Ind. Eng. Chem. Res. 2003, 42, 743-751
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Mathematical Modeling of Dispersion Polymerization of Methyl Methacrylate in Supercritical Carbon Dioxide C. Chatzidoukas, P. Pladis, and C. Kiparissides* Chemical Engineering Department and Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki, P.O. Box 472, 54006 Thessaloniki, Greece
Free-radical polymerization in supercritical carbon dioxide is a heterogeneous process that has recently attracted increased research and industrial interest. In the present paper, a comprehensive mathematical model is developed for the dispersion polymerization of vinyl monomers in supercritical CO2. A detailed kinetic mechanism is employed to describe the molecular weight development in the two-phase polymerization system. The concentrations of the various reacting species in the two phases as well as the pressure variation during polymerization are calculated using the Sanchez-Lacombe equation of state. The predictive capabilities of the present model are demonstrated by comparison of the theoretical predictions with experimental results on monomer conversion, total pressure, and molecular weight averages. 1. Introduction In the plastics manufacturing industry, large quantities of organic solvents are used each year as process aids, cleaning agents, and dispersants. Furthermore, conventional heterogeneous polymerizations of unsaturated monomers are usually conducted in aqueous or organic dispersing media.1 Thus, solvent-intensive polymer industries are looking for alternatives to reduce or/ and eliminate the negative impact of harmful solvent emissions on the environment. Supercritical CO2 represents an environmentally sound alternative to both the use of toxic and flammable organic solvents and the generation of large volumes of aqueous waste. Moreover, the use of supercritical CO2 as a dispersion medium for free-radical polymerizations of vinyl monomers allows the easy isolation of polymer as a free-flowing powder through a simple reduction in the CO2 pressure. Carbon dioxide has a relatively low critical temperature (Tc ) 31.1 °C) and a moderate critical pressure (Pc ) 72 bar). Depending on the temperature and pressure conditions, supercritical CO2 can exhibit “liquidlike” densities and at the same time “gaslike” viscosities. Near its critical point, supercritical CO2 has a dissolving power similar to that of liquids. As with all supercritical fluids, scCO2 offers mass-transfer advantages over conventional solvents because of its low viscosity and surface tension.2,3 Finally, another important fact about supercritical CO2 is that its thermodynamic and transport properties are easily adjustable, as they can be altered by small changes in pressure and/ or temperature. Although polymerization reactions have been carried out under supercritical conditions for many years (e.g., manufacturing of low-density polyethylene in highpressure autoclaves and tubular reactors), only recently has the use of supercritical fluids as inert reaction media received considerable attention. One major drawback of scCO2 is its poor solvency power for most hydrocarbon polymers.2 However, amorphous fluoropolymers and * Correspondence should be sent to: Professor C. Kiparissides. E-mail:
[email protected]. Tel.: 00302310996211. Fax: 0030-2310996198.
silicones exhibit appreciable solubility in scCO2 and, thus, have been employed as stabilizers in the dispersion polymerization of vinyl monomers in scCO2. A free-radical dispersion polymerization is a heterogeneous process during which latex particles are formed in the presence of a suitable stabilizer from an initially homogeneous reaction medium.1 The process is capable of producing particles in the size range of 0.1-15 µm in a single step. Under certain conditions, the resulting particle size distribution can be very narrow. The key component in a scCO2 polymerization recipe is the stabilizer, which can be either a block or graft copolymer [e.g., poly(2-hydroxyethyl methacrylate)-g-poly(1,1-dihydroperfluorooctyl acrylate), (PHEMA-g-PFOA)] or a polymer carrying polymerizable functional groups [e.g., poly(dimethylsiloxane)-methyl acrylate macromonomer (PDMS-mMA)]. In the latter case, the functional groups can participate in the polymerization, and thus, the stabilizer becomes chemically grafted to the polymer particle. Since the first report on polymerization of vinyl monomers in scCO2 by DeSimone and co-workers,4 many papers have been reported dealing with the polymerization of various monomers (e.g., methyl methacrylate, styrene, vinyl pyrrolidone, acrylonitrile, etc.). However, most of the reported studies have been focused on experimental investigations of the effects of process parameters (e.g., pressure, temperature, stabilizer concentration, etc.) and the type of stabilizer on the polymerization kinetics and product morphology. The aim of the present work is the development of a comprehensive mathematical model for the quantitative prediction of the time evolution of monomer conversion and molecular weight distribution (MWD) in the freeradical dispersion polymerization of methyl methacrylate (MMA) in scCO2. In what follows, the kinetic mechanism of MMA dispersion polymerization in scCO2 is presented. On the basis of the postulated kinetic mechanism, detailed molar species balances for monomer, initiator, solvent, and live and dead polymer chains are derived. The equilibrium concentrations of monomer, solvent, and polymer in the two phases are calculated using the Sanchez-Lacombe equation of state. Finally, in the last section, the predictive capabilities of the present model are demonstrated by a
