The reaction locus in supercritical carbon dioxide dispersion polymerization. The case of poly(methyl methacrylate)

Chemical Engineering Science 60 (2005) 377 – 397 www.elsevier.com/locate/ces

The reaction locus in supercritical carbon dioxide dispersion polymerization. The case of poly(methyl methacrylate) P.A. Mueller, G. Storti, M. Morbidelli∗ Swiss Federal Institute of Technology, Institute for Chemical- and Bioengineering, ETH Hönggerberg, HCI, CH-8093 Zürich, Switzerland Received 24 February 2004; received in revised form 15 July 2004; accepted 27 July 2004 Available online 22 October 2004

Abstract The problem of determining the reaction locus in the supercritical carbon dioxide dispersion polymerization of methyl methacrylate is considered. For this, two limit models are comparatively evaluated using experimental data of polymerization kinetics and molecular weight distribution. The two models take opposite assumptions with respect to the relative rate of interphase radical transport with respect to the radical life time, which lead to different relative importance of the polymerization in the continuous or in the dispersed phase. The model parameters have been estimated using independent literature sources so as to ensure genuinely predictive modelling. The results clearly indicate that the interphase transport of the active chains is a key process in determining the reaction locus and it has to be carefully considered in order to reliably simulate any polymerization process of this type. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Dispersion polymerization; Supercritical fluid; Mathematical modelling; Population balance; Interphase mass transport; Kinetic modelling

1. Introduction Dispersion polymerization in supercritical CO2 appears to be a promising alternative to the solvent-intensive, heterogeneous polymerization processes largely in use in industry (Canelas and DeSimone, 1997; Ajzenber et al., 2000). Various monomers, ranging from conventional vinyl monomers to special fluorinated ones, can be polymerized this way (Canelas et al., 1996; Shiho and DeSimone, 2000, 2001; Wang et al., 2003). Moreover, effective stabilizers have been reported and cheaper and better ones are continuously investigated (DeSimone et al., 1994; Schaffer et al., 1996; Giles et al., 2000; Christian and Howdle, 2000; Li et al., 2000). Fundamental research work is still needed in order to better clarify the physico-chemical phenomena underlying this process before a reliable scale-up can be achieved, in particular when a close control of the product quality is required. ∗ Corresponding author. Tel.: +41-1-6323034; fax: +41-1-6321082.

E-mail address: [email protected] (M. Morbidelli). 0009-2509/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.07.122

Mathematical modelling, combined with suitable experimental measurements, can provide a useful tool to elucidate the process mechanisms and their impact on product properties. Very few models have been proposed so far and the only comprehensive one has been recently developed by Chatzidoukas et al. (2003), where the reactions in both the continuous (supercritical) and the dispersed (polymer) phase are accounted for. One aspect which is currently under debate refers to the polymerization loci. Since in general the initiator is soluble both in the dispersed polymer phase and in the continuous supercritical phase, the radicals are formed in both phases. In the case where the chain radical growth and termination process (i.e., the chain life) in the continuous phase is slower than its transport to the dispersed phase, most of the polymer is made in the polymer particles. This is the typical situation for water emulsion polymerization. On the other hand, we could have the opposite situation whereby a good fraction of the polymer is made in the continuous phase, and therefore exhibits different characteristics (e.g. molecular weight) than the polymer made in the dispersed phase. The latter is the

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P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

situation considered by Saraf et al. (2002) for PVDF in order to explain the clear bimodality which is typically found for this polymer in the case of precipitation polymerization. The same assumption has been made in the model mentioned above (Chatzidoukas et al., 2003; Kiparissides et al., 1997) where the growing radicals are assumed to remain segregated in the phase where they have been originally formed through the initiator decomposition. In this work, we investigate this issue by developing two different models, corresponding to two opposite limiting conditions in terms of interphase radical transport. The first one conforms to the previously mentioned segregation limit, thus assuming that interphase radical transport is much slower than the chain life. In the second model it is assumed that the interphase radical transport is extremely fast, so that equilibrium conditions are achieved for the active species. The two models are referred to in the following as “radical segregation” model (RS) and “radical partitioning” model (RP), respectively. It is worth noting that the RP model is different from the previously developed “one-locus” model (Rosell et al., 2001; Bonavoglia et al., 2002) since, even though in both cases the transport rate is infinite, in the latter the radicals were assumed to diffuse only from the continuous to the dispersed phase, coherently with the assumption of no solubility of the polymer chains in the continuous phase. The comparative evaluation of the two models is carried out using our own experimental data for methyl methacrylate dispersion polymerization in supercritical CO2 using AIBN as initiator and Krytox as stabilizer. It is to be noted that the values of all the kinetic parameters in the model have been either taken from independent and reliable literature sources or estimated through fundamental relationships. This allowed us producing truly predictive simulations and therefore achieving a reliable comparative evaluation of the two limiting models.

2. Experimental set-up and procedure All experiments were carried out in the lab-scale unit described in (Rosell et al., 2004). This is a thermostatted, stainless-steel reactor, equipped with a magnetically coupled mechanical stirrer and a mass flow meter used to accurately measure the injected amount of CO2 . Recipe and operating conditions for considered experimental runs are summarized in Table 1. Methyl methacrylate and 2,2’azobis(isobutyronitrile) (AIBN) were obtained from Fluka (Switzerland) and used as received. Carbon dioxide was obtained from Pangas (Switzerland) with analytical grade 4.5 (purity = 99.995%) and used as received. The stabilizer, Krytox 157 FSL, was supplied by Dupont (Switzerland) and used as received. To monitor the time evolution of the reaction, repeated experiments under the same conditions were carried out and interrupted at different time values. The amount of polymer produced was evaluated by gravimetry,

Table 1 Recipe and operating conditions of the experimental runs Initial amounts MMA = 30 g CO2 = 320 ± 2 g AIBN = 0.33 g Krytox = 0.33 g Operating conditions Temperature = 65 ◦ C Pressure = 140 ± 3 bar (initial value) Vreactor = 0.58 L

the particle size by SEM and the complete distribution of molecular weight (MWD) by GPC. More details about these characterization procedures can be found elsewhere (Rosell et al., 2004).

3. Model development Two models representing two opposite extreme conditions in terms of interphase transport of active polymer chains are considered: The radical segregation model (RS): The active chains spend their whole life in the same phase where they have been initiated. This model corresponds to the situation where, in each phase, the characteristic time for radical termination is much shorter than that for interphase mass transport. The radical partitioning model (RP ): The rate of mass transport between polymer particles and continuous phase is assumed to be infinitely fast for all species, so that thermodynamic equilibrium is established at every time during reaction. In general, the interparticle driving forces are such that radicals are transported from the continuous to the dispersed phase. However, the opposite is also possible as in the case of very short radicals produced in polymer particles. A notable example is the desorption of radicals produced by chain transfer to monomer in water emulsion polymerization. This situation corresponds to the case where the characteristic time for radical termination is large compared to that for interphase transport. 3.1. Model assumptions, kinetic scheme and thermodynamic modelling The main model assumptions are as follows: (1) Two reaction loci are considered: the polymer-rich dispersed phase and the CO2 -rich continuous phase. (2) Low molecular weight species (solvent, initiator and monomer) undergo very fast transport between the

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

phases and are assumed to be at interphase equilibrium at all times. (3) The particle nucleation process has not been considered, i.e., a number of particles equal to the final experimental value is assumed to be present from the beginning. (4) In the RP model, a chain-length-dependent equilibrium partition coefficient for polymer chains between continuous and dispersed phase is considered. A few remarks about the last assumption. Rindfleisch and McHugh (1996) demonstrated that high molecular weight poly(methyl methacrylate) (PMMA, Mw = 93, 300 g mol−1 , Mw /Mn = 2.01) is insoluble in pure CO2 up to temperatures of 225 ◦ C and pressures of 2550 bar. A cosolvent effect of MMA for high molecular weight PMMA (Mw = 93, 300 g mol−1 , Mw /Mn = 2.01) was evidenced by Lora and McHugh (1999), but about 1000 bar were required to solubilize 5 wt% of polymer at 65 ◦ C even in the presence of 30 wt% of monomer. Finally, complete insolubility was reported by O’Neill et al. (1998a) for oligomers of 30 monomer units at p = 200 bar and T = 35 ◦ C. Because these pressures are larger than or comparable to typical operating pressures for dispersion polymerization (< 400 bar), the assumption of complete insolubility of polymer chains in the continuous phase whatever their chain length appears to be fully justified. On the other hand, Kumar et al. (1987) measured the solubility of polystyrene oligomers in CO2 at p = 250 bar and T = 60 ◦ C and their data indicate a significant polymer solubilization up to chain lengths around 10 monomer units in pure CO2 . Since PMMA is expected to be more soluble than polystyrene in supercritical CO2 and the monomer is acting as cosolvent, a chain-length-dependent partition coefficient has been introduced in the RP model similar to that found experimentally by Kumar et al. for polystyrene. In order to assess the reliability of this assumption, we discuss later the obtained relationship in the context of the various experimental results mentioned above. The following kinetic scheme is considered, including the most typical kinetic steps of the free-radical polymerization of a vinyl monomer: Initiation:

Propagation: Chain transfer to monomer: Termination:

kdj

2Ij• ,

Ij

−→

Ij• + Mj

−→

Rx,j + Mj

−→

Rx+1,j ,

Rx,j + Mj

−→

Px,j + R1,j ,

Rx,j + Ry,j

−→

Px,j + Py,j ,

Rx,j + Ry,j

−→

Px+y,j .

kIj

kpj

kf mj ktdj ktcj

R1,j ,

Note that, when two subscripts are given, the first one (x or y) indicates the chain length and the second one the phase (j = 1 indicates the continuous phase, j = 2 indicates the disperse polymer-rich phase). This kinetic scheme is applied to both models, with all reactions taking place in both phases.

