1088
Ind. Eng. Chem. Res. 1997, 36, 1088-1094
Nonterminating Polymerizations in Continuous Flow Systems Dong-Min Kim and E. Bruce Nauman* The Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590
The feasibility of manufacturing narrow polydispersity polymers in continuous flow systems is explored. The method of moments is applied to ideal and more realistic kinetic schemes in a variety of reactor types. In most cases, polydispersities approaching those obtained in batch can also be achieved in continuous reactors, but the long-term stability of a tubular reactor with species-dependent, radial diffusion remains in doubt. Introduction Homopolymers with a narrow polydispersity, say xw/ xn < 1.5, are rare in commerce. The narrow polydispersity offers incremental advantages over conventional polymers in applications such as extrusion, but the enhancement in properties is too limited to justify a major increment in selling price. The batch anionic polymerizations used for many block copolymers could certainly be used to produce homopolymers of narrow polydispersity but at too great a cost. Anionic polymerizations demand an unusual degree of purification. Even if the purification costs were manageable, largescale, continuous operation would be needed, and it is commonly believed that continuous processing does not lend itself well to the production of polymers having a narrow polydispersity. The present paper addresses this issue for living polymers in general, although the major focus will be on anionic polymerizations or more recent, nearly living, free-radical polymerizations of vinyl monomers (Georges et al., 1994; Patten et al., 1996). The latter system offers the potential advantage of lower purification costs. The molecular weight distribution of living polymers in a batch reactor was derived by Flory (1940), although the term “living polymer” was first used for anionic polymerizations by Szwarc (1956). The theory has been extended to systems with slow initiation (Gold, 1958), various reactor types (Biesenberger and Tadmor, 1966), termination reactions (Yan and Yuan, 1986a,b), and unequal reactivity of active chains (Saliaya and Kamar, 1995). The present paper repeats the basic results using a particularly simple approach based on the method of moments. It then applies the results to analyze a variety of continuous flow systems. Ideal Kinetics The mathematical treatment of an ideal, nonterminating polymerization begins with the following assumptions: (1) Initiation is fast compared to propagation. (2) There are no chain-transfer or termination reactions. (3) The propagation rate is independent of the chain length. (4) The reactor is batch and isothermal. The polymerization mechanism is
initiation propagation
ki
I + M 98 P1 kp
Pn + M 98 Pn+1
generated instantaneously at the onset of the reaction and remains constant thereafter. After initiation, the monomer consumption is given by
dM ) -kpI0M dt
(3)
The component balance for active chains is
dPn ) kpM(Pn-1 - Pn) dt
(4)
where P0 ) 0. The initial conditions at t ) 0 are
M ) M0
(5)
P1 ) I0
(6)
Pn ) 0
(for n > 1)
(7)
A new variable, ν, is defined as
∫0tkp(M) dt′
(8)
M0 (1 - e-kpI0t) I0
(9)
ν) Then,
ν) and
dPn ) Pn-1 - Pn dν
(10)
with the same initial conditions since ν ) 0 at t ) 0. Note that ν has a physical interpretation as the kinetic chain length. The ith moment of the polymer chain length distribution is defined as ∞
µi )
niPn ∑ n)1
(11)
Summation of eq 10 over all n g 1 gives
dµ0 )0 dν
(12)
dµ1 ) µ0 dν
(13)
(1) Similarly,
(2)
where ki . kp. A quantity, I0, of active chains is S0888-5885(96)00223-0 CCC: $14.00
© 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1089
with the initial conditions of µi ) 0. The solutions are
and
dµ2 ) µ0 + 2µ1 dν
µ0 ) I0(1 - e-ζν)
(14)
The initial conditions are µi ) I0 at ν ) 0 for all i. This gives
µ0 ) I0
(15)
µ1 ) I0(ν + 1)
(16)
µ2 ) I0(ν2 + 3ν + 1)
(17)
{ (
{ ( ) ( xw )
xw ν2 + 3ν + 1 ) xn (ν + 1)2
(20)