10.1021/ie020397a CCC: $25.00 © 2003 American Chemical Society Published on Web 01/25/2003
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Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003
direct comparison of model predictions with reported experimental data on the polymerization of MMA in scCO2. 2. Dispersion Polymerization Kinetics Dispersion polymerization can be viewed as a modified precipitation polymerization process. From a kinetic point of view, the dispersion polymerization of vinyl monomers can be considered to take place in two phases. During the early polymerization stage, primary radicals formed by the thermal decomposition of initiator molecules rapidly react with monomer molecules to produce polymer chains that are usually insoluble in the continuous monomer-CO2 phase. Aggregation of the closely spaced polymer chains results in the formation of primary particle nuclei. In the presence of an amphiphilic stabilizer, the primary particles are stabilized through steric repulsion forces. Typical surfactants consist of a “CO2-philic” segment, usually a siloxane or fluorocarbon, and a “CO2-phobic” moiety that interacts with the polymer. Upon the formation of primary polymer particles, polymerization proceeds in two phases, namely, in the polymer-rich phase and in the continuous monomerCO2 one. It is assumed that, during this polymerization stage, the rate of mass transfer of monomer and CO2 from the continuous phase to the polymer particles is very fast so that the latter is kept, at any time, saturated with monomer and CO2. Thus, the overall polymerization rate is given by the sum of the polymerization rates in the two phases. This stage extends from the time of appearance of the separate polymer phase up to a fractional conversion, Xc, beyond which the polymerization rate in the continuous phase becomes insignificant. Finally, at higher monomer conversions (Xc < X e1.0), polymerization mainly continues only in the polymer-rich phase, which is swollen with monomer and CO2. During this stage, the monomer concentration in the polymer phase decreases as the overall monomer conversion increases. An interesting feature of dispersion polymerization is the possibility of changing the solubility of the polymer in the continuous medium by varying the reactor temperature or/and pressure. In this way, the polymerization rate, the particle size distribution, and the molecular weight characteristics of the produced polymer can be altered. On the basis of the above two-phase kinetic mechanism, the free-radical dispersion polymerization of MMA in scCO2 can be described in terms of the following elementary reactions
initiation
kdj
Ij 98
3. Reactor Design Equations Following the developments of Kiparissides et al.,5 one can easily derive the differential equations describing the conservation of the various molecular species (i.e., monomer, initiator, inhibitor, and the live and dead polymer chains of length x) in each phase in a batch polymerization reactor.
monomer 1 d([M]jVj) Vj
propagation
• Rx,j
+ Mj 98
chain transfer to monomer
• Rx,j
kfmj
kpj
• R1,j
by disproportionation
• • Rx,j + Ry,j 98 Px + Py
by combination
• • Rx,j + Ry,j 98 Px+y
ktcj
kzj
• Rx,j + Zj 98 Px
∑[R•x]j -
x)1
initiator 1 d([I]jVj) ) -kdj[I]j Vj dt
(2)
inhibitor 1 d([Z]jVj)
∞
∑[R•x]j x)1
) -kzj[Z]j
(3)
1 d([S]jVj) )0 Vj dt
(4)
1 d([R ]jVj) ) 2fkdj[I]j - kIj[M]j[R•]j Vj dt
(5)
Vj
dt
solvent
primary radicals •
live polymer chains • 1 d([Rx]jVj)
)
dt
∞
• ]j - [R•x]j) - kfmj[M]j[R•x]j kpj[M]j([Rx-1
termination ktdj
x)1
∞
[R•y]j)δ(x-1) + ∑ y)1
• Rx+1,j
+ Mj 98 Px +
∑
[R•x]j - kfmj[M]j
(kIj[M]j[R•]j + kfmj[M]j
kIj
• R•j + Mj 98 R1,j
dt
∞
) -kpj[M]j
kIj[M]j[R•]j (1)
Vj
2R•j
chain initiation
inhibition
In the above kinetic scheme, the symbols I, M, and Z denote the initiator, monomer, and inhibitor molecules, respectively. Primary radicals formed by the decomposition of the initiator are denoted by the symbol R•. The symbols R•x and Px identify the respective live and dead polymer chains with degree of polymerization x. The symbols kd, kI, kp, kfm, ktd, ktc, and kz denote the rate constants for initiator decomposition, chain initiation, propagation, transfer to monomer, termination by disproportionation, termination by combination, and inhibition, respectively. It should be noted that all of the above elementary reactions can take place either in the continuous monomer-CO2 phase (j ) 1) or/and in the polymer-rich phase (j ) 2).