379

About the description of the interphase equilibrium for low molecular weight species, i.e., monomer, solvent and initiator, the Sanchez–Lacombe model (Sanchez and Lacombe, 1976, 1978) was used for the first two species, while a simple, constant partition coefficient was used for the initiator. Together with the Statistical Association Fluid Theory (SAFT) (Chapman et al., 1989, 1990), these models have been successfully applied to the description of phase behavior and partitioning in high-pressure systems involving polymers (Xiong and Kiran, 1994; Daneshvar et al., 1990; Folie, 1996). However, extremely poor performances of the SAFT equation have been reported by Lora and McHugh (1999) in the prediction of the cloud point of PMMA in a MMA–CO2 mixture. On the other hand, Chatzidoukas et al. (2003) obtained accurate predictions of the pressure evolution during the dispersion polymerization of MMA in scCO2 when using the Sanchez–Lacombe equation. Therefore, the latter equation of state was selected in this work to evaluate phase behavior and interphase partitioning. 3.2. Mass balance equations In the following we consider the mass balances for all the species present in the system. The meaning of all variables is detailed in the notation. Symbols in square brackets indicate molar concentrations, while those without brackets indicate numbers of moles. 3.2.1. Low molecular weight species: material balances and interphase partitioning Since these species are assumed to be at interphase equilibrium, it is convenient to consider balances including both phases, in order to cancel out the interphase mass transport terms: dS
dSj = = 0, dt dt

(1)

dI
dIj = = −kd1 [I1 ]V1 − kd2 [I2 ]V2 , dt dt

(2)

2

j =1 2

j =1

dM
dMj = dt dt 2

j =1

= −2f1 kd1 [I1 ]V1 − 2f2 kd2 [I2 ]V2 ∞
− (kp1 + kf m1 )[M1 ]V1 [Rx,1 ] − (kp2 + kf m2 )[M2 ]V2

x=1 ∞


[Rx,2 ],

(3)

x=1

S = [S1 ]V1 + [S2 ]V2 ,

(4)

I = [I1 ]V1 + [I2 ]V2 ,

(5)

M = [M1 ]V1 + [M2 ]V2 .

(6)

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P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

The concentrations in the two phases are related by the following equilibrium conditions:


1S =
2S ,

(7)


1M =
2M ,

(8)

[I1 ] = KI , [I2 ] j



j =1

j =1 k=j +1



˜ p˜i 1 + ri − + + −1 ˜ T˜i T˜i ˜  1 ln ˜ , × ln(1 − ˜ ) + ri

i=1 j =1

r i Ni , rN

(11)

where Ni is the number of moles of component i, N is the total number of moles in the mixture, while ri and r are the number of lattice sites occupied by component i and by the mixture, respectively. The latter is given by n

−1
 i r= . (12) ri i=1

In Eq. (10) appear the mixture molar density, , pressure p and temperature, T, which can be computed from the Sanchez–Lacombe equation of state in reduced form 

 1 2 ˜ (13) ˜ + p˜ + T ln(1 − ˜ ) + 1 − ˜ = 0, r where the following definitions apply to the reduced variables: pv ∗ p˜ = ∗ , 

TR T˜ = ∗ 

∗

 =

n n

i=1 j =1

i j vij∗ ,

(15)

i j ∗ij ,

(16)

where the new mixture quantities vij∗ and ∗ij are expressed as ∗1/3 ∗1/3 3 + vj vi ∗ vij = , (17) 2

∗ij = (∗i ∗j )1/2 ij

ij =

∗i + ∗j − 2∗ij

. (19) RT The equation of state (13) is applied to each phase and, combined with the constraint of constant reactor volume (Vreactor = V1 + V2 ), allows computing the volumes of the two phases, Vj , and the overall pressure, p.

3.2.2. High molecular weight species: population balance equations (PBE) for active and terminated polymer Since the specific form of these balances is different for the RS and the RP model, they are discussed separately in the following. 3.2.2.1. RS model: As all chains have been assumed to be completely segregated in the phase where they have been formed, the relevant sets of population balance equations (PBEs) for radical and dead chains are written for each phase, separately: dRx,j = kpj [Mj ][Rx−1,j ]Vj (1 − (x − 1)) dt 

  − (kpj + kf mj )[Mj ] + ktdj + ktcj

×

∞


 [Ry,j ] [Rx,j ]Vj

y=1



+ 2fj kdj [Ij ]Vj + kf mj [Mj ]

(14)

being r, v ∗ and ∗ three characteristic parameters, Mm the mixture molecular weight (Mm = yi Mm,i , with yi = i r/ri )

(18)

with ij being an empirical correction factor used to describe the non-ideal binary interaction between components i and j. Note that the interaction parameters ij in Eq. (10) are in fact known functions of the above characteristic parameters

(10)

where n, the component number, is equal to 2 (CO2 and monomer) and 3 (CO2 , monomer and polymer) in phase 1 and 2, respectively. The meaning of all variables in this equation is briefly reviewed here, while full details can be found in the original papers (Sanchez and Lacombe, 1976, 1978). The composition of component i is expressed as the site fraction i , given by:

rv ∗ ˜ = , Mm

n n



v∗ =

(9)

where
i is the chemical potential of component i in phase j. As mentioned above, a simple interphase partition coefficient, KI , is introduced for the initiator, while for monomer and solvent the chemical potentials have been computed in the frame of the Sanchez–Lacombe model as follows: 
i ri  = ln i + 1 − RT r   n n n


i ij − j k j k  + ri ˜ 

i =

and R the ideal gas constant. While for the pure components the characteristic parameters ri , vi∗ and ∗i are usually known, for the mixture they are evaluated through the following mixing rules:

×

∞
y=1

 [Ry,j ]Vj  (x − 1),

(20)

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397 x−1
dPx,j 1 [Ry,j ][Rx−y,j ]Vj = ktcj dt 2 y=1   ∞
+ ktdj [Ry,j ] + kf mj [Mj ] [Rx,j ]Vj

mx , has been assumed equal to that obtained experimentally by Kumar et al. (1987) for polystyrene: log mx = log m1 + (x − 1),

y=1

for j = 1, 2 and x = [1, ∞],

381

(21)

(25)

where m1 is the monomer partition coefficient and is a constant, that have to be estimated for the specific system under examination.

where (x − x0 ) indicates the Kronecker delta function, defined as equal to 1 for x = x0 and zero otherwise.

3.3. Numerical solution

3.2.2.2. RP model: In this case, it is convenient to write overall balances including both phases, as we did for low molecular weight species. The corresponding set of PBEs for the active and dead polymer chains, respectively, are reported in the following:

For both models, an infinite set of mixed algebraicdifferential equations is obtained. Its numerical solution has been performed through the discretization method of Kumar and Ramkrishna (KR) (1996a,b, 1997). Accordingly, a generic distribution nx (t) is partitioned into a series of discrete but contiguous size intervals by applying the following operator:

dRx
dRx,j = dt dt 2

xi+1 −1

Ni (t) =

j =1

= (1 − (x − 1))

−

2


×

[Ry,j ] [Rx,j ]Vj 2
j =1

×

 2fj kdj [Ij ]Vj + kf mj [Mj ] 

[Ry,j ]Vj  ,

(22)

y=1

dPx
dPx,j = dt dt j =1   2 x−1

1 ktcj = [Ry,j ][Rx−y,j ]Vj  2 y=1 j =1   2 ∞

ktdj + [Ry,j ] + kf mj [Mj ] [Rx,j ]Vj 2

j =1

(26)

Therefore, all individuals belonging to class i are “concentrated” to a single pivot value, x¯i , with xi
x¯i < xi+1 . By applying operator (26), the original PBEs, (20) and (21) for the RS model and (22) and (23) for the RP model, are transformed into the so-called discretized equations. In the following, the discretized equations for both models are reported.