The polydispersity has a maximum of 1.25 at xn ) 2 and then approaches 1.0 as the chain length increases. The maximum value for xn is M0/I0, achieved as t f ∞. The average chain length may, of course, be limited by the reaction time rather than stoichiometry. Kinetic Complications
)
(29)
}
(30)
ν+2 2
(31)
2ν2 + 9ν + 6 3(ν + 2)
(32)
xw 2(2ν2 + 9ν + 6) ) xn 3(ν + 2)2
(18)
(19)
)
For the limiting case of case of slow initiation, ζ f 0,
xn )
µ2 ν2 + 3ν + 1 xw ) ) µl ν+1
}
1 (1 - e-ζν) ζ 2 3 2 µ2 ) I0 ν2 + 3 - ν + 1 - + 2 (1 - e-ζν) ζ ζ ζ µ1 ) I0 ν + 1 -
and, in terms of the number and weight average chain lengths,
µ1 xn ) ) ν + 1 µ0
(28)
(33)
The polydispersity has a maximum, 1.375, at ν ) 6 and approaches 1.333 as the chain length increases. However, the actual limit for the polydispersity is more complex than indicated by this simple statement. It depends on the magnitude of the product, ζν, in the limit as ζ f 0 and ν f ∞. If ζν f 0 in the limit, then the limiting polydispersity will be 1.333, but if ζν f ∞ in the limit, the limiting polydispersity will be 1.0. The practical consequence of this statement is that the initiation rate will have very little effect on the polydispersity of high polymers unless it becomes slower than propagation. For example, with xn ) 100 and ζ ) 1 (ki ) kp), the predicted polydispersity is only 1.01. A gradual termination reaction gives rise to the following set of equations:
M dM ) -µ0 dχ M0
(34)
dµ0 ) -βµ0 dχ
(35)
The kinetic chain length is defined as before, eq 8. Then,
dµ1 M ) µ0 - βµ1 dχ M0
(36)
dI ) -ζI dν
dµ2 M M ) µ0 + 2µ1 - βµ2 dχ M0 M0
(37)
The above results are easily modified for the case of slow initiation. Define
ζ)
ki kp
(21)
(22)
and
I ) I0e
-ζν
(23)
where χ ) kpM0t and β ) kt/kp/M0. These moment equations apply to the living polymer. The moments for the dead polymer are given by
dφn ) βµn dχ
The equation governing a chain of l is changed,
dPl ) ζI - P1 dν
(24)
(38)
and the moments for the entire population are given by
but eq 4 holds for n > 1. The moment equations become
ωn ) µn + φn
(39)
dµ0 ) ζI dν
(25)
The initial conditions are M ) M0, µn ) I0, and φn ) 0 at χ ) 0. Two closed-form results are
dµl ) ζI + µ0 dν
(26)
M ) M0 exp[-I0(1 - e-βχ)/βM0]
(40)
ω1 ) I0 + M0 - M
(41)
dµ2 ) ζI + µ0 + 2µ1 dν
and
(27)
1090 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997
polymer chains associate in the liquid phase. The initiation and propagation rates have been found experimentally (Worsfold and Bywaters, 1960; Morton and Fetters, 1975) to be
Ri ) Rp ) -
Figure 1. Polydispersity vs termination-limited chain length for the case of essentially complete monomer conversion, M0/I0 ) (xn)∞.
dI ) kiI1/6M dt
(43)
dM ) kpI01/2M dt
(44)
On the basis of these observed rates, many researchers (Treybig and Anthony, 1978; Cubbon and Margerison, 1962, 1965; Chang et al., 1990, 1993, 1994) have postulated reaction mechanisms. Most of these are so complicated that they prevent a simple model for chain lengths. It seems clear, however, that chain transfer and termination are insignificant in this system at low temperatures and that initiation is faster than propagation. A head-to-head association of the active chains is consistent with the viscosity data (Morton et al., 1961; Morton and Fetters, 1975) and with the dependence of the propagation rate on I01/2. The association takes the form of a fast equilibrium reaction
Pm′ + Pn′ T Am,n
(45)
where Pm′ + Pn′ denote unassociated polymer chains which are presumed to grow individually by bimolecular reactions with the monomer. The batch reaction rate for the formation of nmer is
dPn ) kpM(Pn-1′ - Pn′) dt
(46)
This is identical to eq 10 except that Pn-1′ and Pn′ are less than the total concentration of Pn-l and Pn since most of the chains are in the associated state. We suppose an equilibrium relationship of the form
∑Pn′]2 ) 2K ∑Am,n
[
(47)
where ∑Pn′ is the concentration of all unassociated chains and ∑Am,n is the concentration of the associated pairs. We suppose the equilibrium to be shifted to the right so that K is small and Figure 2. Polydispersity for termination-limited polymerization with an excess of monomer, (xn)∞ ) 1000.