(ktcj +
∞ • ktdj)[Rx]j [R•y]j y)1
∑
- kzj[Z]j[R•x]j (6)
dead polymer chains 1 d([Px]Vj) ) kfmj[M]j[R•x]j + kzj[Z]j[R•x]j + Vj dt ∞
1
∞
• ]j ∑[R•y]j + 2ktcjy)1 ∑[R•y]j[Rx-y y)1
ktdj[R•x]j
(7)
Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 745
dead polymer moments
where δ(x) is Kronecker’s delta function, given by
δ(x) )
{
dµ0
1 if x ) 0 0 if x * 0
)
(17)
φj(kfmj[M]jλ1j + ktcjλ0jλ1j + ktdjλ0jλ1j) ∑ j)1
(18)
2
)
dt
For modeling purposes, it is often computationally cumbersome to calculate the total number chain length distributions (NCLDs) of live and dead polymer chains. Therefore, to reduce the infinite system of molar balance equations (eqs 6 and 7), the method of moments is commonly employed. Accordingly, the moments of the total NCLDs of live and dead polymer chains are defined as5
dµ1
dt
2
)
1
2
)
dt dµ2
(
φj kfmj[M]jλ0j + ktcjλ0j2 + ktdjλ0j2 ∑ 2 j)1
φj ∑ j)1
{
1 kfmj[M]jλ2j + ktcj(2λ0jλ2j + λ1j2) + 2 ktdjλ0jλ2j
∞
λij )
xi[R•x]j, ∑ x)1
i ) 0, 1, 2, ...
(8)
∞
µi )
xi[Px], i ) 0, 1, 2, ... ∑ x)2
(9)
monomer d[M] dX ) -M0 dt dt
)
φj(kpj[M]jλ0j + kfmj[M]jλ0j + kIj[M]j[R•]j)/M0 ∑ j)1
(11)
d[I]
2
φj(kdj[I]j) ∑ j)1
(12)
φj(2fkdj[I]j - kIj[R•]j[M]j) ∑ j)1
(13)
)-
dt primary radicals d[R] dt
2
)
live polymer moments dλ0 dt
2
)
φj{kIj[R•]j[M]j - (ktcj + ktdj)λ0j2} ∑ j)1
(14)
2
)
dt
φj{kIj[R•]j[M]j + kpj[M]jλ0j + ∑ j)1 kfmj[M]j(λ0j - λ1j) - (ktcj + ktdj)λ0jλ1j} (15)
dλ2 dt
number-average molecular weight (NAMW) µ1 + λ11 + λ12 µ1 ≈ MWm Mn ) MWm µ0 + λ01 + λ02 µ0
2
)
(21)
µ2 + λ21 + λ22 µ2 ≈ MWm Mw ) MWm µ1 + λ11 + λ12 µ1
(22)
polydispersity index (PD) PD )
Mw µ2µ0 ) 2 Mn µ
(23)
1
4. Diffusion-Controlled Reactions
initiator
dλ1
The average molecular properties of the polymer (i.e., the number-average molecular weight; the weightaverage molecular weight; and the polydispersity index, which is a measure of the breadth of the MWD) can be expressed in terms of the leading moments of the NCLDs as follows5
2
dt
(20)
weight-average molecular weight (WAMW) (10)
where X is the overall monomer conversion given by
dX
(19)
where φj ) Vj/Vreact refers to the volume fraction of the jth phase. Assuming that the quasi-steady-state approximation (QSSA) for the primary radicals holds true, eq 5 can be replaced by the following algebraic equation
[R•]j ) 2fkdj[I]j/kIj[M]j
where λij is the ith moment of the live polymer NCLD in the jth phase and µi is the ith moment of the NCLD in the polymer phase. The corresponding moment rate functions are subsequently obtained by multiplying each term of eqs 6 and 7 by xi and summing the resulting expressions over the total range of variation of x. In terms of the leading moments of the live and dead polymer NCLDs and the overall monomer conversion, one can easily derive the following reduced system of ordinary differential equations to model the kinetics of MMA polymerization in scCO2
}
φj{kIj[R•]j[M]j + kpj[M]j(2λ1j - λ0j) + ∑ j)1 kfmj[M]j(λ1j - λ2j) - (ktcj + ktdj)λ0jλ2j} (16)
At high monomer conversions, it is well-known that almost all reactions can become diffusion-controlled. Specifically, the initiation, propagation, and termination reactions have been related to the well-known phenomena of cage, gel, and glass effects, respectively. Diffusion-controlled phenomena affecting the termination and propagation reactions occurring in the polymer-rich phase were quantitatively described using a theoretical model developed by Achilias and Kiparissides.6 According to this model, the termination and propagation rate constants are expressed in terms of a reaction-limited term and a diffusion-limited term. The latter depends on the diffusion coefficients of the corresponding species (i.e., polymer and monomer) and an effective reaction radius. Furthermore, the so-called “residual termination” taking place at very high monomer conversions is also taken into account. Diffusion-Controlled Termination Rate Constant. On the basis of the above modeling approach, the termination rate constant is expressed as the sum of two terms, one taking into account the effect of the diffusion of polymer chains, kt,dif, and the other describing the so-called residual termination, kt,res
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kt2 ) kt,dif + kt,res
(24)
The first term on the right-hand side of eq 24 is subsequently written in terms of the intrinsic termination rate constant, kt0, which is equal to the termination rate constant of live polymer chains in the continuous phase, kt1, and the diffusion coefficient of the polymer chains in the polymer-rich phase, Dp