+ (x − 1) ∞


kpj [Mj ][Rx−1,j ]Vj

(kpj + kf mj )[Mj ] + (ktdj + ktcj )

y=1

nx (t).

x=xi

j =1



j =1 ∞


2





3.3.1. RS model • dNi,j • = kpj [Mj ](c(x¯i−2 + 1, x¯i )Ni−2,j dt • • + b(x¯i−1 + 1, x¯i )Ni−1,j + a(x¯i + 1, x¯i )Ni,j )

0,j • − (kpj + kf mj )[Mj ] + (ktdj + ktcj ) Ni,j Vj + (i − 1)(2fj kdj [Ij ]Vj + kf mj [Mj ] 0,j • − kpj [Mj ]Ni,j ), (27) 

k

l  dNi,j 1 1 = ktcj  1 −  (l − k)  dt 2 2 k,l x¯i−2
x¯k +x¯l
(23)

The concentrations in the two phases are related through the partitioning equilibrium constant mx which depends on the chain length, x: [Rx,1 ] [Px,1 ] = = mx . [Rx,2 ] [Px,2 ]

(24)

As mentioned above when listing the model assumptions, the functional form of the interphase partition coefficient,

× b(x¯k + x¯l , x¯i ) +

k

l k,l x¯i
x¯k +x¯l
1−

k,l x¯i−1
x¯k +x¯l
y=1

for x = [1, ∞].

k

l

× c(x¯k + x¯l , x¯i ) +





1−

1 (l − k) 2





 1 (l − k) a(x¯k + x¯l , x¯i )  2

• 1 • × Nk,j Nl,j Vj

0,j • + ktdj + kf mj [Mj ] Ni,j . Vj

(28)

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P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

3.3.2. RP model 2
dNi• • kpj [Mj ](c(x¯i−2 + 1, x¯i )Ni−2,j = dt j =1

• + b(x¯i−1 + 1, x¯i )Ni−1,j 2


• + a(x¯i + 1, x¯i )Ni,j )

(kpj + kf mj )[Mj ] + (ktdj

−

j =1 • × Ni,j + (i − 1)

2


first three moments of the distributions as follows (Kumar and Ramkrishna, 1996a; Peklak et al., 2004):

0,j + ktcj ) Vj

+ c(x¯i + x¯j , x¯k+2 )n¯ 2k+2 = (x¯k + x¯k+1 + x¯k+2 )2 ,

(2fj kdj [Ij ]Vj



j =1

a(x¯i + x¯j , x¯k )n¯ 2k + b(x¯i + x¯j , x¯k+1 )n¯ 2k+1

j =1

• + kf mj [Mj ] 0,j − kpj [Mj ]Ni,j ), 2  1
dNi ktcj  =  dt 2





k

l k,l x¯i−2
x¯k +x¯l
(29)



k

l

+

1−

k,l x¯i
x¯k +x¯l
• • × Nk,j Nl,j

+

2
j =1




k,j =



 1 (l − k) a(x¯k + x¯l , x¯i )  2

1 Vj



0,j • ktdj + kf mj [Mj ] Ni,j . Vj

(30)

In the above Eqs. (29) and (30), the discretized distribution • , and dead chains, N , in each one of the of radicals, Ni,j i,j two phases, are obtained from the overall populations Ni• and Ni by using the equilibrium relationship (24). In particular, the following expressions apply: • Ni,j

  Vj (j − 1)mx + (j − 2) = Rx m x V1 + V 2 x=xi   Vj (j − 1)mx¯i + (j − 2) ≈ Ni• , mx¯i V1 + V2 xi+1 −1




xi+1 −1

Ni,j =




x=xi

N
i=1

(31)

• x¯ik Ni,j ,

(34)

x¯ik Ni,j

(35)

for the active and the dead chains, respectively. This approach implies the approximation that the original chain length distribution is concentrated only at the pivot values being zero elsewhere and its accuracy is again determined by the grid size. After the application of the KR method, a finite set of mixed algebraic-differential equations is obtained. This has been solved by decoupling algebraic and differential equations, using a direct substitution method for the algebraic part and the solver LSODA from the library ODEPACK (Stepleman et al., 1983) for the differential part. Finally, the reconstruction of the entire MWD from the pivot values was done using the following simplified approach (Butté et al., 2002): Rx¯i ,j = Px¯i ,j =

  Vj (j − 1)mx + (j − 2) Px m x V1 + V 2

Vj ((j − 1)mx¯i + (j − 2)) ≈ Ni , mx¯i V1 + V2

N
i=1

× c(x¯k + x¯l , x¯i )

k

l 1 + 1 − (l − k) b(x¯k + x¯l , x¯i ) 2 k,l

(33)

In both models, the moments of the distributions are involved in the discretized equations. The kth order moments of the distribution nx (t) have been estimated from the discretized distributions as follows

k,j =

1 1 − (l − k) 2

x¯i−1
x¯k +x¯l
a(x¯i + x¯j , x¯k ) + b(x¯i + x¯j , x¯k+1 ) + c(x¯i + x¯j , x¯k+2 ) = 1, a(x¯i + x¯j , x¯k )n¯ k + b(x¯i + x¯j , x¯k+1 )n¯ k+1 + c(x¯i + x¯j , x¯k+2 )n¯ k+2 = x¯k + x¯k+1 + x¯k+2 ,

• Ni,j

,

(36)

Ni,j , x¯i+1 − x¯i

(37)

x¯i+1 − x¯i

where the values of Rx¯i and Px¯i , corresponding to the values of Rx and Px at x¯i , are taken as representative of all chains in class i. In order to obtain a continuous MWD, these values have been linearly interpolated. 4. Model parameter evaluation

(32)

where x¯i is the pivot value of class i. Note that the two expressions on the right-hand side of the last two equations are approximations whose inaccuracy becomes negligible when the discretization grid is fine enough. In the discretized equations, three partition coefficients are involved. They have been evaluated in order to preserve the

The reliable evaluation of the large number of parameters is always a critical issue when developing a detailed kinetic model. In this work most of the parameter values have been estimated through independent experimental data taken from the literature, in order to minimize the parameter fitting and to ensure the development of a reliable model. All numerical values and the corresponding sources are listed in Table 2 while their derivation is discussed in the following.

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

383

Table 2 Summary of parameter values and sources Kinetic parameters—dispersed phase f20 = 0.5 kd2 = 1.65 × 10−5 s−1 kp2,0 = 8.30 × 102 L mol−1 s−1 kt2,0 = 3.30 × 107 L mol−1 s−1 kf m2 /kp2 = 5.15 × 10−5 ktd2 /ktc2 = 4.37

(Russell et al., 1988b; Shen et al., 1991) (Odian, 1970), Eq. (63) (Mahabadi and O’Driscoll, 1977; Beuermann et al., 1994), Eqs. (48)–(49) (Mahabadi and O’Driscoll, 1977; Buback and Kowollik, 1998), Eqs. (53)–(54) (Kukulj et al., 1998) (Zammit et al., 1997)

Kinetic parameters—continuous phase f1 = 0.83 kd1 = 6.77 × 10−6 s−1 kp1 = 6.78 × 102 L mol−1 s−1 kt1 = 3.30 × 107 L mol−1 s−1 kf m1 /kp1 = 5.15 × 10−5 ktd1 /ktc1 = 4.37

(Guan et al., 1993) (Guan et al., 1993), Eq. (67) (Quadir et al., 1998), Eq. (66) (Beuermann and Buback, 2002) (Kukulj et al., 1998) (Zammit et al., 1997)

Free-volume parameters D0 = 1.61 × 10−3 cm2 s−1 E = 3.26 × 103 J mol−1 K1M / = 8.15 × 10−4 cm3 g−1 K −1 K1P / = 4.77 × 10−4 cm3 g−1 K −1 K2M = 143 K K2P = 52.4 K Tg,M = 143 K Tg,P = 392 K ∗ = 0.870 cm 3 g−1 VM VS∗ = 0.589 cm3 g−1 VP∗ = 0.757 cm3 g−1 VFrefH ,S = 0.231 cm3 g−1 S = 8.76 × 10−4 K−1 ˜ P = 0.44 MP = 0.60 SP = 0.18

(Faldi et al., 1994), Eq. (43) (Faldi et al., 1994), Eq. (43) (Faldi et al., 1994), Eq. (45) (Faldi et al., 1994), Eq. (47) (Faldi et al., 1994), Eq. (45) (Faldi et al., 1994), Eq. (47) (Faldi et al., 1994), Eq. (45) (Faldi et al., 1994), Eq. (47) (Faldi et al., 1994), Eq. (43) (Alsoy and Duda, 1998; Gupta et al., 2003), Eq. (43) (Faldi et al., 1994), Eq. (43) (Alsoy and Duda, 1998; Gupta et al., 2003), Eq. (46) (Alsoy and Duda, 1998; Gupta et al., 2003), Eq.(46) (Faldi et al., 1994), Eq. (47) (Faldi et al., 1994), Eq. (43) (Zielinski and Duda, 1992), Eq. (43)