∑Am,n = I0/2
There is a limiting chain length achievable with a given kt at long times.
Then, the fraction of the chains that are unassociated is
(xn)∞ )
() ω1 ω0
∞
)1+
M0 (1 - e-I0kp/kt) I0
∑Pn′ ) K1/2I01/2
(42)
When achievement of this limiting chain length coincides with essentially complete conversion of the monomer, e.g., when M0/I0 = (xn)∞, the polydispersity can be reasonably low. For example, with M0/I0 = (xn)∞ ) 1000, the polydispersity in batch is only 1.087. See Figure 1. However, the polydispersity rises sharply when all chains are terminated while there is still monomer present. See Figure 2. Thus, low polydispersity generally requires nearly complete conversion of the monomer. The overall limit on polydispersity when there is an excess of monomer is 2.0, as in many free-radical polymerizations. In a real, anionic polymerization such as styrene in toluene with n-butyllithium, the initiator and growing
(48)
(49)
It is supposed that this overall fraction applies individually to each chain length. Thus,
Pn′ K1/2 ) Pn I 1/2
(50)
dPn kpK1/2 ) M(Pn-1 - Pn) dt I 1/2
(51)
0
and eq 46 becomes
0
This result is consistent with the overall propagation rate, eq 44, and it is now clear that the chain length distribution is the same as the ideal case with a modified
Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1091
propagation constant:
kp′ )
kpK1/2 I01/2
(52)
where kp′ is the apparent propagation rate constant which is measurable from the experiments. The only effect of a fast, equilibrium association is to retard the polymerization rate so that more time is required to achieve the same final results. Slow association reactions can, however, broaden the molecular weight distribution significantly (Veregin et al., 1996; Figini, 1963). The complications considered above are seen to have a small effect on the polydispersities achievable in batch systems. Other complications, such as chain transfer or site-to-site variations in propagation rate, can have a large effect. However, we will show that whatever polydispersities are achievable in batch can also be achieved in continuous flow.
If either factor is eliminated, polydispersities equivalent to those in a batch reactor become possible. Homogeneous CSTRs in Series. The method of moments can be applied to a homogeneous CSTR which is subject to an inflow of living polymer having an arbitrary distribution of chain lengths. The first three moments of the outflowing polymer are given by
µ0,out ) µ0,in ) I0
(62)
µ1,out ) µ1,in + ν1′µ0,out
(63)
µ2,out ) µ2,in + 2ν1′µ1,out + ν1′µ0,out
(64)
Applying these results to J tanks in series gives
µ1,J ) µ1,J-1 + νJ′I0
(65)
µ2,J ) µ2,J-1 + νJ′(I0 + 2µ1,J)
(66)
where νJ′ ) (kpMth)J will usually vary from tank to tank. Algebra gives
Continuous Flow Systems
J
In an ideal PFR (piston or plug-flow reactor), all molecules have the same residence time, and the performance of a PFR is identical to that of a batch reactor. In real flow systems, different molecules have different residence times. These differences may lead to an increase in polydispersity compared to that of a PFR or batch reactor. Homogeneous CSTR. A homogeneous or perfectly mixed CSTR (continuous flow stirred tank reactor) represents a special case where steady-state performance is governed by a set of algebraic equations. For the ideal, nonterminating polymerizations,
µ1,J ) I0(1 + J
µ2,J ) I0(1 + 3
∑ j)1
νj ′ ) ∑ j)1 J
(67) j
∑ ∑ νk′ ) j)1 k)1
νj′ + 2
νj′
(68)
0 ) FM0 - FM - kpI0MV
(53)
These series can be summed when νj varies as a simple function of j. The equal reactivity case, νj′ ) (kpMjht)j ) constant, is important since it minimizes the polydispersity from a series of J stirred tanks. Equal reactivity can be achieved by feeding monomer between tanks or by using an increasing sequence of tank volumes or temperatures. The results for the equal reactivity case are
0 ) FI0 - FP1 - kpP1MV
(54)
µ1,out ) I0(Jν′ + 1)
(69)
0 ) -FPn + kp(Pn-l - Pn)MV
(55)
µ2,out ) I0{J(J + 1)(ν′)2 + 3Jν′ + 1}
(70)
and
The monomer conversion is
M)
M0 1 + kpI0ht
(56)
where ht ) V/F is the mean residence time. The moment equations are
µ0 ) I0
(57)
µ1 ) I0(ν′ + l)
(58)
µ2 ) I0(2(ν′ )2 + 3ν′ + 1)
(59)
where ν′ ) kpMth. The chain lengths are
xw )
xn ) ν′ + 1
(60)
2(ν′ )2 + 3ν′ + 1 ν′ + 1
(61)
The limiting polydispersity for long chain lengths is 2. This result is a consequence of two factors: (1) There is a distribution of residence times, and (2) polymer chains which remain in the system for long times have continued access to monomer and, thus continue to grow.