1 kt,dif
)
rt2 λ0 1 + kt0 3 Dp
(25)
where rt is the termination radius.7 According to the extended free-volume theory of Vrentas and Duda,7 Dp can be approximated by the polymer self-diffusion coefficient
Dp ) (Dp0/Mw2) exp[-γp(ωmV/m/ξmp + ωsV/s /ξsp + ωpV/p)/VFH] (26) with
ξip ) V/i MWi/Vpj
(27)
VFH ) VfpV/pωp + VfmV/mωm + VfsV/s ωs
(28)
and
At very high monomer conversions, the self-diffusion coefficient of the polymer becomes very small, resulting in an unrealistically low value of kt according to eq 25. The reason is that eq 26 does not account for the mobility of the radical chains caused by the monomer propagation reaction. This phenomenon is known as residual termination. To account for this contribution to the overall termination rate constant, a residual termination rate constant, kt,res, which is proportional to the frequency of monomer addition to the radical chain end, is defined
kt,res ) Akp[M]
(29)
where A is a proportionality rate constant.7 Diffusion-Controlled Propagation Reaction. Accordingly, kp2 can be expressed in terms of the intrinsic propagation rate constant, kp0 (i.e., the propagation rate constant in the CO2-monomer phase), and a diffusion term accounting for the diffusion of monomer molecules
1 1 1 ) + kp2 kp0 4πDmrmNA
(30)
In eq 30, rm denotes the radius of a monomer molecule. Accordingly, the monomer diffusion coefficient, Dm, is calculated from the extended free-volume theory as7
Dm ) D0 exp[- γm(ωmV/m + ωsV/s ξmp/ξsp + ωpV/pξmp)/VFH] (31) All symbols used in the preceding equations are explained in the Nomenclature section. 5. Thermodynamic and Phase Equilibrium Calculations One of the major issues regarding the development of a comprehensive mathematical model for a scCO2
dispersion polymerization process, is the calculation of the monomer, CO2, initiator, inhibitor, and polymer concentrations in the two-phase system. As pointed out in section 2 of the present paper, from a kinetic point of view, the dispersion polymerization of vinyl monomers in scCO2 can be considered to take place in three stages. During the first stage, the reaction mixture consists mainly of a homogeneous solution of monomer, CO2, and polymer, provided that the polymer concentration is less than its solubility limit. Note that, according to the recent results of Byun and McHugh,8 the presence of monomer in the continuous scCO2 phase results in an increase in the polymer solubility in CO2 through cosolvency effects. Upon the appearance of a separate polymer phase, the reaction mixture comprises two phases, namely, the polymer-rich phase and the CO2-monomer phase, which are assumed to be in thermodynamic equilibrium. Finally, during the third polymerization stage, polymerization primarily occurs in the monomer-swollen polymer particles. In the present study, each phase was assumed to consist of three main components (i.e., monomer, CO2, and polymer). This means that the concentrations of monomer, CO2, and polymer in the two phases were continuously recalculated as a function of temperature, pressure, and the varying composition of the reaction mixture. The pseudo-equilibrium concentrations of monomer, solvent, and polymer in the two phases were calculated using the Sanchez-Lacombe equation of state. Following the general developments of Sanchez and Lacombe,9-11 the equation of state for a pure fluid or/ and a polymer solution takes the form
˜ )T ˜ 1 - 1 F˜ - ln(1 - F˜ ) - F˜ 2 P r
[(
]
)
(32)
where P ˜, T ˜ , and F˜ are the reduced pressure, temperature, and density of a pure substance, respectively, defined as2
˜ ) P/P/ P
˜ ) T/T/ T
F˜ ) F/F/ ) 1/υ˜ ) V//V
(33)
where T is the absolute temperature (K), P is the pressure (bar), and F is the density (g/cm3). The number of sites (mers) occupied by a molecule in the lattice, r, can be related to the molecular weight of the pure component, MW, according to
r ) (MWP/)/(RT/F/)
(34)
Note that, for a high-molecular-weight polymer, the value of r can be considered infinite. According to McHugh and Krukonis,2 the characteristic parameters P/,T/, F/, and υ/ can be defined as
P/ ) //υ/ V/ ) N(rυ/)
T/ ) //R F/ ) MW/(rυ/)
(35) (36)
where / is the mer-mer interaction energy, ν/ is the close-packed molar volume of a mer, N is the number of molecules, and R is the universal gas constant. For a pure substance, the values of the parameters P/,T/ and F/ can be determined by fitting the Sanchez-Lacombe equation of state to known experimental PVT data.
Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 747
For a polymer-solvent mixture, it is necessary to use a combined mixing rule for the calculation of /mix, υ/mix, and rmix from the corresponding values of the pure components. Thus, using the van der Waals-1 mixing rule, the characteristic close-packed molar volume of a mer in the mixture, υ/mix, can be defined as
υ/mix )
∑i ∑j φiφjυ/ij
parameter
a
υ/ij
)
0.5[(υ/ii
+
υ/jj)](1
- ηij)
(38)
and the parameter ηij accounts for possible deviations of υ/ij from the arithmetic mean value of υ/ii and υ/jj of the pure components. Accordingly, the values of /mix and rmix of the mixture can be calculated by the equations
mix ) (1/υ/mix)
∑ ∑φiφj(iijj)0.5υ/ij(1 - kij) i)1 j)1 r-1 mix
)
φj/rj ∑ j)1
(39) (40)
where kij is a binary interaction parameter for the ith and jth components in the mixture that accounts for deviations of the mixture interaction energy from the geometric mean value of the pure-component energies. Similarly, the volume fraction of the ith component in the mixture, φi, can be expressed in terms of the mass fraction wi and the characteristic value, F/i , of the pure component
φi ) (wi/F/i )/(
wi/F/i ) ∑ j)1
(41)
The calculation of the chemical potential of a component is usually more difficult in a polymer solution than in normal mixtures because of the substantially different sizes of the polymer and solvent molecules. Following the developments of McHugh and Krukonis,2 the chemical potential of the ith component in a multicomponent system can be expressed as
[ ( )] { [∑
µi ) RT ln φi + 1 ri -F˜
2
υ/
ri
+
r
]
φjυ/ij) + / ∑ j)1
φjυ/ij /ij - /
(
j)1
RTυ˜ (1 - F˜ ) ln(1 - F˜ ) +
F˜ ri
]
ln F˜ + Pυ˜ (2
}
φjυ/ij - υ/) ∑ j)1
(42)
i ) 1, 2, 3
CO2
PMMA
305.0 450 5745.0 4500.0 1.71 1.269 5.83 a nMMA-CO2 ) 0.13/0.3 nMMA-PMMA ) -0.10 nCO2-PMMA ) 0.4
Varies with molecular weight according to eq 34.