Thermodynamic parameters MMA ∗ = 3850 J mol−1 v ∗ = 7.66 cm3 mol−1 r = 11.49 PMMA ∗ = 5786 J mol−1 v ∗ = 11.51 cm3 mol−1 ∗ = 1.269 g cm−3 CO2 ∗ = 2709 J mol−1 v ∗ = 5.64 cm3 mol−1 r = 5.30 Binary interactions MMA.CO2 = 0.89 MMA.PMMA = 1 CO2 .PMMA = 1.10 Initiator partitioning KI = 1 Oligomer partitioning = −0.3

by fitting data from Matheson et al., 1949; Kein and Novak, 1985; Boublik et al., 1984 by fitting data from Matheson et al., 1949; Kein and Novak, 1985; Boublik et al., 1984 by fitting data from Matheson et al., 1949; Kein and Novak, 1985; Boublik et al., 1984 (Kiszka et al., 1988) (Kiszka et al., 1988) (Kiszka et al., 1988) by fitting data from Lemmon et al., 2003 by fitting data from Lemmon et al., 2003 by fitting data from Lemmon et al., 2003 by fitting our own data no-interaction assumption by fitting data from (Bonavoglia et al., 2003) cf. Section 4.3 by fitting data from (Kumar et al., 1987)

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P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

4.1. Kinetic parameters—dispersed phase Since the polymer concentration within the particles remains quite large all along the reaction (weight fraction around 0.8), diffusion limitations on the kinetic processes have to be accounted for. In particular, these have been evaluated using a Fickian description of reactant diffusion with diffusion coefficients estimated through the free-volume theory. The obtained relationships are briefly summarized in the following, while more comprehensive treatments are available in the literature (Allen and Patrick, 1974; North, 1964; Noyes, 1961; Litvinenko and Kaminsky, 1994). 4.1.1. Diffusion-limited reaction In order to evaluate the effect of diffusion limitations on the reaction rate between two chemical species, A and B, one can consider the concentration of B around a single A molecule. Assuming steady state, the flux of B molecules towards A is constant which in spherical coordinates leads to (Noyes, 1961):

d dcB r2 = 0, (38) dr dr  cB (r = rAB ) = cB∗ , (39) cB (r → ∞) = cB∞ , where rAB = rA + rB is the collision radius, cB∗ the concentration at the collision radius and cB∞ the bulk concentration (at infinite distance from A). Since the diffusive flux must be equal to the rate of depletion of B by chemical reaction, we have that

dcB  = 4 r 2 DAB = k0 cB∗ NA−1 , (40) dr rAB where DAB is the mutual diffusion coefficient of the reactive molecules, NA the Avogadro’s number and k0 the intrinsic reaction rate constant. Combining the solution of Eq. (38) with the last condition (40), the following expression for cB∗ is obtained: cB∗

cB∞ = . 1 + k0 /4 rAB DAB NA

(41)

Let us now introduce keff , the “observed” rate constant of the reaction, i.e., that corresponding to the reaction rate expressed at the bulk concentration of B, so that  = keff cB∞ NA−1 . Using Eq. (41), keff can be expressed as follows

−1 1 1 + . (42) keff = k0 4 rAB DAB NA Note that three quantities are involved in this equation: the intrinsic chemical reaction rate constant, k0 , the collision radius, rAB , and the mutual diffusion coefficient, DAB . The evaluation of the first two is described separately for each particular reaction later in this section whereas the diffusion coefficient is estimated from the free-volume theory

of Vrentas and Duda (1977a,b,c), Vrentas et al. (1985a,b), Vrentas and Vrentas (1993, 2003), which is now briefly summarized with reference to the prediction of self-diffusion coefficients in ternary systems (Vrentas et al., 1984). 4.1.2. Diffusion coefficients The self-diffusion coefficient of a low molecular weight species such as the monomer in the ternary system monomer/solvent/polymer (M/S/P) can be expressed as follows

E exp DM = D0 exp − RT    M VM∗ + S MS VS∗ + P MP VP∗ × − , (43) VF H where i is the weight fraction of component i, Vi∗ the specific critical hole free volume of component i required for a diffusion jump, D0 a pre-exponential factor, E the critical energy needed by a molecule to overcome the attractive force holding it to its neighbors, and  is an overlap factor accounting for the fact that the same free volume is available to more than one molecule, and ij is the ratio of the molar volume of the jumping unit of component i to that of component j. The average hole free volume per gram of mixture, VF H , can be estimated from those of the individual species with the assumption of volume additivity at any concentration and temperature. Thus, VF H = M VF H ,M + S VF H ,S + P VF H ,P .

(44)

In the original works by Vrentas and Duda, the hole free volume of component i is given by the following expression: VF H ,i K1i = (K2i + T − Tg,i ),  

(45)

where K1i and K2i are the so-called free-volume parameters. For the solvent (CO2 in our case), a formally different but fully equivalent expression has been used: VF H ,S = VFrefH ,S + S (T − T ref ),

(46)

where VFrefH ,S is the hole free volume at the reference temperature T ref and S the coefficient of thermal expansion (Alsoy and Duda, 1998; Gupta et al., 2003). The contribution of the polymer was estimated using original expression (45) but with the following marginal modification: VF H ,P K1P = (K2P + ˜ P (T − Tg,P )),  

(47)

where ˜ P is the ratio of the coefficients of thermal expansion below and above Tg,P , as introduced by Faldi et al. (1994) in order to avoid negative values of VF H ,P . Since the polymer free volume parameters, K1P / and K2P , are typically obtained from measurements carried out above Tg,P , their extrapolation to lower temperatures could in fact lead to negative values of the polymer hole free volume.

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

4.1.3. Propagation For propagation, the value at zero conversion (kp2,0 ) reported in Mahabadi and O’Driscoll (1977) was considered, corrected for the pressure dependence according to Beuermann et al. (1994) as follows: kp2,0 (p0 ) = Ap2 exp(−Ep2 /RT ),

=
V p kp2,0 = kp2,0 (p0 ) exp − (p − p0 ) , RT

(48) (49)

Ap2 = 4.92 × 105 L mol−1 s−1 , Ep2 = 18, 200 J mol−1 , =
Vp = −16.7 cm3 mol−1 , p0 = 1 bar. The value reported in Table 2 corresponds to a pressure of 150 bar, which is representative of the conditions in our experiments. The propagation rate constant kp2 is evaluated using Eq. (42) as follows

−1 1 1 kp2 = + , (50) kp2,0 4 rMP DMP NA where kp2,0 is the value of kp2 at zero conversion reported above, DMP is the mutual diffusion coefficient of a polymer chain end and a monomer molecule and rMP is the radius of interaction for propagation. A sensible estimate of rMP is the size of a monomer molecule (bearing in mind that the free radical chain end is also a monomer unit), which is well estimated by the Lennard–Jones diameter of the monomer, M (Maxwell and Russell, 1993). The mutual diffusion coefficient is given by DMP = DM + DP ,

(51)

where DM is the self-diffusion coefficient of the monomer given by Eq. (43) and DP is the diffusion coefficient of the free radical end of a polymeric active chain. Under conditions of diffusion controlled propagation (i.e., near glassy conditions), monomer diffusion is much faster than polymer diffusion, i.e., DM  DP . In this limit, Eq. (50) reduces to the expression used in the present work:

−1 1 1 kp2 = + . (52) kp2,0 4 M DM NA 4.1.4. Chain transfer to monomer The ratio between the chain transfer to monomer rate coefficient, kf m2 , and the propagation rate coefficient, kp2 , has been assumed equal to the value measured by Kukulj et al. (1998) at 50 ◦ C and low pressure.

=
V t (p − p0 ) , kt2,0 = kt2,0 (p0 ) exp − RT

385



(54) =

At2 = 9.80 × 107 L mol−1 s−1 , Et2 = 2900 J mol−1 ,
Vt = 15.0 cm3 mol−1 , p0 = 1 bar, where the subscript 0 indicates that this expression applies at zero conversion only. Using again Eq. (42) to evaluate kt2 at finite conversion values, one obtains

−1 1 1 xy + , (55) kt2 = kt2,0 4 rxy Dxy NA where Dxy is the mutual diffusion coefficient for two colliding chains with degrees of polymerization x and y, respectively, and rxy is the distance at which termination is assumed to take place instantaneously. To estimate rxy , let us consider two limit cases in terms of chain flexibility. In the limit of chain ends configurationally immobile on the timescale of propagation (rigid chain limit), we can focus on the termination between two monomeric free radicals. In this case (Russell et al., 1988a): rxy = M .

(56)

On the other hand, the maximum value of rxy is found when the chain end is so configurationally flexible on the timescale of propagation that the chain end thoroughly explores all the volume available to it between two propagation events (flexible chain limit). If we assume one node of entanglement every jc monomer units along the chain, the radius of interaction for termination is given by the distance of the chain end from the closest node of entanglement (Russell et al., 1988a): 1/2

rxy = 2aj c .