lim
ν′f∞
µ0µ2 µl2
) 1 + J-l
(71)
This result, first given by Lynn and Huff (1971) but without stating the assumption of equal reactivity, shows that the limiting polydispersity of 2 for a single stirred tank can be significantly reduced by using several tanks in series. Chains remain alive and growing in all the tanks, and an unusually long time spent in one tank will likely be compensated for by a shorter time spent in another tank. It is worth noting that some commercial processes, e.g., the Union Carbide (now Chevron) process for polystyrene, consist of several stirred tanks in series. In ordinary free-radical kinetics, the polymer is dead upon leaving a reactor and a molecule that is unusually long will remain unusually long. The final polydispersity tends to be similar to that achieved in a single CSTR, typically around 2. If a living polymerization used the same reactor configuration, the final polydispersity would be substantially lower. Segregated Reactor. A segregated reactor exhibits a distribution of residence times, but there is no mixing between molecules which have different residence times. Thus, molecules which enter together leave together.
1092 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997
Figure 3. Polydispersity in anionic polymerization in a segregated CSTR (a ) 100).
Figure 4. Polydispersity in anionic polymerization in a segregated laminar flow reactor (a ) 1000).
They behave as a set of PFR’s in parallel with the possibility of a different residence time for each PFR. The outlet concentration of a component is
obtained in batch reactions may translate to say 1.5 in a continuous suspension polymerization. Segregated Laminar Flow. The residence time distribution function for diffusion-free laminar flow in a tube is
Cout )
∫0∞Cbatchf(t) dt
(72)
where f(t) is the residence time density function. Applying this to the moments gives
µi,out )
∫0∞µi,batchf(t) dt
ht 2 2t3
for t g ht /2
f(t) ) 0
for t < ht /2
f(t))
(73)
where ht is the mean residence time. Equation 73 gives the following results:
For the special case of a CSTR,
1 f(t) ) e-t/th ht
2
(74)
µ0 )
The outlet concentration of monomer is unchanged from that in a homogeneous CSTR, eq 56, but the moment equations are different:
µ0 ) I0
{
}
{
µ2 ) I0 (a2 + 3a + 1) -
∫ht∞/2I02tht 3 dt (79)
) I0 2
µ1 ) I0
(75)
∫ht∞/2{a(1 - ek I t) + 1}2tht 3 dt p 0
2
a µ1 ) I0 (a + 1) kpI0ht + 1 2
(78)
) (a + 1)I0
(76) 2
}
a 2a + 3a + kpI0ht + 1 2kpI0ht + 1
2
∫ht∞/22tht 3 dt - aI0∫ht∞/2ek I t 2tht 3 dt p 0
( )
) (a + 1)I0 - 2aI0E3 (77)
The polydispersity approaches a limit of 2 for high values of a ) M0/I0 and low values of kpI0ht. However, for high values of kpI0ht, the polydispersity approaches 1. This is a general consequence of segregation. Local monomer concentrations are depleted in fluid elements with long residence times, and the segregated environment prevents replenishment of monomer by diffusion from other elements. A continuous suspension process represents a physically realizable example of a segregated CSTR. It is perhaps surprising that such a reactor could be used to produce a polymer of narrow polydispersity. See Figure 3. Obviously, water could not be used as the suspending agent in anionic polymerizations, but the long-lived freeradical polymerizations are less sensitive to contaminants (Georges et al., 1994). Polydispersities of 1.3
µ2 ) I0
kpI0ht 2
(80)
∫ht∞/2{a2e-2kI t - (2a2 + 3a)e-kI t + 0
0
ht 2 a2 + 3a + 1} 3 dt 2t k I h t p 0 ) (a2 + 3a + 1)I0 - (4a2 + 6a)I0E3 + 2 2a2I0E3(kpI0ht ) (81)
( )
where E3(x) is the exponential integral. En(x) is defined as ∫1∞e-xt/tn dt (n ) 0, 1, 2, ...) and is a tabulated function. The polydispersity reaches a maximum value of 1.42 at a ) M0/I0 ) 1000. See Figure 4. As kpI0ht is increased, the polydispersity approximates that achieved in a PFR operated at half the mean residence time. The minimum residence time for segregated, laminar flow in a tube is half that of a PFR having the same volume
Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1093
and flow rate. If a PFR operated with ht/2 achieves a low polydispersity (=1), then so will the laminar flow reactor. Laminar Flow with Diffusion. The convective diffusion equation applied to isoviscometric, laminar flow in a tube is
vz(r)
(
)
2 ∂Pn 1 ∂Pn ∂ Pn ) Dn + 2 + kp(Pn-1 - Pn)M ∂z r ∂r ∂r
(82)
where vz(r) ) 2u j (1 - r2/R2) and Dn is the diffusivity of growing polymer chains of length n. Suppose that Dn ) D is constant. Then, the governing equations for the moments are
vz(r)
vz(r)
(
)
2 ∂µ0 1 ∂µ0 ∂ µ0 )D + 2 ∂z r ∂r ∂r
(83)
2 ∂µ1 1 ∂µ1 ∂ µ1 )D + 2 + kpµ0M(r,z) ∂z r ∂r ∂r
(84)
vz(r)
(
(
)
)
2 ∂µ2 1 ∂µ2 ∂ µ2 )D + 2 + kp(µ0 + 2µ1)M(r,z) ∂z r ∂r ∂r (85)
The boundary conditions are
µn ) I0 at z ) 0, all r, n ) 0, 1, 2
(86)
∂µn ) 0 at r ) 0, R, n ) 1, 2 ∂r
(87)
In practice, eq 83 has the trivial solution µ0 ) constant ) I0, but the higher moments are more readily found by numerical solutions. The results show the polydispersity to be intermediate between the PFR and segregated laminar flow reactor, tending toward the PFR as D f ∞. Design Considerations and Complications. The idealized reactors considered to this point suggest that a low polydispersity polymer can be made in a continuous flow system. Many types of flow reactors can closely approximate the polydispersity of a batch system. They achieve this close approach by one of two mechanisms: (1) In segregated flow, chains reach a maximum length and then stop growing due to local depletion of monomer. (2) If monomer remains available to growing chains, low polydispersities can still be obtained if the residence time distribution approximates that of a PFR. The kinetic complications considered earlier can, of course, affect the polydispersity (PD) achievable in a batch system. When a flow reactor is used for a case of nonideal kinetics, we suggest the following as a conservative estimate:
PD ) (PD in a real reactor assuming ideal kinetics) × (PD in a batch reactor for the real kinetics) (89) Thus, the composite polydispersity made in a train of four stirred tanks in series with a nearly living polymerization that gives a polydispersity of 1.3 in batch would be (1.25)(1.3) ) 1.62. The kinetic complications impose a limit on the reactor performance, but more important limitations may result from heat- and mass-transfer considerations. Highly exothermic vinyl polymerizations can give rise to thermal runaways (Bishop, 1971). The large changes
in viscosity cause hydrodynamic instabilities (Mallikarjun and Nauman, 1986). There are design techniques to analyze and manage these complications. In particular, these and other complications can be minimized by using solvent (rather than bulk or near-bulk) polymerizations. With modern environmental requirements, the cost of solvent losses has become negligible, and the capital cost of confinement is largely borne by the cost of confining the monomer (Nauman and Cavanaugh, 1996). It is not yet clear that solvent polymerizations can completely eliminate one type of instability in a living polymer system. The assumption of constant diffusivity used in the analysis of laminar flow reactors was mathematically convenient but physically unrealistic. Diffusivities can be expected to decrease as the chain lengths increase. This leads to a possible scenario where chains near the wall remain near the wall due to their low diffusivity but will continue to have access to monomer due to its comparatively high diffusivity. Without local depletion of monomer, the long chains continue to grow, the polydispersity increases, and the tube eventually plugs. This is a form of instability similar in consequence to the hydrodynamic instabilities exhibited in nonliving, tubular polymerizations of