science Research Council (EPSRC) for their support of the
IRC in Polymer Science and Technology and the Hungarian National Research Foundation (OTKA) for their support of Z.G.M. (Project T-046460). Literature Cited (1) Meira, G. R.; Johnson, A. F. Molecular-weight distribution control in continuous living polymerisation through periodic operation of the monomer feed. Polym. Eng. Sci. 1981, 21, 415. (2) Gosden, R. G.; Auguste, S.; Edwards, H. G. M.; Johnson, A. F.; Meszena, Z. G.; Mohsin, M. A. Living polymerisation reactors. Part I. Modelling and simulation of flow reactors operated under cyclical-steady-state feed conditions for the control of molecular weight distribution. Polym. React. Eng. 1995, 3, 331. (3) Gosden, R. G.; Meszena, Z. G.; Mohsin, M. A.; Auguste, S.; Johnson, A. F. Living polymerisation reactors. II. Theoretical and experimental tests on an algorithm which predicts MWDs from CSTRs with perturbed feeds. Polym. React. Eng. 1997, 5, 45. (4) Meszena, Z. G.; Viczian, Z.; Gosden, R. G.; Mohsin, M. A.; Johnson, A. F. Towards tailored molecular weight distributions through controlled living polymerisation reactors: a simple predictive algorithm. Polym. React. Eng. 1999, 7, 71. (5) Gosden, R. G.; Sahakaro, K.; Johnson, A. F.; Chen, J.; Li, R. F.; Wang, X. Z.; Meszena, Z. G. Living polymerisation reactors: Molecular weight distribution control using inverse neural network models. Polym. React. Eng. 2001, 9, 249. (6) Wulkow, M. Simulation of MWDs in polyreaction kinetics by discrete Galerkin methods. Macromol. Theory Simul. 1996, 5, 393.
Received for review December 19, 2003 Revised manuscript received August 28, 2004 Accepted September 1, 2004 IE034329F