(57)

Considering the flexibility of methacrylates (Russell et al., 1988a), we have used the maximum value of rxy given by the two equations above. As usual, the mutual diffusion coefficient Dxy is given by the sum of the self-diffusion coefficients of the two species: Dxy = Dx + Dy ,

(58)

where Dx refers to the diffusion of the free-radical chain end (not the center of mass of the chain) which, in turn, is given by the sum of two contributions: Dx = Dx,com + DRD .

(59)

4.1.5. Bimolecular termination Mahabadi and O’Driscoll (1977) proposed the following Arrhenius relationship for kt2 at low monomer conversion which again was corrected for the pressure dependence recently reported by Buback and Kowollik (1998):

The first one refers to the motion of the chain end which moves with the chain as a whole, and is characterized by the center-of-mass diffusion coefficient, Dx,com . In a comprehensive work on diffusion coefficients of oligomers in polymeric mixtures, Griffiths et al. (1998) proposed the following universal scaling law for the center-of-mass diffusion coefficient:

kt2,0 (p0 ) = At2 exp(−Et2 /RT ),

Dx,com = DM x −(0.664+2.02 P ) ,

(53)

(60)

386

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

where DM is the self-diffusion coefficient of the monomer as given by Eq. (43). The second contribution is due to the chain end diffusion by propagation (i.e., reaction diffusion), which is characterized by the diffusion coefficient DRD given by the following expression (Soh and Sundberg, 1982): DP ,RD =

kp2 [M2 ]a 2 , 6

(61)

where [M2 ] is the monomer concentration in dispersed phase and a the root-mean-square end-to-end distance divided by the square root of the number of monomer units in the chain. Thus summarizing, the following general expression for the chain-length dependent termination rate coefficient is obtained: xy

kt2 =

−1 1 1 + , 2 kt2,0 8 aj 1/2 c (Dx,com +Dy,com +kp2 [M2 ]a /3)NA (62) where Dx,com and Dy,com are computed through Eq. (60). Finally, the ratio between disproportionation and combination (ktd2 /ktc2 ) measured by Zammit et al. (1997) has been used in Table 2, which implies that disproportionation is the dominant termination mode. 4.1.6. Initiator decomposition The first-order reaction rate constant for the initiator decomposition has been evaluated using the parameter values reported by Odian (1970): kd2 = Ad2 exp(−Ed2 /RT ), Ad2 = 1.97 × 1014 s−1 , Ed2 = 123, 500 J mol−1 .

(63)

The corresponding value at T =65 ◦ C and atmospheric pressure is reported in Table 2. This rate constant is unlikely to be limited by diffusion and, hence, it is not expected to depend on the viscosity of the reaction medium. On the other hand, it is well-known that the efficiency of initiation, f, is diffusion controlled at high conversion. Recently, Shen et al. (1991) reported experimental values of kp obtained directly from the measured radical concentration during polymerization and of kt obtained using the so-called after-effect technique and the electron paramagnetic resonance (ESR) method. Using these data, they determined the value of f as a function of conversion during MMA bulk polymerization, and proposed the following general expression for initiator efficiency accounting for diffusion and combination of the primary radicals in the “cage”: f2 =

DI , r0 k0 + D I

size is comparable), the final expression for the initiator efficiency can be written as follows

−1  1 DM,0 1− , (65) f2 = 1 − DM f20 where f20 and DM,0 are initiator efficiency and monomer diffusion coefficient at zero conversion, respectively. 4.2. Kinetic parameters—continuous phase For the propagation reaction, the parameter values estimated by Quadir et al. (1998) by pulsed laser propagation (PLP) experiments at low pressure have been used: kp1 = Ap1 exp(−Ep1 /RT ),

Ap1 = 5.2 × 106 L mol−1 s−1 ,

Ep1 = 25, 400 J mol−1 .

(66)

These values have been corrected for the pressure dependence using Eq. (49) with the same activation volume. Note that the resulting value of kp1 is about 60% of kp2,0 , as recently measured by Beuermann et al. (1998). For the chain transfer to monomer, the same value of the ratio kf m /kp selected for the dispersed phase has been used. About bimolecular termination, Beuermann and Buback (2002) observed that in the case of MA, there is no difference between kt measured in bulk and in sc-CO2 . Moreover Beuermann et al. (1995) measured by PLP a negligible increase of kt in the free radical solution polymerization of MMA in toluene and 2-butanone with respect to bulk polymerization. Based on these findings, kt1 has been set equal to the kt2 value at zero conversion, kt2,0 . The same value of the ratio between the two bimolecular termination rate constants previously reported for the dispersed phase has been used (cf. Table 2). The rate constant of initiator decomposition is estimated using the data by Guan et al. (1993) in supercritical CO2 : kd1 = Ad1 exp(−Ed1 /RT ), Ad1 = 4.19 × 1015 s−1 , Ed1 = 134, 600 J mol−1

(67)

while neglecting any pressure dependence, as suggested in the original paper. Note that the resulting kd1 value is about 50% of the value in the dispersed phase, kd2 . On the other hand, the same authors reported an initiator efficiency, f1 of 0.83 which is about 2–3 orders of magnitude higher than the value estimated through Eq. (65) for the dispersed phase under typical operating conditions. This difference is so large that the continuous phase is expected to be the main locus for radical production.

(64)

where r0 is the reaction radius of the primary radicals, k0 the rate constant of the primary radical combination and DI the self-diffusion coefficient of initiator. Assuming the latter quantity equal to that of the monomer (since their molecular

4.3. Thermodynamic parameters As mentioned above, the Sanchez–Lacombe equation of state (EOS) has been used to describe the PVT behavior and the phase partitioning of the multicomponent system

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

5. Comparative simulations The two models are now comparatively evaluated, with reference to the experimental runs described in Section 2.

100

10-2

mx

under examination at equilibrium conditions. The pure component parameters involved in the EOS for the monomer have been estimated by fitting pure liquid density data at atmospheric pressure and in the temperature range from 0 to 61 ◦ C (Matheson et al., 1949; Kein and Novak, 1985; Boublik et al., 1984) using the original fitting procedure proposed by Sanchez and Lacombe (1976). The values of the same parameters for CO2 as well as for the polymer have been taken directly from the literature (Kiszka et al., 1988). The pressure–composition isotherm at 65 ◦ C for the binary mixture MMA–CO2 obtained in our lab was used for evaluating the MMA–CO2 interaction parameter, MMA.CO2 . In the fitting procedure we considered only the composition region of interest in this work, i.e., up to a maximum monomer mole fraction in the binary mixture equal to 0.3. The PMMA–CO2 binary interaction parameter PMMA.CO2 was determined by fitting the experimental sorption and swelling data measured in our lab (Bonavoglia et al., 2003). The MMA–PMMA binary interaction parameter MMA.PMMA was set equal to 1, given the chemical structure similarity and complete miscibility of MMA and PMMA (Brandrup and Immergut, 1989). About the initiator, no data of interphase partitioning for AIBN have been found in the literature. However, due to the well-known high solubility of this initiator in organic solvents (e.g. n-hexane) and polymers and to the indication of comparatively high solubility in supercritical carbon dioxide (Medlin, 1995), KI = 1 has been adopted in all calculations. To support this choice, it could be mentioned that reported partitioning data of dyes between supercritical carbon dioxide and different polymers exhibit equipartitioning (KI = 1) when the system pressure is large enough, usually above 100–200 bar (Condo et al., 1996; Kazarian et al., 1998). A similar behavior, characteristic of dyes exhibiting special affinity for the polymer, could be reasonably expected in the case of AIBN. About the polymer interphase partitioning, no data were found in the literature for PMMA. Therefore, it was assumed that the partition coefficient at chain length one, m1 , is equal to that of the monomer estimated at the same conditions of temperature and pressure using the Sanchez–Lacombe equation of state (closed circle in Fig. 1). On the other hand, the chain length dependence (i.e., the slope in the same figure) was estimated so as to reproduce the same slope measured experimentally for polystyrene (open circles in Fig. 1). The resulting mx vs. chain length curve is shown in the same figure by the continuous line and the corresponding parameter value is = −0.3. It is worth mentioning that, experimental results obtained by Bonavoglia et al. (2003) confirmed the reliability of this assumption.

387

10-4

10-6

10-8

10

20

30

40

degree of polymerization, x

Fig. 1. Polymer partition coefficient, mx , as a function of chain length, x. Calculated curves: = −0.3 (solid) and = −0.015 (dashed). Estimated monomer partition coefficient for MMA (•) and experimental values for polystyrene in pure CO2 (◦) (Kumar et al., 1987).