vinyl monomers, but the mechanism is somewhat different and may be more difficult to prevent through the use of solvents. The long-term stability of nonterminating polymerizations in tubular geometries remains an open question. Some form of periodic cleaning, similar in concept to the pressure reductions used in the highpressure, tubular process for polyethylene, may be necessary. However, experimental studies (Kim, 1996) show that relatively infrequent cleaning will be adequate if cleaning is necessary at all. Conclusion It is feasible to manufacture low-polydispersity polymers in reaction trains of the type now used for conventional, free-radical polymerizations. A sequence of four to five CSTR’s in series will give a polydispersity of 1.2-1.25 for an ideal, nonterminating polymerization. Such reaction trains are commonly used for polystyrene. When applied to long-lived, free-radical polymerizations (Georges et al., 1994) that show a polydispersity of 1.3 in batch, the overall polydispersity should not exceed 1.6, which is a significant reduction over the best commercial polystyrenes which have a polydispersity of about 2.2. We note that such conventional polystyrene processes often employ close clearance agitators so that postulated accumulation of high molecular weight polymer at the walls should not cause a problem. Indeed, anionic polymerizations are known to be feasible in a CSTR. See, for example, Priddy et al. (1991, 1993). Substantially lower polydispersities may be achievable using solvent polymerization in a tubular reactor, but the long-term stability of such reactors remains in question. Solvent concentrations of 80-90% are used for solution polyethylene processes and are competitive with gas-solid and bulk (high-pressure) polymerizations, at least for some market segments. High solvent concentrations prevent thermal runaways and can enable large-diameter, adiabatic, tubular reactors for moderately exothermic monomers such as styrene. In short-term operation, such reactors will approach segregated laminar flow and will produce low polydispersity polymers due to local depletion of monomer. Longer term stability is questioned due to a postulated accumulation of high molecular weight polymer near the
1094 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997
wall. The resolution of this question could be experimental, or it could be by numerical solution of the convective diffusion equations. Diffusivities can be expected to vary as a function of viscosity, say as µ-1, and as a function of chain length, say as n-2. The viscosity dependence is easy to handle, but the chain length dependence leads to a challenging problem in numerical analysis. Nomenclature Am,n ) concentration of associated pairs a ) ratio of initial monomer concentration to initiator concentration Dn ) diffusivity En(x) ) exponential integral F ) flow rate f ) residence time density function I ) initiator concentration J ) number of tanks K ) equilibrium constant ki ) initiation rate constant kp ) propagation rate constant M ) monomer concentration n ) number of monomer units Pn ) polymer concentration of chain length n Ri ) initiation rate Rp ) propagation rate ht ) mean residence time V ) reactor volume vz ) axial velocity xn, xw ) number-average and weight-average chain length Greek Symbols β ) dimensionless parameter χ ) dimensionless parameter ν ) kinetic chain length ζ ) ratio of initiation rate to propagation rate φi ) ith moment of dead polymer ωi ) ith moment of entire polymer µi ) ith moment of polymer Abbreviations CSTR ) continuous stirred tank reactor PD ) polydispersity PFR ) plug-flow reactor
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Received for review April 19, 1996 Revised manuscript received June 21, 1996 Accepted June 22, 1996X IE960223R X Abstract published in Advance ACS Abstracts, February 15, 1997.