Table 2. Kinetic Rate Constants for MMA Polymerization in scCO26,19 kpj ) 1.92 × 108 exp(-5.706/RT) (L mol-1 min-1) kfmj/kpj ) 9.48 × 103 exp(-13.88/RT) ktj ) ktcj + ktdj ) 5.4 × 109 exp(-0.701/RT) (L mol-1 min-1) ktcj/ktdj ) 3.3 × 10-4 exp(4090/RT) kdj ) 7 × 1016 exp(-30.640/RT) (min-1) for AIBN Table 3. Physical and Transport Properties and Parameters for Diffusion-Controlled Reactions7 δ ) 6.9 Å Rp ) 4.8 × 10-4 Rm ) 1.0 × 10-3 K-1 Tgm ) 159.15 K Tgp ) 387.15 V/m ) 0.868 cm3/g Xc0 ) 100
re ) 17.0 Å γm ) 0.3 γp ) 0.15 Dm ) 8.1 × 10-9 cm2/s RH ) 1.3 × 10-9Mw0.574 (cm) V/p ) 0.788 cm3/g
One of the most difficult problems in calculating the equilibrium concentrations of the three components in a two-phase system is the determination of the numerical values of the pertinent parameters (i.e., P/, T/, F/, kij, and ηij) appearing in the Sanchez-Lacombe (S-L) equation of state (EOS) model. For pure components, the values of the P/, T/, and F/ parameters are usually determined by fitting the S-L EOS to experimental data obtained under different pressure and temperature conditions. In Table 1, the numerical values of P/, T/, F/, and r for MMA, CO2, and PMMA are reported. The numerical values of the binary interaction parameters kij and ηij for MMA-CO2, MMA-PMMA, and CO2PMMA were obtained by fitting experimental phase equilibrium data obtained in our laboratory or/and reported in the literature (Lora and McHugh15). The calculated values of the respective parameters are reported in Table 1. 6. Simulation Results
+
For a ternary monomer-CO2-polymer mixture at equilibrium, the chemical potentials (µi) for each component in the two phases (i.e., 1, MMA; 2, CO2; and 3, PMMA) must be equal
µ1i (φ11,φ12,φ13) ) µ2i (φ21,φ22,φ23)
MMA
T/ (K) 425.0 P/ (bar) 3508.0 1.339 F/ (g/cm3) r 7.425 kMMA-CO2 ) 0.125/0.183 kMMA-PMMA ) -0.07 kCO2-PMMA ) 0.50/0.41
(37)
where
[
Table 1. Numerical Values of the Parameters Appearing in the Sanchez-Lacombe EOS for the Ternary MMA-CO2-PMMA System
(43)
For the solution of the above phase equilibrium problem, the procedure proposed by Koak and Heidemann12 was applied, in combination with the classical substitution algorithm.13,14
The predictive capabilities of the proposed comprehensive model were demonstrated by a direct comparison of theoretical predictions with experimental data16-18 on monomer conversion, pressure, and average molecular weights. In Table 2, the numerical values of all of the kinetic rate constants employed in the simulation of dispersion polymerization of MMA in scCO2 are reported. The numerical values of the physical and transport properties and parameters used in the calculation of the diffusion-controlled termination and propagation rate constants are shown in Table 3. It should be pointed out that, in all simulations, the initiator partition ratio kI ) [I]2/[I]1 remained constant. One of the major issues in modeling the dispersion polymerization of vinyl monomers in scCO2 is the prediction of the reactor pressure with respect to polymerization time. Experimental data published in the
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Figure 1. Comparison of the simulated final pressure change with experimental data for different initial pressure values.
open literature3,16,17 clearly show that the reactor pressure can either increase or decrease with time in batch polymerizations of MMA in scCO2. In fact, a number of investigators have reported3,16,17 that, when the initial pressure (i.e., at zero monomer conversion) is lower than about 230 bar, the pressure in the batch reactor increases with the progress of polymerization. On the other hand, when the initial pressure is higher than approximately 230 bar, the pressure exhibits a decrease with polymerization time. Lepilleur and Beckmann3 attributed this anomalous pressure variation with respect to the initial pressure in the reactor to the nonideal behavior of the ternary (monomer-CO2polymer) system. In Figure 1, the difference between the final and initial pressures in the reactor (∆P ) Pfinal - Pinitial) is plotted with respect to Pinitial. Discrete points represent experimental measurements of ∆P reported by different research groups for the dispersion polymerization of MMA in scCO2. It should be pointed out that the experimental data on ∆P refer to similar values of the final MMA conversion (i.e., about 90-100%). Note that, in the various kinetic experiments, the MMA/CO2 mass ratio varied in the range of 0.25-0.30. The continuous line in Figure 1 denotes the theoretical prediction of ∆P with respect to Pinitial, assuming a final conversion of 98%. As can be seen, model predictions obtained from the solution of the phase equilibrium problem (see eqs 32-43) for various values of Pinitial are in good agreement with the experimental data on ∆P. Figures 2-7 illustrate typical model predictions for the dispersion polymerization of MMA in scCO2. Discrete points denote experimental results obtained from our laboratory reactor. MMA polymerizations were conducted in a 550-mL high-pressure system17 at different temperatures (65-75 °C) and initial pressures (202-282 bar). Additional information regarding the polymerization procedure and recipe is given elsewhere.17 In Figure 2, the calculated monomer conversion is plotted as a function of polymerization time for two different initial pressures (202 and 235 bar). The point on the graph (marked by the solid square) denotes the final monomer conversion measured experimentally. The mass ratio of MMA/CO2 was 0.247 for 202 bar and 0.297 for 235 bar, and the polymerization was carried out at 65 °C. In Figure 3, the polymerization rates in the two phases as well as the total polymerization rate (i.e., the
Figure 2. Variation of MMA conversion during polymerization in scCO2 for initial pressures of 202 and 235 bar.