Since our final aim is to investigate the relative importance of the two polymerization loci, we are not interested in fitting the parameters of the two models, but rather to evaluate their predictions within the uncertainties in the kinetic parameter values discussed above. Since the two models are identical with respect to PVT as well as to equilibrium partitioning predictions, we consider these results first. Then, predictions in terms of reaction kinetics and polymer microstructure are analyzed in order to discriminate between the two models. 5.1. Pressure and partitioning—both models The monomer consumption by polymerization produces a decrease in the density of the supercritical phase along with a substantial and almost linear increase in pressure, as it is shown in Fig. 2. This clearly indicates a non-ideal behavior of the fluid mixture, since the density of an ideal gas at constant temperature should increase with pressure. In addition, since the density of the monomer is smaller than that of the polymer, a pressure decrease should be expected during the reaction. This means that the system non-ideality is sufficiently large to overtake both these effects. Deviations from ideality are basically due to the entropic term, which leads to a very large excess volume in the supercritical phase. In Fig. 2 it is also seen that the model predictions are in reasonable agreement with the experimental data, with no need for parameter adjustment, thus confirming the reliability of the adopted thermodynamic model. It is worth noting that this behavior is in agreement with experimental findings previously reported in literature (Hsiao et al., 1995; O’Neil et al., 1998a,b; Lepilleur and Beckman, 1997). The partitions of MMA and CO2 between the continuous (solid line) and the dispersed phase (dashed line) are shown in Figs. 3a and b, respectively, in terms of absolute con-

388

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397 4

165

160

partition coefficient

pressure [bar]

3

155

150

145

2

1

140

135

0

0.2

0.4

0.6

0.8

1

0

0

0.2

conversion

0.4

0.6

0.8

1

conversion

Fig. 2. Pressure as a function of conversion for the experimental conditions in Table 1: calculated (line) and experimental data (◦).

Fig. 4. Calculated interphase partition coefficients of MMA (solid) and CO2 (dashed) as a function of conversion.

MMA concentration [mol/L]

0.8

0.6

0.4

0.2

0

0

0.2

(a)

0.4

0.6

0.8

1

conversion

5.2. The RS model

CO2 concentration [mol/L]

14

12

10

8

6

4

(b)

available in the literature to validate these predictions. The interphase partition coefficients of MMA (solid line) and CO2 (dashed line) are shown in Fig. 4. The one of CO2 is larger than 1 at all conversions, indicating preferential solubilization in the supercritical phase whereas MMA is almost equipartitioned. In particular, the concentration of CO2 in the supercritical phase is approximately 3 times larger than that in the polymer phase, and this ratio remains fairly constant through the entire polymerization process. A similar picture applies for the MMA partition coefficient which has a constant value around 0.7 all over the reaction.

0

0.2

0.4

0.6

0.8

1

conversion

Fig. 3. Concentrations of (a) MMA and (b) CO2 in continuous (solid) and dispersed (dashed) phase as a function of conversion.

centrations of each species in the two phases as a function of conversion. It is seen that, as a first approximation, the concentrations of CO2 are practically constant all along the reaction in both phases. No experimental results of partition equilibria of MMA–CO2 mixtures in PMMA are currently

The model predictions in terms of conversion vs. time are shown in Fig. 5 together with the corresponding experimental data. It is evident that the RS model strongly underestimates the rate of polymerization. By analyzing the actual values of the kinetic parameters, the reason for this behavior can be explained as follows. Since the efficiency of the initiator decomposition within the particles is about two orders of magnitude lower than that in the continuous phase and the values of kd in the two phases are comparable, there is hardly any production of radicals within the particles. Moreover, the calculated Mn values in Fig. 6 are quite smaller than the experimental ones. These are consistent with the calculated and experimental MWDs shown in Figs. 7a and b, respectively. The first one is clearly bimodal with the mode at larger values corresponding to the polymer produced in the dispersed phase. This quantity is smaller than that produced in the continuous phase, leading to too small Mn values, while the good agreement with the experimental Mw values is fortuitous. In order to further investigate this conclusion we have reconsidered the equilibrium partitioning of the initiator KI ,

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

389

1

weight distribution

1

conversion [-]

0.8

0.6

0.4

X=0.90

0.8

0.65

0.6

0.40

0.4 0.25 0.2 0

0.2

3

4

5

0

0

1

2

3

4

5

6

6

7

1

time [h]

× 105

weight distribution

X=0.90

Fig. 5. Conversion as a function of time predicted by the RS model (line) and measured experimentally (◦).

5

0.8

0.4

0

0.40 0.25

3

4

5

6

7

log(M)

(b)

Fig. 7. MWD at various conversions predicted by the RS model (a) and measured experimentally (b).

3

2

1

1

0.8

0

0.2

0.4

0.6

0.8

1

conversion Fig. 6. Weight average molecular weight (solid) and number average molecular weight (dashed) as a function of conversion predicted by the RS model. Experimental data: Mw () and Mn ().

whose estimation as mentioned above, is somewhat uncertain. We found that fitting this parameter can produce much better predictions of the MWD. However, the needed value of KI is below 10−6 , thus corresponding to an initiator almost completely insoluble in the continuous phase, which is too far from the situation discussed in Section 4.3 to be accepted. Therefore, we may conclude that the assumption of complete radical segregation is unlikely to be the right one when dealing with the specific polymerization process under examination here.

conversion

average molecular weight [g/mol]

0.65

0.6

0.2

4

0

7

log(M)

(a)

0.6

0.4

0.2

0

0

1

2

3

4

5

6

7

time [h]

Fig. 8. Conversion as a function of time predicted by the RP model (line) and measured experimentally (◦).

5.3. The RP model The conversion vs. time curve predicted by this model is shown in Fig. 8 (solid line) and indicates that the reaction rate is slightly overestimated with respect to the experimen-

390

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397 x 105

1

weight distribution

average molecular weight [g/mol]

5

4

3

X=0.90

0.8 0.65 0.6 0.40 0.4 0.25 0.2

2

0 1

2

3

2.5

3

3.5

4

4

5

6

7

log(M)

(a) 1

X=0.90

0 0.4

0.6

0.8

1

conversion Fig. 9. Weight average molecular weight (solid) and number average molecular weight (dashed) as a function of conversion predicted by the RP model. Experimental data: Mw () and Mn ().

tal data. The same behavior is found in Fig. 9 where the predicted average molecular weights, Mn and Mw (dotted and solid line, respectively), are slightly larger than the experimental ones. The solid lines in Fig. 10a show the calculated MWD at various conversion values, to be compared with the experimental distributions in Fig. 10b. Considering the purely predictive nature of these calculations, with no parameter adjustment, the obtained agreement is indeed remarkable. The reason is that in this case the radicals produced in the continuous phase are rapidly transported in the dispersed phase, which is then the main reaction locus, giving higher polymerization rate and larger average molecular weights than in the case of the RS model. It is worth noting that the calculated MWDs in Fig. 10a are bimodal also in this case, but now the low molecular weight mode is negligible. Actually, a weak tail at low molecular weights is evident in the experimental MWDs shown in Fig. 10b. In order to reproduce such small shoulders, we performed some simulations with smaller values of the parameter . MWDs similar to the experimental one could be obtained (as shown in Fig. 11), but the needed value of was about −0.015 which corresponds to the partition factor shown by the dashed line in Fig. 1. This value is comparable with that of the monomer (i.e., partition factor larger than 0.1) up to a chain length of 56 units, which is clearly a physically unrealistic situation (Rindfleisch and McHugh, 1996; Lora and McHugh, 1999; O’Neill et al., 1998a; Bonavoglia et al., 2003).

6. Discussion The results presented in the previous section indicate that both the RS and the RP model are inadequate to fully de-

weight distribution

0.2

0.8 0.65 0.6 0.40 0.4 0.25 0.2

0 3

4

5

6

7

log(M)

(b)

Fig. 10. MWD at various conversions predicted by the RP model (a) and measured experimentally (b).

X=0.90

1

0.8

weight distribution

0

0.65

0.6 0.40 0.4 0.25 0.2

0 3

4

5

6

7

log(M) Fig. 11. MWD at various conversions predicted by the RP model using the modified parameter value, = −0.015.

scribe the experimental data. This appears very clearly for the RS model since an unrealistic interphase partitioning of the initiator would be required to properly describe the experimental MWD and polymerization kinetics. However, also the RP model cannot be reported as satisfactory since in order to be able to predict the small shoulder in the low

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

molecular weight region of the MWD an unrealistic increase of polymer solubility in the continuous phase would be required. In order to better understand the discrepancies between the RS and RP model and the experimental data, we need to deepen our analysis of the interphase transport of the active radical chains. For this, let us focus on a generic active chain with length x and growing in phase j, which at any given time can undergo one of the following events: propagation, termination and interphase transport. By introducing the overall mass transfer coefficient, Kj , the characteristic time of diffusion out of phase j is given by the following expression 1 , Kj ap,j

diff,j =

(68)

where ap,j = Ap /Vj is the overall particle surface per volume of phase j. On the other hand, the time needed for the addition of one monomer unit in phase j is given by

p,j =

1 . kpj [Mj ]

(69)

Thus, the maximum average length that chains produced in phase j can achieve before diffusing out of the phase can be estimated as the ratio between these two characteristic times: xmax,j =

diff,j kpj [Mj ] = . p,j Kj ap,j

(70)

However, the actual average length of the chains produced in phase j is given by the ratio between the characteristic times of termination and propagation, leading to xact,j =

kpj [Mj ] . ktj [Rj ]

(71)

In the case where xact  xmax , the chains are terminated before they can diffuse out of the phase of origin and we can therefore regard this phase as segregated and apply the RS model. On the other hand, when xact  xmax , the active chains are able to diffuse out of the phase before terminating and we are in the situation assumed by the RP model, where equilibrium interphase partitioning prevails at any time for the active chains. Although this analysis applies to the case where all the kinetic coefficients indicated above are constant during the life of the radical chain, i.e., they are chain length independent, we can extend the same reasoning to the system under consideration in this work and introduce the following characteristic quantities for each phase and each chain length:

1 (x) = 2 (x) =

K1 (x)Ap , kt1 0,1 ∞ 

K2 (x)Ap

y=1

xy

kt2 [Ry,2 ]V2

(72) ,

(73)

391

where the summation in the denominator is used to account for the chain length dependency of the termination rate constant in the dispersed phase. Depending upon the actual values of these  values, four different conditions or operating regions can be identified for the polymerization process: (1) 1 (x) and 2 (x) < 1: The probability of termination is larger than that of diffusion out of the phase for both continuous and dispersed phase. This is the case of the RS model, where the chains terminate in the phase where they were initiated, without any significant chance to be transported to the other phase while active. (2) 1 (x) and 2 (x) > 1: The probability of termination is smaller than that of diffusion out of the phase for both continuous and dispersed phase. This is the case of the RP model, where the chain mobility is so large that termination cannot prevent the achievement of complete equilibration of the chain concentration in the two phases. (3) 1 (x) > 1 and 2 (x) < 1: The chains initiated in phase 1 (continuous) can diffuse to phase 2 but the opposite transport cannot occur because of termination in phase 2 (which is faster than diffusion). The net result is that the active chains initiated in phase 2 are segregated there while those initiated in phase 1 are transported to phase 2 where they terminate. This situation can be regarded as a sort of “irreversible transport” of the radicals from the continuous to the dispersed phase. (4) 1 (x) < 1 and 2 (x) > 1: The situation is the opposite to that in the previous case. The chains initiated in phase 2 (dispersed) can diffuse to phase 1 but the opposite transport does not occur because of termination in phase 1 (which is faster than diffusion). Therefore, active chains initiated in phase 1 are segregated there and those initiated in phase 2 are transported to phase 1 where they terminate. This corresponds to an “irreversible transport” of the radicals from the dispersed to the continuous phase. All four operating regions are sketched in the plane 1 .2 in Fig. 12 thus obtaining a kind of “master plot“ for radical partitioning, which is of general validity since in principle it applies to any two-phase polymerization process. The four quadrants obtained when drawing the two lines at 1 =1 and 2 =1 have been enumerated clockwise from I to IV starting from the quadrant in the origin of the plane. Quadrants I and III correspond to the operating region of the RS and RP model, respectively, while quadrants II and IV identify the regions where irreversible transport is dominant, from phase 2 to phase 1 in quadrant II and viceversa in quadrant IV. As an example, a typical emulsion polymerization would be located in quadrant IV since, due to the practical insolubility of the polymer chains in the continuous (aqueous) phase, an irreversible mechanism of radical transport prevails. On the other hand, a typical precipitation polymerization should be located in quadrant I, since, due to the large reactivity of the

392

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

where Dx,2 is given by Eq. (60) and Dx,1 was estimated using the correlation proposed by Lusis and Ratcliff (1968):



vCO2 1/3 8.52 × 10−10 T vCO2 Dx,1 = 1.40 + 1/3 vx vx
CO v

104

sc

102

pol

sc

pol

2

CO2

Ω2

(77)

100

II

III

I

IV sc

x=1

pol

10-2

x=10 x 10-4 10-4

10-2

100

102

104

Ω1 Fig. 12. 1 .2 calculated path at X = 50% using the RP model predic tions ().

growing chains, they are typically able to grow and terminate in the continuous phase before moving to the other phase. Let us now examine our specific polymerization process in order to find out where it is located in the master plot in Fig. 12. Since in this case chain length-dependent kinetic parameters are involved, we can compute the complete path in the 1 –2 plane covered as the chain length x changes from 1 to ∞. For this we need to compute the mass transfer coefficients Kj , which were not involved in the RS and RP models developed above. Using the two film theory, it is readily found that K1 = K2 =

1 mx + k1 k2

−1

1 1 + m x k1 k2

1 (x) = ,

−1

(74)

,

(75)

where k1 and k2 are the local transport coefficients in the continuous and the dispersed phase respectively, while mx = [Rx,1 ]/[Rx,2 ] is the equilibrium interphase partitioning coefficient defined by Eq. (25) and shown in Fig. 1. The local transport coefficients, kj (x), have been evaluated using the following relationship: kj (x) =

Dx,j , rp

with Dx,1 in cm2 s−1 , T in K,
CO2 viscosity of CO2 in poises (obtained from the NIST chemical database), vCO2 molar volume of CO2 in cm3 mol−1 and vx molar volume of a chain of length x, both estimated from the group contribution method of Le Bas (Reid and Sherwood, 1966). It is worth noting that in order to compute the  values in Eqs. (72) and (73) we need appropriate values for the volumes and specific surfaces of the different phases as well as for the active chain concentration. Since the RP model appeared to predict the system behavior in a more reliable way, the values of all these quantities as calculated by this model have been used. The 1 –2 values calculated at different chain lengths are shown in Fig. 12 at intermediate conversion, X = 0.5. It is clear that the calculated path covers two regions only in the  plane: regions I and IV. In particular, we see that points representative of very short chains (x
10) lie in region I, indicating that these radicals are segregated because of their relatively large solubility in the continuous phase. However, at increasing chain length, the chain solubility decreases and the corresponding  pairs enter region IV, thus indicating that an irreversible transport from the continuous to the dispersed phase is dominant. In addition, it can be noted that above a certain chain length, 1 remains practically constant as the chain length increases further, while 2 decreases strongly. The largest chain length considered in Fig. 12 is x = 28. If we consider longer chains, the approximation of no solubility in the continuous phase, i.e., mx = 0 can be taken. Using Eqs. (72) and (73) it is readily seen that this leads to

(76)

k1 (x)Ap Dx,1 4 rp Np = , kt1 0,1 kt1 0,1

2 (x) = 0.

(78) (79)

This result indicates that for systems where the polymer chains exhibit scarce solubility in the continuous phase, which is the case of most of the common two-phase polymerization system, 2 is well below 1 and radical partitioning regime is determined by the value of 1 and typically corresponds to either segregation (region I, 1 < 1) or irreversible transport from the continuous to the dispersed phase (region IV, 1 > 1). Let us now reconsider the model performance discussed in Section 5 in the frame of the calculated 1 –2 curve shown in Fig. 12. It is now clear why model RS does not reproduce the experimental data: it is adequate only for very short chain lengths, which are a very minor fraction of the final product. In particular, the RS model predicts that the

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

longer radicals produced in the continuous phase terminate in the same phase, while instead the master plot in Fig. 12 tells us that they move in the dispersed phase, where they produce more polymer with larger average molecular weight in agreement with the experimental observations. More attention has to be dedicated to the RP model, which should be applicable only in quadrant III of Fig. 12 and therefore should give even poorer predictions than the RS model. We have instead seen above that its predictions are in many respects, i.e., the rate of polymerization and the average molecular weights, in rather good agreement with the experimental findings. The RP model assumes that in both phases of the polymerization system the radicals can achieve equilibrium partitioning conditions. The master plot in Fig. 12 says that this assumption is completely wrong for the dispersed phase since the corresponding 2 -values are actually much smaller than one, and therefore the radicals are segregated in this phase. However, the fact that most of the radicals have almost negligible solubility in the continuous phase makes it irrelevant whether the interphase transport is fast or slow: the radicals do not diffuse outside the polymer particles for thermodynamic reasons. For the continuous phase, the value 1 > 1 in Fig. 12 for most of the radical lengths implies fast transport in agreement with the RP model. From these observations it follows that in practice the RP model fails to predict only the behavior of the shortest active chains, that is those that the master plot shows to remain in the continuous phase (i.e., 1 < 1), while the RP model predicts them to move in the polymer particles. This is the reason why this model cannot predict the slight shoulder at the low molecular weights of the MWD observed experimentally. Going now back to the general validity of the master plot in Fig. 12 to predict the radical segregation regime of a given two-phase polymerization system, we see that we have to include thermodynamic considerations. In particular, if the radicals are not soluble in the continuous phase, i.e., mx  1, the parameter 2 loses its importance and this phase can be described equally well with a RS or RP model. The only question is what is the value of 1 , which determines whether we should use a RS model (1 < 1) or a RP model (1 > 1). Conversely, in the case where the radicals are not soluble in the dispersed phase, i.e., mx  1, the only relevant parameter is 2 and we use a RS model if 2 < 1 and a RP model if 2 > 1.