Figure 3. Change of polymerization rate in the two phases during polymerization.
Figure 4. Variation of the monomer concentration in the two phases.
sum of the two polymerization rates) are depicted with respect to polymerization time. As can be seen, the polymerization rate in the continuous monomer-CO2 phase exhibits a monotonic decrease with time as a result of the decrease of the monomer concentration in phase 1 (also see Figure 4). On the other hand, the polymerization rate in the polymer-rich phase initially increases to a maximum value and then slowly decreases with polymerization time. The initial increase of the polymerization rate in the polymer-rich phase can be attributed to both the increase of the polymer mass (i.e., increase of polymer reaction volume) and the presence of gel effect (see Figure 5). It should be pointed out that only the propagation and termination reactions occurring in the polymer-rich phase can become diffu-
Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 749
Figure 5. Variation of the gel-effect function in the two phases.
Figure 6. Variation of pressure during polymerization. Comparison of model prediction with experimental data of Bouldouka et al.17 for initial pressures of 202 and 235 bar.
sion-controlled because of the restricted mobility of the polymer chains (caused by the increased viscosity of the polymer phase). Note that the rate of monomer transfer from phase 1 to phase 2 increases as the amount of polymer phase increases. However, because of the reduction of the monomer concentration in phase 1, the polymerization rate in the polymer-rich phase starts decreasing as the monomer in phase 2 is consumed. Figure 4 shows the MMA and CO2 concentrations in the two phases during polymerization. In Figure 5, the gel-effect function (i.e., the ratio of the termination rate constant at a given polymer concentration to the value of kt at zero polymer concentration) is plotted with respect to polymerization time. Note that the termination rate constant in the continuous monomer-CO2 phase, kt1, is equal to the intrinsic value of kt0 (i.e., kt1/kt0 )1). On the other hand, the geleffect function in the polymer-rich phase initially takes a very low value (i.e., ∼2 × 10-4) upon the appearance of the separate polymer phase as the polymer mass fraction in phase 2 is relatively high (i.e., ∼0.82). A comparison between model predictions and experimental measurements of the total reactor pressure is shown in Figure 6. It is apparent that the model predictions are in very good agreement with the experimental data. Note that the model clearly predicts the decrease in total reactor pressure with polymerization time when the initial pressure is 235 bar and the increase when the initial pressure is 202 bar.
Figure 7. Variation of number- and weight-average molecular weights with time.
Figure 8. Variation of MMA conversion during polymerization for an initial pressure of 205 bar. Comparison of model prediction with experimental data of O’Neill.18
In Figure 7, the calculated number- and weightaverage molecular weights are plotted with respect to polymerization time. The discrete point on the graph denotes the experimentally determined value of viscosity-average molecular weight of the final PMMA. The observed continuous increase in the values of Mn and Mw with respect to polymerization time is attributed to the gel effect in the polymer-rich phase. The predictive capabilities of the present mathematical model were further tested by a comparison of model predictions with experimental data on monomer conversion, pressure, and molecular weight averages reported by O’Neil et al.18 and Hsiao et al.16 In Figures 8 and 9 and 10-12, model predictions, shown by the continuous lines, are compared with experimental measurements, denoted by the discrete points. It should be pointed out that, in all simulations, the numerical values of the thermophysical parameters and the kinetic rate constants remained the same (see Tables 1-3). In Figure 8, the evolution of the monomer conversion with respect to polymerization time is shown. Discrete points denote experimental measurements, reported by O’Neill et al.18 The polymerization of MMA in scCO2 was carried out in a 50-cm3 high-pressure cell at 65 °C and an initial pressure of Pinitial ) 205 bar. The MMA/CO2 mass ratio was equal to 0.25. Apparently, there is a good agreement between model predictions (continuous line) and the experimental measurements. In Figure 9, model predictions and experimental measurements on number- and weight-average molecular weights are plotted
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Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003
Figure 9. Variation of number- and weight-average molecular weights with time. Comparison of model prediction with experimental data of O’Neill.18
Figure 12. Variation of number- and weight-average molecular weights with time. Comparison of model prediction with experimental data of Hsiao et al.16
7. Conclusions
Figure 10. Variation of MMA conversion during polymerization for an initial pressure of 193 bar. Comparison of model prediction with experimental data of Hsiao et al.16
In the present work, a comprehensive mathematical model was developed for the free-radical dispersion polymerization of MMA in scCO2. A detailed kinetic mechanism was considered, and the general rate functions for the production of live and dead polymer chains were developed. Dynamic material balance equations were derived to describe the conservation of the various molecular species (i.e., monomer, initiator, and moments of the live and dead number chain length distributions) in the two phases. The equilibrium monomer and CO2 concentrations in the two-phase system were calculated using the Sanchez-Lacombe equation of state. A generalized free-volume model was employed to describe the variation of the propagation and termination rate constants with the polymer concentration in the polymerrich phase. Finally, the predictive capabilities of the present model were successfully demonstrated by comparisons of model predictions with experimental data on monomer conversion, reactor pressure, and molecular weight averages. Acknowledgment We gratefully acknowledge the EU for supporting this research under the BRITE/EURAM Project BE 97 4520 (SUPERPOL). Nomenclature
Figure 11. Variation of pressure during polymerization.
with respect to polymerization time. As can be seen, the model predictions are in very good agreement with the experimental measurements, despite the uncertainty in the polymerization conditions and in the measurements of Mn and Mw. Finally, in Figures 10-12, model predictions are compared with experimental measurements of the final conversion, pressure, Mn, and Mw reported by Hsiao et al.16 It is evident that the model predictions are in good agreement with the experimental data, which further demonstrates the predictive capabilities of the proposed mathematical model.
A ) proportionality constant (L/mol) Dm ) monomer diffusion coefficent (m2/s) Dp ) polymer diffusion coefficent (m2/s) f ) initiator efficiency [I]j ) concentration of initiator in the jth phase (mol/L) kdj ) decomposition rate constant of initiator in the jth phase (min-1) kfmj ) chain transfer to monomer rate constant in the jth phase (L mol-1 min-1) kIj ) chain initiation rate constant of initiator I in the jth phase (L mol-1 min-1) kij ) binary interaction parameters between components i and j kpj ) propagation rate constant in the jth phase (L mol-1 min-1) kp,0 ) propagation rate coefficient without any diffusional contribution (L mol-1 min-1) kto ) overall termination rate coefficient (L mol-1 min-1) ktcj ) termination by combination rate constant in the jth phase (L mol-1 min-1)
Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 751 ktdj ) termination by disproportionation rate constant in the jth phase (L mol-1 min-1) kt,dif ) termination rate constant term due to diffusion effects (L mol-1 min-1) kt,res ) residual termination rate constant term (L mol-1 min-1) kz,j ) inhibition rate constant in the jth phase (L mol-1 min-1) M0 ) initial concentration of MMA (mol/L) [M]j ) concentration of MMA in the jth phase (mol/L) Mn ) number-average molecular weight (g/mol) Mv ) viscosity-average molecular weight (g/mol) Mw ) weight-average molecular weight (g/mol) MW ) molecular weight in the Sanchez-Lacombe EOS (g/ mol) MWm ) molecular weight of MMA (g/mol) N ) number of molecules NA ) avogadro’s number (6.023 × 1023 molecules/mol) ni,j ) binary interaction parameters between components i and j P ) pressure (bar) P ˜ ) reduced pressure PD ) polydispersity index [Px] ) concentration of dead polymer chains concentration (mol/L) r ) number of sites occupied by a molecule occupies in the lattice R ) universal constant for gases (cal mol-1 K-1) [R•] ) concentration of primary radicals (mol/L) [R•x] ) concentration of live polymer chains concentration (mol/L) rm ) radius of a monomer molecule (m) rt ) termination radius (m) [S]j ) concentration of CO2 in the jth phase (mol/L) t ) time (min) T ) temperature (K) T ˜ ) reduced temperature v/ ) closed-packed molar volume (L/mol) Vj ) mixture volume in the jth phase (L) V/i ) critical hole free volume of component i (cm3/g) Vpj ) molar volume of the polymer jumping unit (cm3/g) VFH ) average free volume of the mixture (cm3/g) Vreact ) reactor volume (L) wi ) weight fraction of component i X ) monomer conversion [Z]j ) concentration of inhibitor in the jth phase (mol/L) Greek Letters ξip ) ratio of the critical molar volume of component i to the critical molar volume of the polymer γ ) overlap factor δ ) Kronecker’s delta / ) interaction energy (J/mol) λkj ) kth moment of the molecular weight distribution of live polymer radicals in the jth phase µk ) kth moment of dead polymer chains µi ) chemical potential of the ith component F˜ ) reduced density φi ) volume fraction of component i ω ) weight fraction Subscripts and Superscripts ) characteristic parameter of the Sanchez-Lacombe EOS m ) monomer mix ) mixture /
p ) polymer s ) CO2 x ) number of structural units of active or dead polymer chains
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Received for review May 31, 2002 Revised manuscript received September 27, 2002 Accepted October 2, 2002 IE020397A