radicals can diffuse out of their phase of origin before being terminated. For this to occur, two models, one based on the assumption that the growing radicals are segregated in the phase of origin (RS), and the other on the assumption that they are partitioned between the two phases (RP) have been developed and compared with experimental data in the case of supercritical dispersion polymerization of methyl methacrylate. Due to the large impact of the so-called cage effect, most radicals are produced in the continuous phase and they are (i) all terminated in the continuous phase in the RS model and (ii) completely transferred to the particle phase in the RP model due to the low solubility of the polymer in the continuous phase. As a result, the experimental molecular weight is largely underestimated by the first model and reasonably well reproduced by the second one. This indicates a closer approach of the RP model to the real system behavior. However, even the RP model is not fully satisfactory since it fails to predict the occurrence of a slight shoulder in the low molecular weight region of the experimental MWD. These results have been explained by introducing a suitable master plot based on two parameters defined as the ratio between characteristic time of termination and that of interphase mass transport in the continuous and in the dispersed phase, respectively. By analyzing the experimental system behavior on this master plot it has been possible to explain the model deviations from the experimental data, and to conclude that for this specific system a complete model, which accounts in detail for the interphase transport and solubility of the active chains in each phase as a function of chain length is needed to reproduce the experimental observations. The developed procedure can be used in general for any two-phase polymerization system to determine the radical segregation regime and then the relative importance of the different polymerization loci. An interesting example is the precipitation polymerization of VDF in scCO2 where the shoulder in the MWD indicating the presence of two polymerization loci is more pronounced. Sufficient data are probably currently lacking for a reliable estimation.

Notation a

7. Conclusions a A typical open problem in two-phase heterogeneous polymerization is the determination of the relevant polymerization loci which effects not only the rate of the polymerization process but also the product MWD, particularly with respect to the possible occurrence of bimodalities. In this work we have developed a simple master plot which allows estimating the radical segregation regime prevailing in each phase. This means to identify whether or not the

393

ap,j Ap Adj Apj

partition coefficient in the frame of the KRmethod, cf. Eq. (33) rms end-to-end distance per square root of monomer units, cm overall particle surface per volume of phase j, m−1 overall particle surface, m2 pre-exponential factor of initiator decomposition in phase j, s−1 pre-exponential factor of propagation in phase j, L mol−1 s−1

394

Atj b c ci ci∗ ci∞ D0 Di Di,0 Di,com Di,RD Dij E Edj Epj Etj fj 0 fj I Ij [Ij ] Ij• jc kj k0 keff kdj kf mj kIj kpj kp2,0 ktj xy ktj kt2,0

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

pre-exponential factor of termination in phase j, L mol−1 s−1 partition coefficient in the frame of the KRmethod, cf. Eq. (33) partition coefficient in the frame of the KRmethod, cf. Eq. (33) concentration of component i, mol L−1 concentration of component i at the collision radius, mol L−1 bulk concentration of component i, mol L−1 pre-exponential factor, cf. Eq. (43), cm2 s−1 self-diffusion coefficient of component i, cm2 s−1 self-diffusion coefficient of component i at zero conversion, cm2 s−1 center-of-mass diffusion coefficient of componenti, cm2 s−1 reaction diffusion coefficient of component i, cm2 s−1 mutual diffusion coefficient of components i and j, cm2 s−1 critical energy to overcome attractive forces, cf. Eq. (43), J mol−1 activation energy of initiator decomposition in phase j, J mol−1 activation energy of propagation in phase j, J mol−1 activation energy of termination in phase j, J mol−1 initiator efficiency factor in phase j at zero conversion initiator efficiency factor in phase j total amount of initiator, mol amount of initiator in phase j, mol initiator concentration in phase j, mol L−1 activated initiator in phase j entanglement spacing local transport coefficient in phase j, m s−1 intrinsic bimolecular chemical rate constant, L mol−1 s−1 observed bimolecular chemical rate constant, L mol−1 s−1 initiator decomposition rate constant in phase j, s−1 chain transfer to monomer rate constant in phase j,L mol−1 s−1 initiation rate constant in phase j, L mol−1 s−1 propagation rate constant in phase j, L mol−1 s−1 propagation rate constant in dispersed phase at zero conversion, L mol−1 s−1 termination rate constant in phase j, L mol−1 s−1 termination rate constant in phase j between chains of length x and y, L mol−1 s−1 termination rate constant in dispersed phase at zero conversion, L mol−1 s−1

ktcj ktdj Kj K1i K2i KI mx M Mj [Mj ] Mm,i Mn Mw n N NA Ni Ni,j Ni• • Ni,j Np p p0 p˜ p˜i Px Px,j [Px,j ] r ri rij rp R Rx Rx,j [Rj ] [Rx,j ] S Sj [Sj ] t T Tgi

termination by combination rate constant in phase j, L mol−1 s−1 termination by disproportionation rate constant in phase j, L mol−1 s−1 overall transport coefficient referred to phase j, m s−1 free volume parameter of component i, cm3 g−1 K −1 free volume parameter of component i, K initiator partition coefficient partition coefficient of chains of length x total amount of monomer, mol amount of monomer inphase j, mol monomer concentration in phase j, mol L−1 molecular weight of species i, g mol−1 number average molecular weight, g mol−1 weight average molecular weight, g mol−1 number of components total number of moles in mixture, mol Avogadro number, mol−1 total population of dead chains of class i, mol population of dead chains of class i in phase j, mol total population of active chains of class i, mol population of active chains of class i in phase j, mol overall number of particles pressure, bar reference pressure, bar reduced pressure, cf. Eq. (14) reduced pressure of component i, cf. Eq. (14) total amount of dead chains of length x, mol amount of dead chains of length x in phase j, mol dead chain concentration of length x in phase j, mol L−1 number of lattice sites occupied by the mixture, cf. Eq. (12) number of lattice sites occupied by component i, cf. Eq. (12) collision radius for an encounter between i andj particle radius, cm ideal gas constant, J mol−1 K −1 total amount of active chains of length x, mol amount of active chains of length x in phase j, mol total active chain concentration in phase j, mol L−1 active chain concentration of length x in phase j, mol L−1 total amount of solvent, mol amount of solvent in phase j, mol solvent concentration in phase j, mol L−1 time, s temperature, K glass transition temperature of component i, K

P.A. Mueller et al. / Chemical Engineering Science 60 (2005) 377 – 397

T˜ T˜i vi V1 V2 Vi∗ VF H VF H ,i =
Vp =
Vt

x, y xmax,j xact,j xi yi

reduced temperature, cf. Eq. (14) reduced temperature of component i, cf. Eq. (14) molar volume of component i, cm3 mol−1 volume of the continuous phase, L volume of the dispersed phase, L specific critical hole free volume of component i, cm3 g−1 average specific hole free volume of mixture, cm3 g−1 specific hole free volume of component i, cm3 g−1 activation volume of propagation, cm3 mol−1 activation volume of termination, cm3 mol−1 chain length maximum average length of chains produced in phase j actual average length of chains produced in phase j pivot value representing length of chains in class i mole fraction of component i

Greek letters

i ˜ i  (x − x0 ) ∗ ∗i

¯ k,j
¯ k,j
i j
i ij ij  ˜ ˜i ∗ i diff,j p,j ∗ ∗i

chain partitioning parameter, cf. Eq. (25) coefficient of thermal expansion of component i, K −1 ratio of coefficients of thermal expansion below and above Tg of i overlap factor, cf. Eq. (43) Kronecker delta function characteristic interaction energy of mixture, cf. Eqs. (16), (18), J mol−1 characteristic interaction energy of component i, cf. Eqs. (16), (18), J mol−1 kth order moment of activechain distribution in phase j, mol kth order moment of dead chain distribution in phase j, mol dynamic viscosity of component i, poise chemical potential of component i in phase j, J mol−1 binary interaction parameter, cf. Eq. (18) ratio between molar volumes of jumping units of i and j, cf. Eq. (43) density, g cm−3 reduced density, cf. Eq. (14) reduced density of component i, cf. Eq. (14) characteristic density, g cm−3 Lennard–Jones diameter of component i, cm characteristic time of diffusion out of phase j, s characteristic time of propagation in phase j, s characteristic volume of mixture, cf. Eqs. (15), (17), cm3 mol−1 characteristic volume of component i, cf. Eqs. (15),(17), cm3 mol−1

 i ij j

i

395

diffusive flux, cf. Eq. (40), mol s−1 site fraction of component i, cf. Eq. (11) interaction parameter, cf. Eq. (19) ratio between characteristic times of termination and interphase mass transport in phase j weight fraction of component i

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